university of rwanda scientiﬁc conference week mathematical statistics and applications ·...

University of Rwanda Scientific Conference Week

Mathematical Statistics and Applications

BOOK OF ABSTRACTS

Kigali, Rwanda 14-16 June 2017

Contents

Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Abstract - Invited Speakers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Narayanaswamy Balakrishnan – Cure Rate Modeling and Applications . . . . . . . . 15Youssef Ouknine – Optimal stopping with f -expectations: the irregular case . . . . . . 15Jianxin Pan – Regularization of Covariance Structures . . . . . . . . . . . . . . . . . . 15Simo Puntanen and Augustyn Markiewicz – Linear su�ciency in linear estimation and

prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Abstract - Contributed Speakers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Benson Ade Afere and Ekele Alih – On the reduction of global error of multivariate

higher-order product polynomial kernels . . . . . . . . . . . . . . . . . . . . . . . . 17Godwin Norense Osarumwense Asemota – Jump Resonance in Wind-Felled Plantains 18Richard O. Awichi – Spatio-temporal Predictions by Multivariate Singular Spectrum

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Kidanemariam Alem Berhie – Identification of risk factors associated with under five

mortality in Ethiopia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Mahamat Ali Issaka, Ali S. Dabye and Lamine Gueye – On detection of epileptic

seizure with an approach based on power spectral density with an AR model . . . . 20Denis Kagyera Katakara and Faraimunashe Chirove –Mathematical Modelling of Hep-

atitis E Virus (HEV) and Chronic Myeloid Leukemia (CML) Co-infection Dynamics 20Timothy Kevin Kuria Kamanu Kernel density estimation: Predicting microRNA

gene boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Rodnellin Onesime Malouata – The successive Co-Inertia Analysis type 3 . . . . . . 22Isaac Mugume, Michel D. S. Mesquita, Yazidhi Bamutaze, Didier Ntwali, Daniel Waiswa,

Charles Basalirwa, Joachim Reuder, Revocatus Twinomuhangi, Fredrick Tumwine,Triphonia Jacob Ngailo and Bob Alex Ogwang – A new technique to improve en-semble rainfall prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Stanislas Muhinyuza, Taras Bodnar and Mathias Lindholm – A test on the location ofthe tangency portfolio on the set of feasible portfolios. . . . . . . . . . . . . . . . . 24

Jean-Paul Murara, Anatoliy Malyarenko, Ying Ni and Sergei Silvestrov – Pricing Eu-ropean Options under two-dimensional Black-Scholes Partial Di↵erential Equationby using the Crank-Nicholson Finite Di↵erence Method . . . . . . . . . . . . . . . . 24

Nicholas Mwilu Mutothya and Joseph A. M. Ottieno – Distributions Arising fromPolya - Aeppli Birth Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Joseph Ivivi Mwaniki – On skewed, leptokurtic returns and pentanomial lattice optionvaluation via minimal entropy martingale measure . . . . . . . . . . . . . . . . . . 26

Lukman Abiodun Nafiu and Umaru Waniyos Hamidu – Prevalence of five-child-killerdiseases and under-five mortality in Adamawa state, Nigeria . . . . . . . . . . . . . 26

John M. Ndiritu – Empirical modeling and forecasting of exchange rate dynamics inKenya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Innocent Ngaruye, Dietrich von Rosen and Martin Singull – Small area estimationunder a multivariate linear model for incomplete repeated measures data . . . . . . 27

Gothatamang Patrick Nthoiwa, Ester Bale and Mosiamise Mokgolele – Time seriesmethods for water level forecasting of Gaborone dam, Botswana . . . . . . . . . . . 28

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2 CONTENTS

Gothatamang Patrick Nthoiwa, Michael Zwikiti and Mosiamise Mokgolele – Method-ological comparison of missing data techniques used in completely randomized blockdesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Davis Bundi Ntwiga – Interaction Dynamics in a Social Network Using Hidden MarkovModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Joseph Nzabanita Introduction to the 2-fold growth curve model . . . . . . . . . . . . 29Everlyne Odero and Fredrick Onyango – Modeling the e↵ects of interference in fertility

rate of Kenya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Rose Auma Odhiambo and Joseph A. M. Ottieno – Normal mixtures and posterior

distributions with their moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Nelson Onyango and Samuel Mwalili – Obtaining HIV sub-national indicator estimates:

A Small Area Estimation approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Idah Orowe and Joseph A. M. Ottieno – Multi-State Transition Models with Censoring

in Vertical Transmission of HIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Joseph A. M. Ottieno – Discrete mixtures of order statistics from an exponential

distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Andrea Otwande and Joseph A. M. Ottieno – Binomial mixtures based on Beta prior

distributions and their generalizations with application to group screening design . 33Daniel Pan and Thomas Hline – The search for ultimate truth in inexact science - Can

statisticians and clinicians work together? . . . . . . . . . . . . . . . . . . . . . . . 34Jolanta Pielaszkiewicz – On negative (spectral) moments for some functions of Wishart

matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Mohamed Riad Remita and Thara Belhamra – Graphical approach for claims reserving 35Bernardo Joao Rota and Thomas Laitila – On the Use of Auxiliary Variables and

Models in Estimation in Surveys with Nonresponse . . . . . . . . . . . . . . . . . . 36Harouna Sangare and Gane Samb Lo – A general strong law of large numbers and

applications to associated sequences and to EVT . . . . . . . . . . . . . . . . . . . 37Rachel Sarguta and Joseph A. M. Ottieno – Four routes to mixed Poisson distributions 37Amadou Sawadogo and Simplice Dossou-Gbete – Ties in One Block Comparison Ex-

periments: A Generalization of the Mallows-Bradley-Terry Ranking Model . . . . . 38Cheikh Tidiane Seck and Gane Samb Lo – Uniform in Bandwidth Consistency for

Transformation Kernel Estimators of Copula . . . . . . . . . . . . . . . . . . . . . 38Abdu Mohammed Seid, Tesfahun Berhanie and Lassi Roininen –Hierarchical Bayesian

Inversion Model for Daily Maximum Temperature Variability . . . . . . . . . . . . 39Adilson Silva, Miguel Fonseca and Martin Singull – Sub-D facing the worst ”one-way”

designs scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Martin Singull – More on estimation of the mean matrix in a Growth Curve model in

high dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Sandrine Unezeza and Jean Baptiste Habyarimana – External Economic Shock and

Food Price Volatility in Rwanda: Evidence from ARCH and GARCH models . . . . 40Denise Uwamariya and Emelyn Umunoza Gasana – Multivariate Analysis of Rwanda

Economic Indicators using Vector Autoregressive (VAR) Model . . . . . . . . . . . 41Denise Uwamariya and Denis Ndanguza – Modeling and forecasting maize production

in Rwanda with Markov Chain Monte Carlo methods (MCMC) . . . . . . . . . . . 42Dietrich von Rosen – Testing bilinear hypothesis in bilinear models . . . . . . . . . . 42Tatjana von Rosen and Dietrich von Rosen – Estimation in Generalized Reduced Rank

Regression Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Moses Wamalwa Wakoli and Joseph A. M. Ottieno – A link between Poisson and

exponential mixtures through Laplace transform . . . . . . . . . . . . . . . . . . . . 43Patrick G. O. Weke and Idah Orowe – Simplified Optimal and Relay Linear Unbiased

Estimation of Parameters of Logistic Distribution . . . . . . . . . . . . . . . . . . . 44Patrick G. O. Weke and Idah Orowe – Stochastic Claims Reserving in Short-Term

Insurance Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Invited Speakers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

CONTENTS 3

Contributed Speakers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4 CONTENTS

Preface

The interaction between mathematicians and statisticians reveals to be an e↵ective approach whendealing with applications of statistics. The aim objective of this thematic area is to bring togethersta↵ from academia and other researchers, sharing an interest in variety of aspects related tothe mathematical and statistical sciences with applications and o↵er them a possibility to discusscurrent developments in these subjects.

Moreover, this is in line with the objectives of the UR- African Centre of Excellence for DataSciences (ACE-DS). The Centre is focusing on data sciences involving the collection, analysis andtransmission of data for facilitating decision-making. One of its specific objectives is to stimulatecollaboration between academics, partners and stakeholders. It will be an occasion for this centreto launch its activities.

The following are the sub-themes of interest:

• Statistical inference

• Probability models

• Multivariate statistics

• Stochastic process

• Design of experiments

• Mixed models

• Biostatistics

• Bayesian statistics

• Statistical learning / Machine learning

• Stochastic Models in finance and insurance

• Applied econometrics

Conference Proceeding

The conference proceeding will be published in a special volume of the journal Afrika Statistika(http://univi.net/jas/). Afrika Statistika is a journal that promotes a scientific culture in prob-ability, statistics, econometrics, operational research and related topics around the World andparticularly in Africa. All submitted papers will be reviewed and papers should have become ac-cepted for presentation at the conference. More information will soon be available on the conferencewebsite.

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Program

The 1st Annual Scientific Conference Week 2017 will be held at the prestigious Kigali Conferenceand Exhibition Village.

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ProvisionalAgenda:URScientificConferenceWeek2017ThematicArea5:MathematicalStatisticsandApplications

Time Item Room

07:30–08:00 Registration Entrance/Protocol

08:00–08:45 OfficialOpening

08:45–09:45 KeynoteSpeaker:ProfessorJeffreyD.Sacks,“RebrandingResearchforSustainableDevelopment”.09:45–10:00 Coffee/TeaBreak

10:00-10:45

SESSION(5-A+B)- Invitedspeaker:NarayanaswamyBalakrishnan,“CureRateModelingandApplications.”

Chair:Prof.MartinSingullRapporteur:DrBanziW.

10:45–12:30

PanelParallelsession(5-A)- JosephA.M.Ottieno,“Discretemixturesoforderstatisticsfromanexponentialdistribution.”- Andrea Otwande and Joseph A. M. Ottieno, “Binomial mixtures based on Beta prior distributions and their

generalizationswithapplicationtogroupscreeningdesign.”- DietrichvonRosen,“Testingbilinearhypothesisinbilinearmodels.”- IsaacMugume,MichelD.S.Mesquita,“Anewtechniquetoimproveensemblerainfallprediction.”

Chair:ProfessorJohnMangoRapporteur:MrCyprienMugemangano

12:30–14:00 Lunch

14:00–16:00

PanelParallelsession(5-A)- CheikhTidianeSeckandGaneSambLo,“UniforminBandwidthConsistencyforTransformationKernelEstimatorsof

Copula.”- TimothyKevinKuriaKamanu,“Kerneldensityestimation:PredictingmicroRNAgeneboundaries.”- BernardoJoaoRotaandThomasLaitila,“OntheUseofAuxiliaryVariablesandModelsinEstimationinSurveyswith

Nonresponse.”- NicholasMwiluMutothyaandJosephA.M.Ottieno,“DistributionsArisingfromPolya-AeppliBirthProcess.”

Chair:DrNdanguzaDenisRapporteur:EmelyneUmunoza

ThematicArea5B:MathematicalStatisticsandApplications

Time Item

07:30–08:00 Registration Entrance/Protocol

08:00–08:45 OfficialOpening

08:45–09:45 KeynoteSpeaker:ProfessorJeffreyD.Sacks,“RebrandingResearchforSustainableDevelopment”.09:45–10:00 Coffee/TeaBreak

10:00-10:45

SESSION(5-A+B)- Invitedspeaker:NarayanaswamyBalakrishnan,“CureRateModelingandApplications.”

Chair:Prof.MartinS.Rapporteurr:DrBanziWellars

10:45–12:30

PanelParallelsession(5-B)- Jolanta Pielaszkiewicz, “On negative (spectral) moments for some functions of Wishart matrices.”- Mahamat Ali Issaka, Ali S. Dabye and Lamine Gueye “On detection of epileptic seizure with an approach based on power

spectral density with an AR model.”- Innocent Ngaruye,DietrichvonRosenandMartinSingull,”Smallareaestimationunderamultivariatelinearmodelfor

incompleterepeatedmeasuresdata”- Jean-Paul Murara, “Pricing European Options under two-dimensional Black-Scholes Partial Differential Equation by using the

Crank-Nicholson Finite Difference Method.”

Chair:DrJosephNzabanitaRapporteur:DeniseUwamaliya

12:30–14:00 Lunch

14:00–16:00PanelParallelsession(5-B)

- John M. Ndiritu, “Empirical modeling and forecasting of exchange rate dynamics in Kenya.”- Rachel Sarguta and Joseph A. M. Ottieno “Four routes to mixed Poisson distributions.”- Mohammed Riad Remita and Thara Belhamra: Graphical approach for claims reserving- GothatamangPatrickNthoiwa,EsterBaleandMosiamiseMokgolele,“Timeseriesmethodsforwaterlevelforecasting

ofGaboronedam,Botswana.”- RichardO.Awichi,”Spatio-temporalPredictionsbyMultivariateSingularSpectrumAnalysis”

Chair: Jolanta Pielaszkiewicz Rapporteur:CharlesRuranga

ThematicArea5A:MathematicalStatisticsandApplications

8:30-9:15 SESSION(5-A+B)- Invited Speaker : Simo Puntanen, “Linear sufficiency in linear estimation and prediction.”

RoomChair:Prof.BengtOveTuressonRapporteur:InnocentNgaruye

9:15-10:00

SESSION(5-A+B)- Invitedspeaker:YoussefOuknine,“Optimalstoppingwithf-expectations:theirregularcase.”

10:00-10:45 Coffee/TeaBreak

10:45–12:30

PanelParallelsession(5-A)- Denise Uwamariya and Emelyne Umunoza Gasana, “Multivariate Analysis of Rwanda Economic Indicators

usingVectorAutoregressive(VAR)Model.”- NelsonOnyangoandSamuelMwalili,“ObtainingHIVsub-nationalindicatorestimates:ASmallAreaEstimation

approach.”- TatjanavonRosenandDietrichvonRosen,“EstimationinGeneralizedReducedRankRegressionModels.”- GodwinNorenseOsarumwenseAsemota“JumpResonanceinWind-FelledPlantains.”

Chair:DrNdengoMarcelRapporteur:NsabimanajeanPaul

12:30–14:00 Lunch

14:00–16:30

PanelParallelsession(5-A)- PatrickG.O.WekeandIdahOrowe,“SimplifiedOptimalandRelayLinearUnbiasedEstimationofParameters

ofLogisticDistribution.”

- Richard Opaka AWICHI, “Spatio-temporal Predictions by Multivariate Singular” - Benson Ade Afere and Ekele Alih, “On the reduction of global error ofmultivariate higher-order product

polynomialkernels.”- LukmanAbiodunNafiuandUmaruWaniyosHamidu,“Prevalenceoffive-childkillerdiseasesandunder-five

mortalityinAdamawastate,Nigeria.”- DanielPan,“Thesearchforultimatetruthininexactscience”

Chair:Dr.SylvesterRugeihamuRapporteur:NkurunzizaAlexandre

ThematicArea5B:MathematicalStatisticsandApplications

8:30-9:15 SESSION(5-A+B)- Invited Speaker : Simo Puntanen, “Linear sufficiency in linear estimation and prediction.”

RoomChair:Prof.BengtOveTuressonRapporteur:InnocentNgaruye

9:15-10:00

SESSION(5-A+B)- Invitedspeaker:YoussefOuknine,“Optimalstoppingwithf-expectations:theirregularcase.”

10:00–10:45 Coffee/TeaBreak

10:45–12:30

PanelParallelsession(5-B)- JosephNzabanita,”Introductiontothe2-foldgrowthcurvemodel”- JosephIviviMwaniki,”Onskewed,leptokurticreturnsandpentanomiallatticeoptionvaluationviaminimal

entropymartingalemeasure”- DavisBundiNtwiga,”InteractionDynamicsinaSocialNetworkUsingHiddenMarkovModel”- Rose Auma Odhiambo and Joseph A. M. Ottieno: Normal mixtures and posterior distributions with their

moments

Chair:Prof.PatrickWekeRapporteur:NiyigenaJeandeDieu

12:30–14:00 Lunch

14:00–16:30

PanelParallelsession(5-B)- StanislasMuhinyuza,TarasBodnarandMathiasLindholm:Atestonthelocationofthetangencyportfolioon

thesetoffeasibleportfolios- GothatamangPatrickNthoiwa,Michael Zwikiti andMosiamiseMokgolele :Methodological comparison of

missingdatatechniquesusedincompletelyrandomizedblockdesign- Sandrine Unezeza and Jean Baptiste Habyarimana : External Economic Shock and Food Price Volatility in

Rwanda:EvidencefromARCHandGARCHmodels- AmadouSawadogo:TiesinOneBlockComparisonExperiments:AGeneralizationoftheMallows-Bradley-Terry

RankingModel

Chair:DrBanziWellarsRapporteur:NsabimanaJeanPaul

ThematicArea5&B:MathematicalStatisticsandApplicationsTime Item Room

9:00:10:00 - Invitedspeaker:JianxinPan:RegularizationofCovariancestructures Chair:DrDenisNdanguzaRapporteur:Nsabimana

Jeanpaul10:00-10:30 Coffee/TeaBreak 10:30-12:30 PanelParallelsession(5-A)

- Denis Kagyera Katakara and Faraimunashe Chirove, “Mathematical Modelling of Hepatitis E Virus (HEV) and Chronic Myeloid Leukemia (CML) Co-infection Dynamics.”

- Rodnellin Onesime Malouata, “The successive Co-Inertia Analysis type 3.”- Patrick G. O. Weke and Idah Orowe, “Stochastic Claims Reserving in Short-Term Insurance Contracts.” - MartinSingull,MoreonestimationofthemeanmatrixinaGrowthCurvemodelinhighdimensions

Chair:DrJosephNzabanitaRapporteur:ByukusengeBeatrice

10:30–12:30

PanelParallelsession(5-B)- IdahOroweandJosephA.M.Ottieno,“Multi-StateTransitionModelswithCensoringinVerticalTransmission

ofHIV.”- DeniseUwamariyaandDenisNdanguza,“ModelingandforecastingmaizeproductioninRwandawith

MarkovChainMonteCarlomethods(MCMC).”- HarounaSangaréandGaneSambLo,“Ageneralstronglawoflargenumbersandapplicationstoassociated

sequencesandtoEVT.”- EverlyneOderoandFredrickOnyango,“ModelingtheeffectsofinterferenceinfertilityrateofKenya.”- KidanemariamAlemBerhie,“IdentificationofriskfactorsassociatedwithunderfivemortalityinEthiopia.”

Chair:Prof.PaulVaderlindRapporteur:NdayambajeFelix

12:30–14:00 Lunch PanelParallelsession(5-A+B)

14:00–15:30

- AbduMohammedSEID,TesfahunBerhanieandLassiRoininen, “HierarchicalBayesian InversionModel forDailyMaximumTemperatureVariability.”

- MosesWamalwaWakoliandJosephA.M.Ottieno,“AlinkbetweenPoissonandexponentialmixturesthroughLaplacetransform.”

- Adilson Silva, Miguel Fonseca and Martin Singull, "Sub-D facing the worst “one-way” designs scenarios”

Chair:DrMinaniFrodualdRapporteur:InnocentNgaruye

.”

14 PROGRAM

Abstract - Invited Speakers

Cure Rate Modeling and Applications

Narayanaswamy BalakrishnanMcMaster University, Hamilton, Canada

Abstract

In this talk, after giving a brief introduction to basic cure rate modelling, I will introduce a flexiblefamily of cure rate models in the framework of competing causes. I will then describe the maximumlikelihood estimation of the model parameters and illustrate it with a cutaneous melanoma data.Next, I will discuss a formal EM-algorithm approach for the estimation of model parameters anduse it to perform model discrimination, and finally revisit the analysis of cutaneous melanoma datafrom this viewpoint. I will finally conclude the talk with some extensions of the work that havebeen carried out recently.

Optimal stopping with f-expectations: the irregular case

Youssef OuknineCadi Ayyad University, Morocco

Abstract

We consider the optimal stopping problem with non-linear f -expectation (induced by a BSDE)without making any regularity assumptions on the pay-o↵ process ⇠. We show that the valuefamily can be aggregated by an optional process Y . We characterize the process Y as the Ef -Snellenvelope of ⇠. We also establish an infinitesimal characterization of the value process Y in termsof a Reflected BSDE with ⇠ as the obstacle. This characterization is established by first showingexistence and uniqueness for the Reflected BSDE with irregular obstacle and also a comparisontheorem. The predictible case is also given using the so called Mertens decomposition and Snellenvelope smallest predictible strong supermartingales greater than pay-o↵ process ⇠.

Regularization of Covariance Structures

Jianxin PanUniversity of Manchester, UK

Abstract

The need to estimate structured covariance matrices arises in a variety of applications and theproblem is widely studied in statistics. In this talk, a new method is introduced for regularizingthe covariance structure of a given covariance matrix whose underlying structure may be blurred byrandom noise, particularly when the dimension of the covariance matrix is high. The regularization

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16 ABSTRACT - INVITED SPEAKERS

is made by choosing an optimal structure from an available class of covariance structures in termsof minimizing the discrepancy, defined via the entropy loss function and Frobenius norm, betweenthe given matrix and the class. A range of potential candidate structures comprising tri-diagonalToeplitz, compound symmetry, AR(1), and banded Toeplitz are considered. It is shown thatfor the first three structures local or global minimizers of the discrepancy can be computed byone-dimensional optimization, while for the fourth structure Newton’s method enables e�cientcomputation of the global minimizer. Simulation studies are conducted, showing that the proposednew approach provides a reliable way to regularize covariance structures for both low- and high-dimensional problems. The approach is also applied to real data analysis, demonstrating theusefulness of the proposed approach in practice.

Linear su�ciency in linear estimation and prediction

Simo Puntanen1 and Augustyn Markiewicz21University of Tampere, Tampere, Finland2Poznan University of Life Sciences, Poznan, Poland

Abstract

A linear statistic Fy is called linearly su�cient, or shortly BLUE-su�cient, for the estimableparametric function of K� under the linear model M = {y,X�,V} if there exists a matrix A suchthat AFy is the best linear unbiased estimator, BLUE, for K�. Similarly, Fy is called linearlyprediction su�cient, or shortly BLUP-su�cient, for the new “future” observation y⇤, say, if thereexists a matrix A such that AFy is the best linear unbiased predictor, BLUP, for y⇤. The newobservation y⇤ is satisfying y⇤ = X⇤�+ e⇤, where � is the same as in M , the parametric functionX⇤� is estimable, and the covariance matrix between e⇤ and y is known. Our purpose is to predicty⇤ on the basis of y. The concept of linear su�ciency is strongly connected to the transformedmodel T = {Fy,FX�,FVF0}: If Fy is linearly su�cient for K� under M , then the BLUEs ofK� are the same under M and T .

The concept of linear su�ciency was essentially introduced in early 1980s by [1, 2]. In thispaper/talk we generalize their results in the spirit of recent papers [3] and [4]. In particular, wepay attention to the linear su�ciency of Fy with respect to y⇤, X⇤� and e⇤ and the mutualrelations between these su�ciencies. We also introduce new upper bounds for the Euclideandistances between the BLUPs under the original model M and the transformed model T .

Keywords: Best linear unbiased estimator, Best linear unbiased predictor, Linear su�ciency,Transformed linear model.

References

[1] Baksalary, J.K. & Kala, R. (1981). Linear transformations preserving best linear unbiasedestimators in a general Gauss–Marko↵ model. The Annals of Statistics, 9:913–916.

[2] Baksalary, J.K. & Kala, R. (1986). Linear su�ciency with respect to a given vector of parametricfunctions. Journal of Statistical Planning and Inference, 14:331–338.

[3] Kala, R., Puntanen, S. & Tian, Y. (2017). Some notes on linear su�ciency. Statistical Papers,58:1–17.

[4] Markiewicz, A. & Puntanen, S. (2017). Further properties of the linear su�ciency in the par-titioned linear model. Matrices, Statistics and Big Data, Proceedings of the 25th InternationalWorkshop on Matrices and Statistics, IWMS-2016, Funchal, Madeira, Portugal, 6–9 June 2016.Springer, in press.

Abstract - Contributed Speakers

On the reduction of global error of multivariate higher-orderproduct polynomial kernels

Benson Ade Afere1 and Ekele Alih11Federal Polytechnic, Idah, Nigeria

Abstract

A higher-order kernel has the features of both negative and positive kernels. The advantage of thisover the lower-order kernel is that it leads to faster rate of convergence. Thus, in this paper, wepresented the reduction of global error of multivariate higher-order product polynomial kernels.The family of product polynomial multivariate higher-order kernels is constructed. A generalizedscheme for determining the global error of any kernel in this family is proposed. A Monte Carloexperiment is performed using six di↵erent data sets and it was observed that our scheme is e�cienteven if the data set departs from the standard normal distribution; and thus have higher rate ofconvergence.

Keywords: Density estimation, Higher-order kernels, Multivariate product polynomial kernel,Generalized global error, Convergence rate.

References

[1] Cacoullos, T. (1966). Estimation of a multivariate density, Annals of the Institute of Statis-tical Mathematics 18, 179 - 189.

[2] Deheuvels, P. (1977). Estimation non parametriani de la densite par histogrammes generalisesII, Publications de l’insitute de statistique de l’universite de Paris 22, 1 - 25.

[3] Epanechnikov, V. A. (1969). Nonparametric estimation of a multivariate probability density,Theory of Probabilty and its Applications 14, 153 - 158.

[4] Hirukawa, M. & Sakudo, M. (2014). Nonnegative bias reduction methods for density esti-mation using asymmetric kernels, Computational Statistics and Data Analysis 75 , 112 -123.

[5] Jones, M. C. & Foster, P. J. (1993). Generalized Jackknifing and higher-order kernels, Journalof Nonparametric Statistics 3, 81 - 94.

[6] Mynbaev, K. & Martins-Filho, C. (2016). Reducing bias in nonparametric density estimationvia bandwidth dependent kernels: L1 view, Statistics and Probability Letters, 1 - 6.

[7] Orava, J. (2011). k-Nearest neigbour kernrl density estimation the choice of optimal k. TatraMountains Mathematical Publications 50, 39 - 50.

[8] Osemwenkhae, J. E. (2003). Higher-order forms in kernel density estimation, Ph.D Thesis,University of Benin, Benin City, Nigeria.

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18 ABSTRACT - CONTRIBUTED SPEAKERS

[9] Parzen, E. (1962). On the estimation of a probability density function and the mode, Annalsof Mathematical Statistics 33, 1065 - 1076.

[10] Ruppert, D., Sheather, S. J. & Wand, M. P. (1995). An e↵ective bandwidth selector for localleast squares regression, Journal of the American Statistical Association. Vol. 90, No. 432,1257 - 1270.

[11] Scott, D. W. (1992). Multivariate Density Estimation: Theory, practice and visualization,New York: John Wiley & Sons Inc.

[12] Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis, London:Chapman and Hall.

[13] Wand, M. P. & Jones, M. C. (1995). Kernel Smoothing, London: Chapman and Hall.

Jump Resonance in Wind-Felled Plantains

Godwin Norense Osarumwense Asemota11University of Rwanda, Kigali, Rwanda

Abstract

In this paper, jump resonance was applied to wind-felled plantains, which budded on the plantainpseudo-stem side when guyed about 60� to the horizontal to obtain the jump function. Du�ng’smodel, describing function and Chebyshev polynomials were used to obtain the best fit of ap-proximants to the developed plantains jump function. Traditionally, wind-damaged plantains andbanana pseudo-stems are cut for future vegetative growth, leading to heavy and perennial losses toindividuals, households, communities, nations and even regions. The motivation for this study wasthe possibility of using jump resonance discontinuity as a plantain wind-damaged salvage process.Furthermore, it was to develop plantain growth equation and determine the harmonic content re-sponsible for the observed stable switching mode for plantain side shoot outgrowth. The resultsshow that the polynomial growth equation for plantains was of order 14 with 0.5 percent error. Inaddition, the first harmonic content in the plantain jump resonance function was absent and onlythe jump-up mode of the switching mechanism was stable, leading to plantain pseudo-stem sideshoot outgrowth. Therefore, a sinusoid plus bias (steady or d.c.) signal input could be used toimprove algorithm accuracy in future research.

Keywords: Chebyshev polynomials, Describing function, Du�ng’s model, Harmonic, Jump-upmode, Switching.

Spatio-temporal Predictions by Multivariate Singular Spec-trum Analysis

Richard O. Awichi11Busitema University, Tororo, Uganda

Abstract

In this paper, I present a method for utilizing the usually intrinsic spatial information in spatialdata sets to improve the quality of temporal predictions within the framework of singular spectrumanalysis (SSA) techniques. The SSA-based techniques constitute a model free approach to time se-ries analysis and ordinarily, SSA can be applied to any time series with a notable structure. Indeedit has a wide area of application including social sciences, medical sciences, finance, environmentalsciences, mathematics, dynamical systems and economics. SSA has two broad aims:

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i) To make a decomposition of the original series into a sum of a small number of independentand interpretable components such as a slowly varying trend, oscillatory components and astructure-less noise.

ii) To reconstruct the decomposed series for further analysis in the absence of the noise compo-nent.

Multivariate singular spectrum analysis (MSSA) is an extension of SSA to multivariate statisticsand takes advantage of the delay procedure to obtain a similar formulation as SSA though withlarger matrices for multivariate data. In situations where spatial data is an important focus ofinvestigation, it is not uncommon to have attributes whose values change with space and timeand an accurate prediction is thus important. The usual question asked is whether the intrinsiclocation parameters in spatial data can improve data analysis of such data sets. The proposedmethod is based on the inverse distance technique and is exemplified on rainfall (climate) data.Results show that the proposed technique of incorporating spatial dependence into MSSA analysisleads to improved quality of statistical inference thus more reliable predictions.

Keywords: Time Series Analysis, MSSA, Inverse Distance Weighting, Spatial Dependence.

Identification of risk factors associated with under five mor-tality in Ethiopia

Kidanemariam Alem Berhie11University of Gondar, Gondar, Ethiopia

Abstract

An Ethiopian child is 30 times more likely to die by his or her fifth birthday than a child in WesternEurope [1]. Therefore, understanding the determinants of under-five mortality is helpful to narrowthe gap between di↵erent social classes.

The survey collected information from a total of 16,515 women aged 15-49 years out of which9917 women were considered in this study. To meet the research objectives, descriptive statistics,Poisson, negative binomial, zero inflated Poisson and zero-inflated negative binomial regressionmodels were used for data analysis.

The descriptive statistical results showed that nationally 34.9% of mothers had faced at leastone under five death. Based on the result of the zero inflated Poisson regression model, region, placeof delivery, source of drinking water, family size, wealth index, occupation of fathers, employmentstatus of mother, availability of toilet facility, type of birth, birth order and vaccination of childrenwere found to be significant determinant of under-five mortality.

Then, the researcher revealed that e↵orts should be done regarding family planning services,using mass media, family planning workers and health centers, in order to have a family with abalanced number of children. The head o�ces should work to properly raise the awareness ofparents for vaccination and e↵orts should be made to improve access to pipe drinking water andtoilet facility.

Keywords: Under Five Mortality, Poisson, Negative binomial, Zero-inflated Poisson, Zero-inflatedNegative binomial.

References

[1] Mubiana M. and Bernard P. (2008). Di↵erentials in under-five mortality in Zambia: persistentgaps in child survival interventions. Medical journal of Zambia, Volume 35 number 4.


On detection of epileptic seizure with an approach based onpower spectral density with an AR model

Mahamat Ali Issaka1,2, Ali S. Dabye1,2 and Lamine Gueye31Universite de N’Djamena, N’Djamena, Chad2Universite Gaston Berger, Saint-louis, Senegal3Centre Hospitalier Universitaire, Dakar, Senegal

Abstract

In this manuscript, we considered an autoregressive (AR) model combined with spectral analysisand segmentation criterion in order to analyze electroencephalograph (EEG) recording data col-lected from normal and epileptic subjects. With the goal to localize epileptic seizures from EEGdata, we proposed methods on how to calculate the power spectral density (PSD) for each of the128 time series generated from EEG via estimating parameters in an AR(p) model, where p is de-termined by AIC and BIC as the examination of EEG signals is often done with visual inspectionof the rhythm (delta, theta, alpha, beta, gamma) by neurologist practitioners.The accuracy of the detection is estimated to 70% with the sensitivity of 80.56% according to theinterpretation of neurologist.

Keywords: Spectral Analysis, Change-point, Epilepsy, EEG.

References

[1] Abdulhamit Subasi (2007), Selection of optimal AR spectral estimation method for EEG signalsusing Cramer-Rao bound, Computers in Biology and Medicine, 37, 183� 194.

[2] O. Faust, R.U. Acharya , A.R. Allen ,C.M.Lin (2008), Analysis of EEG signals during epilepticand alcoholic states using AR modeling techniques,ITBM-RBM , 29, 44� 52.

[3] S.M. Kay (1993), Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice-Hall, New Jersey.

[4] Wayne A. Fuller (1996), Introduction to Statistical Times Series, Wiley Series in Probabilityand Statistics.

Mathematical Modelling of Hepatitis E Virus (HEV) andChronic Myeloid Leukemia (CML) Co-infection Dynamics

Denis Kagyera Katakara1 and Faraimunashe Chirove21Mbarara University of Science and Technology, Mbarara, Uganda2Kwa-Zulu University, Natal, South Africa

Abstract

There are major advances which have been made to understand HEV and CML transmissiondynamics but none of these have considered the e↵ects of transmission parameters on the burdenof HEV on CML prevalence in a co-infection scenario. We formulated a mathematical model forthe co-infection of HEV and CML using a system of ordinary di↵erential equations, in order tounderstand the e↵ects of the co-infection on HEV and CML and vice versa in a human population.The model was analysed and steady state conditions were derived. Our results showed that thedisease free equilibrium was both locally stable and globally stable if the basic reproduction number,R0h 1 and unstable if the basic reproduction number, R0h > 1. Our results also suggestthat, (i) HEV reduces the CML infectives and accelerates the co-infection, (ii) CML enhances the

21

progression of both HEV infection and the co-infection and, (iii) there is an increase in HEV-CMLburden due to co-infection compared to single infections of either HEV or CML.

Keywords: CML, HEV, Basic reproduction number, Co-infection.

References

[1] J. Y. T. Mugisha, B. Nannyonga, D. J. T Sumpter, L. S. Luboobi (2012). The Dynamics, Causesand Possible Prevention of Hepatitis E Outbreaks. PLoS ONE 7.7: e41135. doi:10.1371/journal.pone.0041135.

[2] M. Helen, K. Natasha Li (2004). A mathematical model for CML and T cell interaction. Journalof Theoretical Biology, 227: 513- 523.

Kernel density estimation: Predicting microRNA gene bound-aries

Timothy Kevin Kuria Kamanu11University of Nairobi, Nairobi, Kenya

Abstract

Gene-finding or prediction of up- and down-stream gene boundaries is important in understandingand annotating the structure and function of genomic DNA during normal and aberrant states e.g.diseases. Most gene-finding methods focus on identifying protein-coding genes from repeat-maskedDNA sequences often in a given species [1, 2]. However, approximately 97% of genomic DNA inhigher eukaryotes comprises non-coding and repetitive genomic DNA that encode important non-coding regulatory RNA (ncRNA) genes including microRNAs (miRNA). miRNA are emergingas important dominant regulators of genome integrity and protein expression that co-ordinatethe onset, development and progression of cancer and other incurable diseases [3]. FunctionalmiRNA products are excised from well-defined ⇡ 70-110nt monocistronic and/or polycistronicgene sequences that have characteristic stem-loop/hairpin-like secondary structures.

We propose an novel ab initio method for predicting miRNA gene boundaries from any genomicDNA sequence in any species. The method implements kernel density estimation and machinelearning given miRNA genes in di↵erent miRNA families [4]. A genetic algorithm was used tooptimize classifiers that exploit the physics and symmetry of hairpin-like gene sequences to unveilpreferred nucleotides and palindromic motifs that may guarantee stable RNA secondary structures.The classifiers can be used to detect and quantify motif signals on potent genomic sequences usingsliding windows thus resulting in multiple and nested window score profiles. Numerical derivativesof densities that were estimated using Epanechnikov, triangular, bi-weight and Gaussian kernelswere used to investigate the mode (most probable detection window) given centered window scoreprofiles and corresponding boundaries. Our method is significant for annotating miRNA genes inrepetitive genomic regions. The results indicate that: miRNA gene boundaries can accurately bepredicted and that annotated extrinsic upstream and downstream miRNA gene boundaries maybe understated by approximately 10nt and 20nt, respectively - and this may obscures importantflanking regions that characterize the genes or their functional products.

Keywords: Computational Biology, Genomics, MicroRNA, Gene boundaries, Kernel smoothing,Gene finding.

References

[1] Slaetor, R. D. (2010). An overview of the current status of eukaryote gene prediction strategies.Gene. 461:1-4.


[2] Yandell, M. & Ence, D. (2012). A beginner’s guide to eukaryotic genome annotation. NatureReviews Genetics. 13:329-342.

[3] Kamanu, T. K. K., Radovanovic, A., Archer, J. A. C. & Bajic,V. B. (2013). Exploration ofmiRNA families for hypotheses generation. Scientific Reports 3:2940,1-8.

[4] Wand, M.P. & Jones, M.C. (1994). Kernel Smoothing. Chapman & Hall/CRC Monographs onStatistics & Applied Probability, Taylor & Francis.

The successive Co-Inertia Analysis type 3

Rodnellin Onesime Malouata11Marien Ngouabi University, Brazzaville, Republic of Congo

Abstract

We interest in ecological surveys with temporal components by coupling a faunistical table and amesological table. This notion of coupling between two tables has been studied in the co-inertiaanalysis (Lafosse and Hanafi, 1997). when the two tables are partitioned in line, the link betweentwo multi-tables has been studied in STATICO (Simier et al., 1999). It is well known that theweighting coe�cients for determining the compromise may be contrary signs in some cases andrender the latter interpretable (Thioulouse and Chessel, 1987). This observation has given riseto alternative techniques which maximize the sum of the covariances and the sum of the squaresof the covariances between the components of the couple’s tables. In this work we propose theextension of the criterion of co-inertia analysis by maximizing the product of two double sums ofthe squares of the covariances between the component of one of the tables and the variables of theother table of the couple (Niere, 2014). The method can be called successive Co-Inertia Analysistype 3 (ACIs3). The objective of this method is to characterize the stability of the relations ob-served between two groups of descriptors.An example of implementation of the method is presented as part of an ecological study. Wecompare the ACIs3 approach to the STATICO analysis, which amounts to seeking the average ofthe co-structures. In this particular situation, the two methods lead to similar interpretations.

Keywords: Multi-tables, Co-inertia analysis, Common structure, STATICO.

References

[1] Lafosse, R., and Hanafi, M. (1997). Concordance d’un tableau avec K tableaux: definition deK + 1 uples synthetiques. Revue de Statistique Appliquee, XLV(4):111–126.

[2] Niere, L. (2014). Proposition d’analyses de lien entre deux multi-tableaux verticaux: Methodeset applications. These de doctorat. Universite Marien Ngouabi..

[3] Simier, M., Blanc, L., Pellegrin, F. and Nandris, D. (1999). Approche simultanee de k couplesde tableaux: Application a l’etude des relations pathologie vegetale-environnement. Revue deStatistique Appliquee, 47:31–46.

[4] Thioulouse, J., and Chessel, D. (1987). Les analyses multitableaux en ecologie factorielle. i:De la typologie d’etat a la typologie de fonctionnement par l’analyse triadique. Acta Oecologica,Oecologia Generalis, 8:463–480.

23

A new technique to improve ensemble rainfall prediction

Isaac Mugume1, Michel D. S. Mesquita2, Yazidhi Bamutaze1, Didier Ntwali3, Daniel Waiswa1,Charles Basalirwa1, Joachim Reuder4, Revocatus Twinomuhangi1, Fredrick Tumwine1, TriphoniaJacob Ngailo5 and Bob Alex Ogwang61Makerere University, Kampala, Uganda2Uni Research Climate, Bjerknes Centre for Climate Research, Bergen, Norway3Institute of Atmospheric Physics, University of Chinese Academy of Sciences, Beijing, China &Rwanda Meteorological Agency4Geophysical Institute, University of Bergen, Bergen, Norway5Department of General Studies, Dar as Salaam Institute of Technology, Dar as salaam, Tanzania6Uganda National Meteorological Authority, Kampala, Uganda

Abstract

Accurate and timely rainfall prediction enhances productivity and can aid proper planning insectors such as agriculture, health, transport and water resources [1]. This study is aimed atimproving rainfall prediction using ensemble methods. It first assesses the performance of sixconvective schemes (Kain–Fritsch (KF); Betts-Miller-Janjic (BMJ); Grell-Fretas (GF); Grell 3Densemble (G3); New-Tiedke (NT) and Grell-Devenyi (GD))[3] using the root mean square error(RMSE) and mean error (ME)[2] focusing on the March-May 2013 rainfall period over Uganda. 18ensemble members are generated from the three best performing convective schemes (i.e. KF, GF& G3). The performance of three ensemble methods (i.e. ensemble mean (EM); ensemble meananalogue (EMA) and multi–member analogue ensemble (MAEM)) is also analyzed using the RMSEand ME. The EM presented a smaller RMSE compared to individual schemes (EM:10.02; KF:23.96;BMJ:26.04; GF:25.85; G3:24.07; NT:29.13 & GD:26.27) and a better bias (EM:-1.28; KF:-1.62;BMJ:-4.04; GF:-3.90; G3:-3.62; NT:-5.41 & GD:-4.07). The EMA and MAEM presented 13/21stations & 17/21 stations respectively with smaller RMSE compared to EM thus demonstratingadditional improvement in predictive performance. The MAEM is a new approach proposed anddescribed in the study.

Keywords: Quantitative rainfall prediction, Multi–analogue ensemble mean, Convective schemes.

References

[1] Ogwang, B. A., Chen, H., Li, X. and Gao, C. (2014). The influence of topography on EastAfrican October to December climate: sensitivity experiments with RegCM4. Advances in Me-teorology, 1–14.

[2] Mugume, I., Basalirwa, C., Waiswa, D., Reuder, J., Mesquita, M. d. S., Tao, S. and Ngailo, T.J. (2016). Comparison of Parametric and Nonparametric Methods for Analyzing the Bias of aNumerical Model. Modelling and Simulation in Engineering, 1–9.

[3] Mayor, Y. G. and Mesquita, M. D. S. (2015). Numerical simulations of the 1 May 2012 deep con-vection event over Cuba: sensitivity to cumulus and microphysical schemes in a high–resolutionmodel. Advances in Meteorology, 1–11.


A test on the location of the tangency portfolio on the set offeasible portfolios.

Stanislas Muhinyuza1,2, Taras Bodnar1 and Mathias Lindholm1

1Stockholm University, Stockholm, Sweden2University of Rwanda, Kigali, Rwanda

Abstract

Due to the problem of parameter uncertainty, specifying the location of the tangency portfolio onthe set of feasible portfolios becomes a challenging task. The set of feasible portfolios is a parabolain the mean-variance space with optimal portfolios lying on its upper part. Using the statisticaltest theory, we make a decision if the tangency portfolio is mean-variance e�cient, i.e if it belongsto the upper limb of the e�cient frontier. In the opposite case, the investor would prefer to investinto the global minimum variance portfolio which lies in the vertex of the set of feasible portfolios.Assuming that the portfolio asset returns are independent and multivariate normally distributed,we provide a test on the location of the tangency portfolio on the set of feasible portfolios. Thedistribution of the test statistic is derived under both hypotheses, using which we analyse thepower of the test and construct a confidence interval. Moreover, we conduct the out-of-sampleperformance of the portfolio determined by implementing the suggested test. The robustness tothe assumption of normality is investigated via an extensive simulation study. In an empiricalstudy we apply the developed theory to real data.

Keywords: Tangency portfolio, Feasible portfolios, Test theory, Power function, Out-of-sampleperformance.

Pricing European Options under two-dimensional Black-ScholesPartial Di↵erential Equation by using the Crank-NicholsonFinite Di↵erence Method

Jean-Paul Murara1,2, Anatoliy Malyarenko1, Ying Ni1 and Sergei Silvestrov11Malardalen University, Vasteras, Sweden2University of Rwanda, Kigali, Rwanda

Abstract

In the option pricing process, Black-Scholes in 1973 solved a partial di↵erential equation andintroduced a model to determine the price of European Options. Many researchers improvedBlack-Scholes model afterwards. Christo↵ersen proved in 2009 that models with two stochasticvolatilities capture better the skewness and the smiles of the volatilities, meaning that they canmore accurately determine the options prices. Chiarella and Ziveyi in 2013 and Canhanga et al.in 2014 used the model introduced by Christo↵ersen to determine European and American optionprices respectively.While dealing with many problems in financial engineering, the application of Partial Di↵erentialEquations (PDEs) is fundamental to explain the changes that occur in the evolved systems. Somefamilies of this type of equations are known to have the so-called classical solutions. Others canbe transformed into simple PDEs, for example by using scaling methods, Laplace and Fouriertransforms, afterwards one can compute their solutions. Moreover, in many cases the PDEs thatcharacterize the real life problems do not have known forms of solutions. In this occasion, numericalmethods are considered in order to obtain the approximate solutions of the PDEs.In the present paper, we consider the option pricing problems that involves a two-dimensionalBlack-Scholes PDE as the one obtained by Canhanga et al. in 2014, and instead of solving it bythe approximation approach presented by Conze in 2010 we perform the Crank - Nicholson finitedi↵erence method. Comparing examples are included in the paper.

25

Keywords: Stochastic Volatility, Two-dimensional Black-Scholes PDE, Crank-Nicholson FiniteDi↵erence Method.

References

[1] Chiarella C. and Ziveyi J. (2011). Two Stochastic Volatility Processes - American OptionPricing. Quantitative Finance Research Centre, number:292.

[2] Canhanga B., Malyarenko A., Ni Y., and Silvestrov S. (2014). Perturbation Methods for PricingEuropean Options in a Model with Two Stochastic Volatilities. In SMTDA 2014 Proceedings ,ISAST: Internationa Society for the Advancement of Science and Technology, number:157-168.

Distributions Arising from Polya - Aeppli Birth Process

Nicholas Mwilu Mutothya1 and Joseph A. M. Ottieno21Taita Taveta University, Voi, Kenya2University of Nairobi, Nairobi, Kenya

Abstract

Many real life processes realize non-negative real valued event that evolve stochastically. Identi-fication of the distribution associated with each process helps in describing and in-depth under-standing of the behavior and characteristics of the process. A birth process is a phenomenon wherethe number of event occuring with time is non decreasing integer. In this paper, distributions willbe constructed for three variables namely; size of the population at any time t, X(t), independentincrements X(t)�X(s) where t > s, and the waiting time upto nth event.

[1] is among the early researchers who developed di↵erence di↵erential equations for a birthand death process. The di↵erence di↵erential equations for single birth process are derived withthe assumption that within an infinitesimal time interval only one event occurs or no event. Toextend the idea of single birth process to many real life processes, multiple events are allowed tooccur within an infinitesimal time interval; this leads to multiple birth process.[2] considered four approaches to solving di↵erence di↵erential equation for simple birth and deathprocess. In this paper various mathematical techniques; Iteration Method, Probability GeneratingFunction and Laplace Transform will be used to solve multiple birth process di↵erence di↵erentialequations. Further, suppose the parameter in birth process vary taking a particular distribution,then we have a single or multiple mixed birth process. A birth process with gamma mixing lead toa polya (Mixed Poisson) process. [3] has studied Polya-Aeppli birth process with gamma mixingdistribution leading to inflated-parameter polya process. Mixture of birth process with othermixing distribution will be obtained.

Keywords: Pure Birth, Independent Increments, Waiting Time, Di↵erence Di↵erential Equations,Mixed Distribution, Polya-Aeppli.

References

[1] Kendall, D.G. (1949). Stochastic Processes and Population Growth. Journal of the Royal Sta-tistical Society, 11.2, 230:282.

[2] Morgan, B.J.T. (2005). Four Approaches to Solving Linear Birth - and - Death (and Similar)Processes. International Journal of Mathematics Education in Science and Technology., 10.1,51:64.

[3] Minkova, L.D. (2011). I-Polya Process and Applications,. Commun.statist. - Theory and Meth-ods, 40, 2849:2855.


On skewed, leptokurtic returns and pentanomial lattice op-tion valuation via minimal entropy martingale measure

Joseph Ivivi Mwaniki11University of Nairobi, Nairobi, Kenya

Abstract

This article develops, a lattice-based approach for pricing contingent claims when parameters gov-erning the logs of the underlying asset dynamics are modelled by generalized hyperbolic distributionand normal inverse Gaussian distribution. The pentanomial lattice is constructed using a momentmatching procedure. Moment generating functions of Generalized hyperbolic distribution and Nor-mal inverse Gaussian distribution are utilized to compute probabilities and jump parameters underhistorical measure P. Minimal entropy martingale measure (MEMM) is used to value Europeancall option with a view of comparing the results with some of the existing benchmark model suchas Black Scholes model. Empirical data from S&P500 index, RUTSELL2000 index and RUI1000index are used to demonstrate how the model works. There is a significant di↵erence especiallyfor long term maturity (six months and above) type of contracts, the proposed model outperformthe benchmark model, while performing poorly at short term contracts. Pentanomial NIG modelsseems to outperform the other models especially for long dated maturities.

Keywords: Pentanomial lattice, Generalized hyperbolic distribution, Normal Inverse Gaussian,Minimal entropy martingale measure, European call option.

Prevalence of five-child-killer diseases and under-five mortal-ity in Adamawa state, Nigeria

Lukman Abiodun Nafiu1 and Umaru Waniyos Hamidu21Islamic University in Uganda, Mbale, Uganda2Adamawa State Polytechnic, Yola, Nigeria

Abstract

This study investigates the prevalence of the five-child killer diseases and its cause e↵ect on under-five mortality. It uses an entirely quantitative approach with secondary data between 2001 and2015 obtained from the data bank of Adamawa State Primary Health Care Development Agency(PHCDA). Data was collected regarding the number of children immunized and number of childrenthat were infected but later died due to Pneumonia, Diarrhoea, Measles, Tetanus, Polio and theoverall under-five mortality irrespective of diseases within that time frame. The study measuresprevalence rate per a thousand live birth and uses Newey-West regression tool for analyzing anddeveloping a model. The results indicate that the prevalence rates have generally been decreasingwith Pneumonia recording the highest prevalence and Tetanus recording the lowest prevalence.Polio was excluded from the analysis because it did not register any incidences or deaths. The re-gression model shows that there is a strong positive and significant relationship between Pneumoniaand under-five mortality. The model also shows a weak positive and non-significant relationshipbetween diarrhoea and under-five mortality. Furthermore, there was a strong positive but non-significant relationship between measles and under-five mortality and a negative non-significantrelationship between tetanus and under-five mortality. The child killer diseases explained 60.92%cause e↵ect on overall under-five mortality and the model is statistically significant. The studyrecommends that government needs to implement the Global Action Plan for Pneumonia and Di-arrhoea (GAPPD) as campaigned by WHO and UNICEF, adequate nutrition should be given tothese children and children infected with HIV/AIDs should be given daily vaccines to reduce therisk of contracting these five-child-killer-diseases.

Keywords: Five-child killer diseases, Prevalence, Nutrition, Mortality, Under-five children.

27

Empirical modeling and forecasting of exchange rate dynam-ics in Kenya

John M. Ndiritu11University of Nairobi, Nairobi, Kenya

Abstract

The performance of the Kenyan Shilling against other foreign currency is important in the macroe-conomics state of the economy. Exchange rate influences the international trade balance betweenthe two nations being considered. This paper uses the Box-Jenkins approach to fit IntegratedAutoregressive Moving Average (ARIMA)[1] model to the Kenya Shillings/USD exchange ratedata. Further Autoregressive Conditional Heteroscedasticity (ARCH)[2] family of models is usedto analyze the exchange rate returns of the data. The best realized model is used to forecast futureKenya Shillings/USD exchange rate.

Keywords: Exchange rate, ARIMA, ARCH, Forecasting.

References

[1] Shumway Robert H. & Sto↵er David S., (2011). Time Series Analysis and it Application withR examples. Springer NY.

[2] Francq Christian & Jean-Michel Zakoyan , (2010). GARCH Models Structure, Statistical In-ference and Financial Applications. John Wiley & Sons Ltd

Small area estimation under a multivariate linear model forincomplete repeated measures data

Innocent Ngaruye1,2, Dietrich von Rosen1,3 and Martin Singull11Linkoping university, Linkoping, Sweden2University of Rwanda, Kigali, Rwanda3Swedish University of Agricultural Sciences, Uppsala, Sweden

Abstract

In this talk, the issue of analysis of multivariate repeated measures data that follow a monotonicsample pattern for small area estimation is addressed. Random e↵ects growth curve models withcovariates for both complete and incomplete data are formulated. A conditional likelihood basedapproach is proposed for estimation of the mean parameters and covariances. Further, the predic-tion of random e↵ects and predicted small area means are also discussed. The proposed techniquesmay be useful for small area estimation under longitudinal surveys with grouped response unitsand drop outs.

Keywords: Conditional likelihood, Multivariate linear model, Monotone sample, Repeated mea-sures data


Time series methods for water level forecasting of Gaboronedam, Botswana

Gothatamang Patrick Nthoiwa1, Ester Bale1 and Mosiamise Mokgolele1

1Botswana College of Agriculture, Gaborone, Botswana

Abstract

Botswana has predominantly sub-tropical climate. As a result the country is largely arid to semi-arid. The rainfall season is in the summer months which start from October to march which tendto be erratic and unpredictable. The average annual rainfall is about 500mm per annum in theextreme northeast, to less than 250mm per annum in the other parts of the country. Gaboronedam is located in southern part of Botswana ,After the dam was opened and filled, the averagewater levels began dropping. In part, this was due to a cyclical change in rainfall, In part it wasdue to growth of the city and growing per-capita demand for water as the population became morea✏uent, using water for purposes such as filling swimming pools and washing cars. By the end of2004 the reservoir was just 27% full and the government was forced to impose harsh restrictions onwater use .In the drought-prone country, the water supply is a constant concern. In this paper timeseries is used to for Modeling and forecasting of water dam level as they are a major componentof e↵ective planning and management of water resources. This study uses 45 year record of thedaily water levels at the dam from 1966 to 2011. The desired statistical interval was divided intotwo parts and statistics for 1966 to 2004 were used for modeling and statistics from 2005 to 2010were used for validation of the model. The PACF plot shows definite significant values at lag 1, 12,and 13.A A few other partial autocorrelations may be significant also.A The ACF has significantautocorrelations at lag 1 and 12.A This indicates that an ARIMA(0,1,1)⇥(0,1,1). Using thismodel, the Dam level was predicted for up to 60 months.

Keywords: Time Series, Reservoir Level, ARIMA, Forecasting.

Methodological comparison of missing data techniques usedin completely randomized block design

Gothatamang Patrick Nthoiwa1, Michael Zwikiti1 and Mosiamise Mokgolele1

1Botswana College of Agriculture, Gaborone, Botswana

Abstract

The problem of missing data in statistical analysis is one that the field of experimental designhas failed to adequately address despite its potential to significantly a↵ect results and subsequentsubstantive conclusions. The purpose of this study is to evaluate the practical application ofmissing data techniques Complete randomized design. It will focus on multiple imputations, adhoc imputation, Em algorithm and multiple imputation. These are used to estimate parametersin a complete dataset and data missing at random, then sets of results will be compared usingstandard error of di↵erence and standard error of means to determine the best method. It isconcluded that the under the multiple imputation (MCMC) method is the best for the treatmentof missing data for analysis of variance. Complete case analysis showed lower variation valuesthou is not the best method for treating missing data as it can substantially lower the sample size,leading to a severe lack of power and does not properly reflect statistical uncertainty.

Keywords: Completely randomised block design, Missing data, Multiple imputation, Adhocmethods, Complete case.

29

Interaction Dynamics in a Social Network Using Hidden MarkovModel

Davis Bundi Ntwiga11University of Nairobi, Nairobi, Kenya

Abstract

Agents interactions in a social network are dynamic, stochastic in nature and evolve over time.These interactions can be reinforced, decay with time or new friendships created. In this paper,we enhance research in interaction dynamics of agents in a social network through the use of aprobabilistic model, the hidden Markov model [1],[3]. The transition matrix is estimated throughsimulated data with three processes of forgetting, reinforcement and exploration. An agent canforget current interactions, reinforce it or explore new avenues of interactions.. Singular valuedecomposition estimates the observation matrix and initial state probability through rank reductionand error estimation of the transition matrix respectively, [2]. Hidden Markov model estimatesthe states of the agents and observations emitted from the interactions of the agents in the socialnetwork. The level of dependency amongst the agents in the network through interactions isvery high. The first 50% portion of rank reduction has higher coe�cient of variation and higherstochasticity as compared to the second 50% portion. At the error estimation of 50% and above,dependency levels decrease. Therefore, agents interactions in a social network account for between20% and 50% of all the active interactions in the network. The other portion represents noise dueto agents strategies and interaction intensity as they change their transition positions in the socialnetwork.

Keywords: Agents, social network, Hidden Markov model, Interactions, Singular value decom-position.

References

[1] Ehab, E., and Sassone, V. (2013). A HMM based reputation model. Advances in SecurityInformation and Communication Networks, 381: 111-121

[2] Kalman, D. (1996). A singularly valuable decomposition: The SVD of a matrix. The CollegeMathematics Journal, 27 (1): 1-23

[3] Raghavan, V., Steeg, G., Galstyan, A., and Tartakovsky, A.G. (2013). Modeling temporalactivity patterns in dynamic social networks. arXiv:1305.1980vl

Introduction to the 2-fold growth curve model

Joseph Nzabanita11University of Rwanda, Kigali, Rwanda

Abstract

There is a growing interest in the analysis of multi-way data. In some studies the inference aboutthe dependencies in three-way data is done using the third order tensor normal model, wherethe focus is on the estimation of the variance-covariance matrix which has a Kronecker productstructure. Little attention is paid to the structure of the mean, though, there is a potential toimprove the analysis by assuming a structured mean. We will introduce a 2-fold growth curvemodel by assuming a trilinear structure for the mean in the tensor normal model and propose analgorithm for estimating parameters. Simulation study to illustrate the quality of the proposedalgorithm will be presented.


Keywords: Growth curve model, Kronecker product structure, Maximum likelihood estimators,Multi-way data, tensor normal model, Trilinear regression

Modeling the e↵ects of interference in fertility rate of Kenya

Everlyne Odero1 and Fredrick Onyango21Masinde Muliro University of Science and Technology, Kakamega, Kenya2Maseno University, Kisumu, Kenya

Abstract

Many studies have been done on fertility for many years. However, very little has been docu-mented in the existing literature concerning modeling of fertility in the presence of interference,yet interference to fertility is a common phenomenon. In this study fertility data sets for Kenyawere modeled both before and after interference. The parameters of the model were estimated bythe maximum likelihood estimation method. Using Akaike’s Information Criteria, (AIC), it wasestablished that amongst the distributions fitted; Gamma, Weibull and Lognormal, Gamma gavethe best fit for the Kenya fertility rate data and interference simply shifts the Gamma distributionparameters. The result of this study would help the Governments to understand fully the e↵ectof interference on fertility rate and plan for it. Demographers would also benefit from this studysince it can be used to project population growth after an interference.

Keywords: Fertility, Interference, Fecundity, Kenya.

References

[1] Barret R. E.,Bogue J. D. and Anderson D. L. (1997).The population of the United States. 3rd

edition.Compendium of data functions. Population Studies, 14: 148-162

[2] Brass W. (1960). The graduation of fertility distributions by polynomial functions. PopulationStudies, 14: 148-162.

[3] Gage T.B. (2000) The Age specific fecundity of mammalian population: A test of three mathe-matical models.Paper

[4] Guarcello, L., F. Mealli and F. C. Rosati (2002). Household Vulnerability and Child Labor: TheE↵ect of Shocks, Credit Rationing and Insurance. Working Paper, UCW.

Normal mixtures and posterior distributions with their mo-ments

Rose Auma Odhiambo1 and Joseph A. M. Ottieno11University of Nairobi, Nairobi, Kenya

Abstract

Normal mixtures are derived from conditionalN(µ,�2) distribution where the conditioning variable(�2)varies, Linden (2001). Distributions chosen for variance appear to be very limited. Nadarajah(2012) uses sixteen mixing distributions to derive normal mixtures and the corresponding estima-tion procedures are derived by method of moments.This work aims to construct normal mixtures from conditional N(µ,�2) distributions for caseswhere µ = 0 and µ is a function of the variance. We use eighteen mixing distributions for bothcases. A formula for moments,posterior distributions and posterior moments is obtained.

Keywords: Variance, Moments, Posterior, Normal, Mgf, Special function.

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References

[1] Nadarajah, S. (2012) . Models for stock returns. Quantitative Finance, 12:411-424.

[2] Linden, M. (2001) . A model for stock returns distribution. International Journal of Financeand Economics, 6:149.

Obtaining HIV sub-national indicator estimates: A SmallArea Estimation approach

Nelson Onyango1 and Samuel Mwalili21University of Nairobi, Nairobi, Kenya2US Centre for Disease Control, Nairobi, Kenya

Abstract

Aims: The aim of this study is to obtain informative and precise sub-national or county estimatesof HIV prevalence data (and later sub-counties) within Kenya. Ante-natal Care (ANC) sentinelsurveillance and population based surveys provide complementary information, thereby provid-ing a clear picture of the level, trends, and distribution of HIV infection. This analysis howeveris based on population based surveys including Kenya AIDS Indicator Survey (KAIS 2014) andKenya Demographic and Health Surveillance (KDHS) data, among other data archives.Methodology: Estimates for smaller areas or regional units are increasingly being sought for re-source planning in sub-national governments. Kenya for instance, has since been divided intosemi-autonomous county governments, with economic resources that need to be planned for. Directdomain estimates from sample survey techniques are mainly design based and may lack precision.Larger samples are often required to attain better estimates, but this can be very costly, requiringlarger budgets that most small regions may not a↵ord [2]. Small area estimates (SAE) are a hybrid,having both design based and model based assumptions and capable of incorporating uncertaintyin domain estimates and information from ancillary data. This enables better precision from SAE(informative confidence estimates) than domain estimates [1]. The model based approach alsoenables incorporating spatial patterns in the data, hence enabling improved estimation of domainsthat have smaller sample sizes.Results: Results show that standard errors under SAE are smaller that direct Domain estimates ofthe HIV prevalence, precision is more improved in the SAE’s as the confidence intervals for countyestimates are reduced considerably. The estimation process under SAE is thus more informativeand useful for planning purposes.Discussion: The challenge with HIV prevalence or incidence estimation is the lack of proper datasets for population demographics and HIV related data. ANC data are often subjective, onlyconsidering the reproductive female population. While national population based surveys such asAIS and DHS typically have national coverage and generate data for women and men in urban andrural areas, they also have limitations such as the potential for bias introduced by non-responseand the exclusion from the sampling frame of population groups at high risk of HIV infection. Asa result, available data sets often have cases of missing information or are generally aggregateddatasets. Furthermore, although the Fay Herriot model allows for incorporating spatial patternsin data, its a challenge to define true spatial correlations between data sets.Conclusions: When sample sizes are small, as is often the case with small domains, estimates fromsurvey data are greatly improved when design based methods from survey estimation are blendedwith model based methods such as Small Area Estimation that allow for estimation of uncertaintyand use of covariates in yielding the final data estimates.

Keywords: Small Area Estimates, Domain Estimation, HIV prevalence.


References

[1] Polettini, S. (2016). A generalised semiparametric bayesian Fay Herriot model for Small AreaEstimation shrinking both means and variances. Bayesian Analysis, TBA:TBA

[2] Wanjoya, A. K., N. Torelli, and G. Datta. (2012). Small area estimation: An application of aflexible fay-herriot method. Journal of Agriculture Science and Technology, 14,1:75-85.

Multi-State Transition Models with Censoring in VerticalTransmission of HIV

Idah Orowe1 and Joseph A. M. Ottieno11University of Nairobi, Nairobi, Kenya

Abstract

The objective is to derive transition probabilities and transition intensities for multi-state modelsfor a child born infected with HIV and a child born healthy. To obtain the results number ofmathematical tools have been used. Particularly, Kolmogorov forward di↵erential equations forobtaining transition probabilities are derived. Generator matix approach for solving the di↵erentialequations is used. Maximum likelihood method for estimating transition rates is applied. As resultsformula for transition probabilities and transition intensities are obtained for the following statemodels:Two State Model: Has infected and Dead states. It is a left censoring model;Three State Models: For left censoring we have Infected, Aids and Dead states. For right censoringwe have Healthy, Infected and Dead states;Four State Model: For right censoring we have Healthy, Infected, Aids and Dead states;Five State Model: For right censoring we have Healthy, Exclusive Breastfed/ Non Breastfed,Infected, Aids and Dead states.We notice that as the number of states increases, the size of the generator matrix also increasesand hence the more complex it becomes to construct and di↵erentiate the transition probabilities.For further research, multi-state models should include covariates for better estimates of transitionintensities. Apart from the generator matrix approach, other methods for solving the di↵erentialequations could be tried. Methods for graduating probabilities should be applied.

Keywords: Multi-state, Transition probabilities, Intensities, Vertical transmission, Censoring.

Discrete mixtures of order statistics from an exponential dis-tribution

Joseph A. M. Ottieno11University of Nairobi, Nairobi, Kenya

Abstract

ObjectivesThe objective of this work is to construct probability density functions, survival functions and haz-ard functions of discrete mixtures of minimum and maximum order statistics from an exponentialdistribution with parameter � > 0. Thesecond objective is to obtain rth moments of the mixtures.MethodologyThe mixing distributions used are zero-truncated power series; namely, zero-truncated Poisson, bi-nomial, geometric and negative binomial distributions. The logarithmic series mixing distributionis also used.

33

The beta generated approach is used to obtain the distribution of the ith order statistic.The rth moments of the mixtures are obtained by direct integration and by the method of momentsof the mixing distributions.The factorial moments of the mixing distributions are obtained through the derivatives of theprobability generating functions, pgfs.ResultsThe minimum order statistic from an exponential distribution with parameter � is also an expo-nential distribution with parameter n� where � > 0 and n is a positive integer.The pdfs, survival functions and hazard functions of the mixtures are in explicit forms while therth moments are in summation forms.The rth moment of the mixture is in general expressed in terms of the rth moment of the reciprocalof the discrete mixing distribution.The distribution of the maximum order statistic is an exponentiated exponential distribution withparameters � > 0 and n being a positive integer.The pdfs and survival functions have been obtained by direct summation and by using factorialmoment approach. The results are in explicit forms. The rth moments of the mixtures are ex-pressed in terms of summations of factorial moments of the mixing distributions.DiscussionIn this paper, discrete mixtures of order statistics from an exponential distribution has been dis-cussed. This is the case where � > 0 has been fixed and n being a positive integer is varying.The new mixed distributions have not been named.ConclusionFurther work for discrete mixtures of order statistics is in estimation and applications. Some workon continuous mixtures of order statistics from an exponential distribution, where n is fixed and� varies is in progress.

Keywords: Discrete mixtures, Order statistics, Zero-Truncated, Power Series.

Binomial mixtures based on Beta prior distributions andtheir generalizations with application to group screening de-sign

Andrea Otwande1 and Joseph A. M. Ottieno11Jaramogi Oginga Odinga University of Science and Technology, Kisumu, Kenya

Abstract

A mixed distribution can be obtained when two individual distributions are mixed together. Theintegral of the product of the mixture is determined for continuous mixture and for discrete casethe sum of the product is determined. This study focusses on Beta-binomial mixture whose origindates back to the year 1948 when Skellam mixed a binomial distribution with its parameters beingprobability of success taking beta distribution. The binomial distribution is mixed with betadistribution and its generalizations as prior to obtain various mixed distributions.The generalized beta mixing distribution would involve parametrization of beta distribution usingp = ✓

1+✓ . This parametrization would change the domain of the classical beta distribution from[0, 1] to [0,1], thus expanding further the limits of the mixing distribution.Direct approaches of construction, i.e., direct integration methods and methods of moment wouldbe applied and proved to produce the same result.Bayesian statistical inferences are applicable where both the distribution and its parameters areconsidered as random variables. In this case Bayesian inference is applied to the mixture bydetermining the posterior distribution of the constructed mixture.Expressed in recursive forms, the beta-binomial mixtures will be fitted to the recursive modelssuch as Panjer-Willmot (1982) and Hesselager’s (1992) where applicable and their correspondingdi↵erential equations determined.


Beta-binomial mixtures will be applied in group-screening designs where population N is dividedinto g groups of size k and each group is tested for at least one individual being positive with aprobability p varying according to a given distribution.[1]

Keywords: Binomial mixtures, Group screening design.

References

[1] Skellam, J. G. (1948). A probability distribution derived from the binomial distribution byregarding the probability of success as a variable between the set of trials. Journal of Royalstatistical society. Series B, Vol 10:257-261

[2] Alanko, T. and Du↵y, J. C. (1996). Compound Binomial distributions for modeling consump-tion data. Journal of the Royal Statistical society. Series D (The Statistician), Vol 45, No.3:269-286.

[3] Libby, D. I. and Novick, M. R. (1982). Multivariate Generalized beta distributions with appli-cations to utility assessment. Journal of educational statistics. Vol 9:163-179.

The search for ultimate truth in inexact science - Can statis-ticians and clinicians work together?

Daniel Pan1 and Thomas Hine21Academic Cardiology, Castle Hill Hospital, Cottingham, United Kingdom2Academic Emergency Medicine, Hull Royal Infirmary, Hull, United Kingdom

Abstract

”Doctors are men who prescribe medicines of which they know little, to cure diseases of which theyknow less, in human beings of whom they know nothing” - Voltaire

Clinicians and statisticians have more in common than most would think. Both are logical gamblers- recommending rational decisions based on inferences from the maximum amount of data available.Conclusions drawn from studies in high impact journals constantly have an immeasurable impacton populations across the world.

Therefore, until the day a perfect diagnostic test or treatment exists, it is dangerous for doc-tors to be statistically illiterate (since they would not be able to communicate risk properly topatients), and for statisticians to be simply disinterested technicians, providing clinical colleagueswith pointless numerical output. In this talk, the speaker takes the audience through a brief his-tory of the use of statistics in medicine, focusing on the use of good (and bad) statistics and theirconsequences. Case studies of patients seen are used to demonstrate the di↵erence between statis-tical significance and clinical significance. Preliminary data from one of the biggest observationalAcute Heart Failure studies in Europe will also be presented to demonstrate practical obstaclesfaced when statisticians and clinicians work together.

It is easy to make perfect decisions with perfect information - medicine asks the clinician tomake perfect decisions with imperfect information. The statistician is therefore instrumental inhelping doctors make the least biased, most accurate decisions for patients more often.

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On negative (spectral) moments for some functions of Wishartmatrices

Jolanta Pielaszkiewicz11Linnaeus University, Vaxjo, Sweden

Abstract

The goal is to present results on negative moments of spectral distribution for functions of Wishartmatrix including relation between the moment generating function of matrix and its inverse andclosed form expression for R-transform (free cumulant generation function) of Inverse Wishartmatrix. We discuss the usage of property of freeness (free independence), by application of R-transform, in order to obtain expressions on functions of free matrices. The talk will be illustratedwith comparison of theoretical and simulations results.

Keywords: Negative moments, Wishart matrix, Trace, Freeness, R-tranform, Inverted WishartDistribution.

References

[1] Cressie, N., Davis, A. S., Folks, J. L., Policello II, G.E. (1981). The Moment-Generating Func-tion and Negative Integer Moments. The American Statistician, 35:3,148-150.

[2] Pielaszkiewicz, J. (2015). Contributions to High-Dimensional Analysis under Kolmogorov Con-dition. Linkopings Studies in Science and Technology. Dissertation No. 1724.

[3] Pielaszkiewicz, J., von Rosen, D., Singull, M. (2017). On E⇥Qk

i=0 Tr{Wmi}⇤, where W ⇠

Wp(I, n) Communications in Statistics - Theory and Methods, 46, 2990-3005.

[4] von Rosen, D. (1997). On Moments of the Inverted Wishart Distribution. Statistics, 30:3,259-278.

Graphical approach for claims reserving

Mohamed Riad Remita1,2 and Thara Belhamra11Badji Mokhtar University, Annaba, Algeria2Laboratory of Probability and Statistics (LaPS), Badji Mokhtar University, Annaba, Algeria

Abstract

In this paper, we give a geometrical aspect of the standard theory of claim reserving using incre-mental claims. We estimate development factors using graphical technique, for that we need tocalculate the tangents for each development year, then we estimate the lower circles using observedcircles and tangents. This leads to estimations of the yearly and total provisions, [2], [3] and [4].

The results are exactly the same as those calculated by chain-Ladder method and the stochasticincremental approach [1]. Some examples are given to illustrate this method.

Keywords: IBNR, Chain Ladder, Incremental Approach, Graphical Approach, Provision.

References

[1] I. Chorfi, M. R. Remita, Stochastic Incremental Approach for Modelling the Claims Reserves,International Mathematical Forum, 8 (2013), 807-828.

[2] P. England, R. Verral, Geometrically Designed, Stochastic claims reservering in general insur-ance, British Actuarial Journal, 8 (2002), 443-544.


[3] T. Mack, Distribution-free calculation of the standard error of chain-ladder reserve estimates,ASTIN Bulletin, 23 (1993), 213-225.

[4] A.E. Renshaw, R.J. Verrall, A stochastic model underlying the chain-ladder technique, BritishActuar. J., 4 (1998), 903-923.

On the Use of Auxiliary Variables and Models in Estimationin Surveys with Nonresponse

Bernardo Joao Rota1 and Thomas Laitila1,21Eduardo Mondlane University, Maputo, Mozambique2Orebro University, Orebro, Sweden

Abstract

This paper contains a discussion on two alternative weighting procedures, that is, weighting withand without explicitly modeling of the response mechanism, known as the direct weighting and theweighting approaches respectively. The generalized regression estimator benchmarks the weightingmethods, whereas a general double-weighted Horvitz-Thompson estimator represents the direct-weighting approach. A general reliance on the strength of the correlation between the auxiliaryvariables, the response behavior and the study variables prevailing mostly in weighting approachesis shown to be inappropriate in some cases, that is, such reliace increases the bias of the resultingestimator. Conversely, the traditional use of simple models in representing the true responsebehavior is addressed through an example in which it is shown to be adequate only under veryrestrictive assumptions. Results presented for both weighting procedures reveal the need to developnew tools and methodologies when performing estimation in surveys with nonresponse.

Keywords: Auxiliary Variables, Nonresponse, Regression Estimator, Weighting.

References

[1] Bethlehem, J.G. (1988). Reduction of nonresponse bias through regression estimation. Journalof O�cial Statistics, 4:251-260.

[2] Estevao, V.M. and Sarndal, C.-E. (2000). A functional form approach to calibration. Journalof O�cial Statistics, 16:379-399.

[3] Kim, J. K. and Riddles, M. K. (2012). Some theory for propensity-scoreadjustment estimatorsin survey sampling. Survey Methodology, 38:157-165.

[4] Sarndal, C.-E. and Lundstrom, S. (2005).Estimation in Surveys with Nonresponse. Wiley, NewYork.

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A general strong law of large numbers and applications toassociated sequences and to EVT

Harouna Sangare1,2 and Gane Samb Lo1,3

1Universite Gaston Berger, Saint-Louis, Senegal2Universite des Sciences Techniques et Technologies de Bamako (USTTB), Bamako, Mali3Universite Pierre et Marie Curie, Paris, France

Abstract

The purpose of this paper is to establish a general strong law of large numbers (SLLN) for arbitrarysequences of random variables (rv’s) based on the squared indice method and to provide applica-tions to SLLN of associated sequences. This SLLN is compared to those based on the Hajek-Renyitype inequality. Nontrivial examples are given. An interesting issue that is related to extremevalue theory (EVT) is handled here.

Keywords: Positive Dependence, Association, Negatively Associated, Hajek-Renyi Inequality,Strong Law of Large Numbers, Extreme Value Theory, Hill’s Estimator.

Four routes to mixed Poisson distributions

Rachel Sarguta1 and Joseph A. M. Ottieno1

1University of Nairobi, Nairobi, Kenya

Abstract

The objective of this work is to construct mixed Poisson distributions in four ways; namely inexplicit form, in terms of special functions, in recursive forms and in terms of transforms. Bydirect integration Poisson mixtures and their moments are obtained explicitly. A gamma functionand its properties are frequently used in such situations. Moments are also derived by conditionalexpectation approach. Confluent hypergeometric function and modified Bessel function of the thirdkind are the special functions used in expressing Poisson mixtures. Moreover integration by partstechnique is used for recursive forms. Laplace and Mellin transforms are the expectation formsused to construct Poisson mixtures while the probability generating function, another transform,has been used to obtain moments.

Inverse Gamma, beta and transmuted exponential mixing distributions lead to Poisson mix-tures in terms of modified Bessel function of the third kind, confluent hypergeometric functionand in explicit form respectively. The first and the third mixing distributions are also for thetransforms. Moments of a mixed Poisson distribution are expressed in terms of moments of themixing distribution through explicit and transform routes. Di↵erential equations in probabilitygenerating functions have been derived for recursive models.

We stress that only a few mixed Poisson distributions can be expressed explicitly. One way ofcircumventing this problem is to use recursive models. An advantage of using integration by partsover the other methods for deriving recursive models is that it does not need any condition to besatisfied like in other recursive models. One bottle-neck of the transform approach is, in manycases, to di↵erentiate Laplace transform many times.

We refer to literature where the other mixing distributions have been used for the four routesand identities based on Poisson mixtures have been derived. We also mention proven link betweenPoisson and exponential mixtures through Laplace transform.

Keywords: Explicit, Recursive, Transform, Special Function Routes, Moments


Ties in One Block Comparison Experiments: A Generaliza-tion of the Mallows-Bradley-Terry Ranking Model

Amadou Sawadogo1 and Simplice Dossou-Gbete21University of Felix Houphouet-Boigny, Abidjan, Ivory Coast2University of Pau et des Pays de l’Adour, Pau, France

Abstract

This study is concerned with the extension of the Mallows-Bradley-Terry [1] ranking model for oneblock comparison consisting of all the items of interest to situations which allow an expression of nopreference. We consider a modification of the Mallows-Bradley-Terry ranking model by introducingan additional parameter, called an index of discrimination [2], in the model. This permits ties inthe model. The maximum likelihood estimates of the parameters are found using a Maximization-Minimization algorithm [3]: the evaluation of the mathematical expectations involved in the log-likelihood equation is obtained by generating samples of Monte Carlo Markov chain [4] from thestationary distribution. In addition, a simulation study for asymptotic properties assessment hasbeen made. The proposed method is applied to analyze data election.

Keywords: Babington Smith model, Mallows-Bradley-Terry model, rank data, maximum likeli-hood estimation, MM algorithm, MCMC.

References

[1] Critchlow, E.D., and Fligner, A. M. (1991). Paired comparison, triple comparison, and rankingexperiments as generalized linear models, and their implementation on GLIM., Psychometrika,3:517–533.

[2] Davidson, R.R. (1970). On extending the Bradley-Terry Model to accommodate ties in pairedcomparison experiments. Journal of the American Statistical Association, 329:317–328.

[3] Hunter, D.R.(2004). MM algorithms for generalized Bradley-Terry Models. The Annals ofStatistics, 1:384-406.

[4] Green, P.J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesianmodel determination. Biometrika, 4:711–732.

Uniform in Bandwidth Consistency for Transformation Ker-nel Estimators of Copula

Cheikh Tidiane Seck1 and Gane Samb Lo1,21Universite Alioune Diop, Bambey, Senegal2Universte Gaston Berger, Saint-Louis, Senegal

Abstract

In this paper we establish uniform in bandwidth consistency for the transformation kernel estimatorof the copula function introduced in [2]. We follow the approach developed in [1] which employsgeneral empirical process methods to establish uniform in bandwidth consistency of a wide classof kernel-type function estimators including those for densities and distribution functions. We willprove a uniform in bandwidth strong law of the iterated logarithm for the maximal deviation ofthis estimator from its expectation. We then show, as the sample size n goes to infinity, thatthe bias of the estimator converges to zero, uniformly in the bandwidth h varying over a suitableinterval depending on n. A practical method of selecting the optimal bandwidth based on a cross-validation criterion is presented. Finally, we provide conclusive simulation experiments showingthe performance in finite samples of the estimator constructed with this optimal bandwidth.

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Keywords: Copula function, Nonparametric estimation, Transformation kernel estimator, Uni-form in bandwidth consistency.

References

[1] Mason, D. M. and Swanepoel, J. H. W. (2010). A general result on the uniform in bandwidthconsistency of kernel-type function estimators. Sociedadde Estadistica e Investigation Operativa2010., DOI 10.1007/S11749-010-0188-0

[2] Omelka, M., Gijbels, I. and Veraverbeke, N. (2009). Improved kernel estimators of copulas:weak convergence and goodness-of-fit testing. The Annals of Statistics., vol. 37, 5B:3023-3058

Hierarchical Bayesian Inversion Model for Daily MaximumTemperature Variability

Abdu Mohammed Seid1, Tesfahun Berhanie1 and Lassi Roininen21Bahir Dar University, Bahir Dar, Ethiopia2Imperial College London, London, United Kingdom

Abstract

In this paper, we have modeled the trend, seasonal and other fluctuation e↵ects of daily maximumair temperature using data from Bahir Dar Meteorological station, Ethiopia 2010-2014. The studyaims to understand the temporal behavior of air temperature through modeling and forecastingone-step ahead. Bayesian hierarchical model with Gaussian Markov random field prior in a MarkovChain Monte Carlo (MCMC) framework is used for the study. The uncertainty of the predictionis reduced by incorporating temporal correlations and seasonal variance in the model. The studyreveals that the proposed model describes well the dynamics of daily maximum air temperaturewith 95% posterior interval. The comparison of estimated posterior mean of the model with thecalendar day data sample mean shows the model output to be a perfect fit with the model creatinga smooth process through the mean samples. The one-step ahead forecast is validated with actualmeasurement data from same year that is not part of the training set. In the prediction, the actualtemperature values of the year that is used for validation lies within the 95% posterior interval.The auto-correlation and analysis of the residuals indicates that the standardized residuals arenormally distributed and white noises.

Keywords: Bayesian Inference, Hyper-prior, MCMC, Gaussian Process.

Sub-D facing the worst ”one-way” designs scenarios

Adilson Silva1,2, Miguel Fonseca1 and Martin Singull3

1University of Cape Verde, Praia, Cape Verde2New University of Lisbon, Lisbon, Portugal3Linkoping university, Linkoping, Sweden

Abstract

Sub-D is an estimator for variance components in mixed linear models developed and named bySilva [1]; the name is due to the use of the concept of subdiagonalization of a matrix also proposedby the authors.

In any estimation process, the worst scenarios faced by any estimator occur when the informa-tions are almost entirely lost, requesting, oftentimes, the data collection done again, elevating sothe cost for the ongoing research.


This work aims to deduce the Sub-D’s mean square error (MSE) for the “one-way” designs andshow through simulations that it can face the worst “one-way” designs scenarios and yet producingbetter estimates then those based on ANOVA and maximum likelihood (the most frequent ones).The MSE for the di↵erent scenarios simulated as well as the comparison tables will be presentedand discussed.

Keywords: Sub-D, MSE, “One-way” designs, Lost of information.

References

[1] Silva, A. (2017). Variance Components Estimation in Mixed Linear Models. PhD Thesis.

More on estimation of the mean matrix in a Growth Curvemodel in high dimensions

Martin Singull11Linkoping university, Linkoping, Sweden

Abstract

We consider the problem of estimating the mean matrix in a general multivariate linear model,the so-called Growth Curve model, when the p ⇥ N observation matrix is normally distributed.The maximum likelihood estimator (MLE) for the mean is a weighted estimator with the inverseof the sample covariance matrix which is unstable for large p close to N and singular for p largerthan N . One can modify the MLE to an unweighted estimator and it can be shown that theunweighted estimator is in fact the MLE under certain conditions and examples when this occurswill be discussed.

When these conditions are not fulfilled it is not obvious which estimator that should be chosen.We will present a statistical test for choosing estimator given that the eigenvectors of the covariancematrix are known. This is, for example the case, when the covariance matrix is a circular Toeplitzor the special case intraclass covariance structure.

Keywords: GMANOVA, Growth Curve model, high dimension, circular Toeplitz

References

[1] Srivastava, M. S. and Singull, M. (2017). Test for the mean matrix in a Growth Curve modelfor high dimensions. Communications in Statistics - Theory and Methods 46(13):6668-6683.

External Economic Shock and Food Price Volatility in Rwanda:Evidence from ARCH and GARCH models

Sandrine Unezeza1 and Jean Baptiste Habyarimana11University of Rwanda, Kigali, Rwanda

Abstract

In our paper we adopt and explain autoregressive conditional heteroscedasticity (ARCH) model[1] and generalized autoregressive conditional heteroscedasticity (GARCH) model [2] models toinvestigate how variations in the price of crude oil in Rwanda and at world market, as externaleconomic shock, a↵ect the price of cereals at domestics market in Rwanda. Empirical resultsemanating from our paper show that all selected series, cereal price series and crude oil priceseries, are linearly related and they su↵er from a high relative variability. Therefore, price returns

41

analysis reveal that there is evidence of volatility in cereal price returns clustering from variations inthe price of crude oil at both domestic Rwandan market and world market. ARCH and GARCHresults demonstrate that volatility in the market price of cereals at Rwanda market is stronglyinfluenced by own past squared residuals and by past squared residuals in the price of crude oil atboth domestic market and world market. Furthermore, scedastic functions reveal that the upwardmovements in the price of crude oil in Rwanda had led to the period of high price in cereals (maizeand wheat). To reduce ine�ciencies in domestic cereal distribution system while improving cerealmarket e�ciency and ensuring food security, this paper suggests that trade policy should take intoconsideration the relationship between the price of cereals and the price of crude oil. This paperalso suggests that, as there exists a negative relationship between the devaluation of Rwandancurrency and the price of crude oil in Rwanda, stabilization of Rwandan Francs would lead tocrude oil price stability as a key factor to improve the e�ciency of cereal distribution system inthe country.

Keywords: External economic shock, Food price volatility, ARCH and GARCH models.

References

[1] Engle, R.F. (1982). Autoregressive Conditional Heteroscedasticity with Estimates of the Vari-ance of United Kingdom Inflation. Econometrica, 50: 987-1007.

[2] Bollerslev, T. (1986). Generalised Autoregressive Conditional Heteroskedasticity. Journal ofEconometrics, 31: 307-327.

Multivariate Analysis of Rwanda Economic Indicators usingVector Autoregressive (VAR) Model

Denise Uwamariya1 and Emelyn Umunoza Gasana11University of Rwanda, Kigali, Rwanda

Abstract

Rwanda’s economy has been growing fast due to important economic and structural reforms overthe last decade. Consumer Price Index (CPI), Exchange Rate and Nominal Growth DomesticProduct (NGDP) constitute some of the major economic indicators in emerging market economiesthat require monetary authorities to elaborate tools and policies to prevent high volatility inprices. Thus, understanding CPI, exchange rate and NGDP dynamics is a key to the design offund programs to help stabilize the economy of a developing country such as Rwanda. In thisstudy, secondary data from the National Bank of Rwanda, depicting quarterly time series of the 3indicators from 1997Q1 to 2014Q4 has been used. Appropriate Vector Autoregressive (VAR) modelwith maximum 3 lagged values for the underlying variables was selected, based on the smallestvalue of Bayesian Schwartz information criterion and diagnostic tests for disturbances performed.The Granger causality and Impulse response function analysis confirmes that in the fitted VARmodel, CPI, exchange rate and NGDP are endogeneous variables, each one related to its laggedvalues and/or of the lag values of other variables. In addition, the results suggest that the CPI isnot directly related to the exchange but to the NGDP.

Keywords: VAR model, Exchange rate, Nominal GDP, CPI.


Modeling and forecasting maize production in Rwanda withMarkov Chain Monte Carlo methods (MCMC)

Denise Uwamariya1 and Denis Ndanguza11University of Rwanda, Kigali, Rwanda

Abstract

Rwanda is the country whose economy relies on agriculture. Therefore, forecast in agriculturesector is very important in Rwanda for future plan. In our study, secondary annual data fromMINAGRI1, spanning from 1960 to 2014 have been used. In the analysis, appropriate model isselected based on the appearance of ACF and PACF of the transformed data. In addition to that,we use the fitted model to provide a four year forecasts of maize production from 2015 to 2018.Through Box-Jenkins methodology, the appropriate model is ARIMA (1,2,1) and fit the data at91%. From the results and forecast, it is seen that the production of maize in Rwanda will have anincreasing trend in the future. To strengthen the model, we also use the MCMC algorithm as analternative methods in parameters estimation. Diagnostics prove the chains’ convergence which isthe sign of an accurate model.

Keywords: Maize, Model, Box-Jenkins, Forecast, MCMC.

Testing bilinear hypothesis in bilinear models

Dietrich von Rosen11Swedish University of Agricultural Sciences, Uppsala, Sweden

Abstract

Let WH0 and WH1 be two independently distributed Wishart matrices which build up Wilks ⇤,i.e.

⇤ =|WH0 +WH1 |

|WH1 |.

The matrices appear when testing H0: BG = 0 versus H1: B unrestricted in a MANOVA model,i.e.

X = BC+E,

X is a random matrix which represents the observations, C and G are known matrices, andE ⇠ Np,n(0,⌃, I), where B and ⌃ are unknown parameter matrices. The distribution of ⇤ equalsa product of independent beta-distributed variables. When approximating the distribution severalapproaches are available, where the most commonly applied uses approximations of the gamma-function.

Let the GMANOVA model be given by

X = ABC+E,

where in addition to the MANOVA model a known matrix A has been introduced.Remarkable is an old classical result which states that the likelihood ratio test for testing in

a GMANOVA model H0: FBG = 0, where F and G are known, versus H1: B unrestricted alsofollows a Wilks ⇤ distribution.

It is remarkable since the maximum likelihood estimators in the MANOVA and GMANOVAare very di↵erent. The talk will derive the distribution in a somewhat di↵erent way than whatusually is applied which also sheds some light on some conditional arguments.

Keywords: Growth Curve model, Likelihood ratio tests.

43

Estimation in Generalized Reduced Rank Regression Models

Tatjana von Rosen1 and Dietrich von Rosen2,31Stockholm University, Stockholm, Sweden2Swedish University of Agricultural Sciences, Uppsala, Sweden3Linkoping university, Linkoping, Sweden

Abstract

Nowadays, the data collected in empirical studies have relatively complex structures though oftendemanding a parsimonious modelling. This can be achieved for example through imposing rankconstraints on the regression coe�cient matrices. The reduced rank regression structure alsoprovides a theoretical interpretation in terms of latent variables. In this work, we consider theGrowth Curve model with rank restrictions on the matrix of regression coe�cients as well as onthe covariance matrix. The likelihood based estimators for the mean parameters and covariancematrix in this type of models will be derived.

Keywords: Growth Curve model, Reduced rank covariance matrix.

References

[1] Kollo, T., von Rosen, D. (2005). Advanced multivariate statistics with matrices. Dordrecht:Springer.

[2] Pottho↵, R.F. and Roy, S.N. (1964). A generalized multivariate analysis of variance modeluseful especially for growth curve problems. Biometrika, 51:313-326.

[3] Srivastava, M.S., and von Rosen, D. (2002). Regression models with unknown singular covari-ance matrix. Linear algebra and its applications, 354:255-273.

A link between Poisson and exponential mixtures throughLaplace transform

Moses Wamalwa Wakoli1 and Joseph A. M. Ottieno21Technical University of Kenya, Nairobi, Kenya2University of Nairobi, Nairobi, Kenya

Abstract

Specific objectives of this work are to show that:

1. a mixed Poisson distribution expressed in terms of a Laplace transform can also be expressedin terms of a hazard function of an exponential mixture;

2. A compound Poisson distribution can be expressed in terms of a hazard function of anexponential mixture;

3. A convolution of mixed Poisson distribution can be expressed in terms of a sum of hazardfunctions of exponential mixtures;

To accomplish that task the Laplace transform and probability generating function are used. Haz-ard functions and survival functions are the functions of survival time used. Infinite divisibility isalso a property of importance.The number of mentioned below results are presented.

• The probability generating function, the mean and the variance of an infinitely divisiblemixed Poisson distribution are expressed in terms of a hazard function of an exponentialmixture.


• A compound Poisson distribution which is an infinitely divisible mixed Poisson distributionis expressed in recursive form and in terms of the probability mass function of the iid randomvariables and a cumulative hazard function of the exponential mixture.

• The pgf of a mixed Poisson distribution expressed in terms of a sum of two hazard functionsof exponential mixtures is a product of two pgfs expressed in terms of two individual hazardfunctions of exponential mixtures.

• The pgfs are infinitely divisible. The recursive form of the convolution of compound Poissondistributions is in terms of the sum of hazard functions.

In our talk the Hofmann distributions form a class of mixed Poisson distributions defined in termsof a class of hazard functions of exponential mixtures are discussed. Hazard functions which aresums of two hazard functions are also discussed. We conclude that there is need to identify otherclasses of hazard functions and to identify exponential mixtures whose hazard functions are sumsof hazard functions.

Keywords: Laplace transform, Pgf, Infinite divisibility, Hazard function.

Simplified Optimal and Relay Linear Unbiased Estimation ofParameters of Logistic Distribution

Patrick G. O. Weke1 and Idah A. Orowe11University of Nairobi, Nairobi, Kenya

Abstract

We present some simplified methods for estimating parameters of logistic distributions. The logisticfunction is one of the oldest growth functions and has been found useful in a variety of applications.Initially discovered and used as a model for growth of human population and organismic growth, ithas since been used in studies of physiochemical phenomenon, geological studies and psychologicalstudies. The logistic function has also been used in bioassay, medical diagnosis and public health,among others.

A simplified linear estimation method that is based on selected order statistics whose ranks aredetermined by pairs of spacings is developed. These spacings are values within the interval [0, 1]such that the inverse of the cumulative distribution function at large samples quantities is equalto the value of spacings. Simplified optimal linear estimators based on various asymptotically bestoptimal spacings and their corresponding real-valued coe�cients and selected order statistics arepresented.

The so-called relay linear unbiased estimation (abbreviated as RLUE ) method for the estimationof the scale parameter of the logistic distribution is considered. This method is based on spacingsthat are in the form of a reducing geometric series with a constant factor appropriately chosen.The spacing of RLUE have the form as ci :2�(i+1);i=1(1)m. These spacings are used to cover aprescribed sequence of ranks of order statistics given by 1.5 ⇥ 2m n < 3 ⇥ 2m � 1,m = 2(1)5and their respective optimal coe�cients are determined by applying optimization techniques. TheRLUE method is di↵erent from other traditional linear estimation methods whose spacings aredetermined by optimization methods. These methods are applied to both complete and Type-IIcensored samples and the results compared with those of asymptotic best linear estimation andbest linear unbiased estimation.

The expectation of the sum of consecutive AL-variables formula has been used widely in the struc-ture of above estimators, and relation for the determination of ranks of order statistics by spacingis i = [nc+ 0.5] instead of i = [nc] + 1. In addition to bias and e�ciency, the methods developedhere have the advantage of producing reasonable good results, quick and easy computations andfinally there is no reference made for pre-existing tables for the computation of the estimator. Loses

45

in the e�ciencies of the estimators are very small and they are compromised with the convenienceof using these simplified estimation methods.

Keywords: Order Statistics, Logistic Distribution, Expectation of the Sum of Consecutive AL-Variables, Simplified Linear Estimation, Relay Linear Unbiased Estimation.

Stochastic Claims Reserving in Short-Term Insurance Con-tracts

Patrick G. O. Weke1 and Idah Orowe11University of Nairobi, Nairobi, Kenya

Abstract

Claims reserving for general insurance business has developed significantly over the recent past.There has always been a slight mystery in short-term insurance contracts of how to go aboutreserving for claims, which have not yet come in, and are still in some sense of figment of thefuture. Stochastic models for triangular data are derived and applied to claims reserving data.The standard actuarial technique is given a sound statistical foundation and considered as a linearmodel. The chain ladder technique and two-way analysis of variance are employed for purposes ofestimating and predicting the IBNR claims reserves. Insurance claims variables are non-normallydistributed and therefore a measure that will capture the dependence among the variable betterthan the usual correlation is employed. One such method is the use of copulas.

List of Participants

Invited Speakers

1. Narayanaswamy BalakrishnanMcMaster University, Hamilton, [email protected]

2. Youssef OuknineCadi Ayyad University, [email protected]

3. Jianxin PanUniversity of Manchester, [email protected]

4. Simo PuntanenUniversity of Tampere, Tampere, [email protected]

Contributed Speakers

5. Benson Ade AfereFederal Polytechnic, Idah, [email protected]

6. Godwin Norense Osarumwense AsemotaUniversity of Rwanda, Kigali, Rwanda

7. Richard AwichiBusitema University, Tororo, [email protected]

8. Kidanemariam Alem BerhieUniversity of Gondar, Gondar, [email protected]

9. Mahamat Ali IssakaUniversite de N’Djamena, N’Djamena, ChadUniversite Gaston Berger, Saint-louis, [email protected]

10. Denis Kagyera KatakaraMbarara University of Science and Technology, Mbarara, [email protected]

11. Timothy Kevin Kuria KamanuUniversity of Nairobi, Nairobi, [email protected]

47

48 LIST OF PARTICIPANTS

12. Rodnellin Onesime MalouataMarien Ngouabi University, Brazzaville, Republic of [email protected]

13. Isaac MugumeMakerere University, Kampala, [email protected]

14. Stanislas MuhinyuzaStockholm University, Stockholm, SwedenUniversity of Rwanda, Kigali, [email protected]

15. Jean-Paul MuraraMalardalen University, Vasteras, SwedenUniversity of Rwanda, Kigali, [email protected]

16. Nicholas Mwilu MutothyaTaita Taveta University, Voi, [email protected]

17. Joseph Ivivi MwanikiUniversity of Nairobi, Nairobi, [email protected]

18. Lukman Abiodun NafiuIslamic University in Uganda, Mbale, [email protected], [email protected]

19. John M. NdirituUniversity of Nairobi, Nairobi, [email protected]

20. Innocent NgaruyeLinkoping university, Linkoping, SwedenUniversity of Rwanda, Kigali, [email protected]

21. Gothatamang Patrick NthoiwaBotswana College of Agriculture, Gaborone, [email protected]

22. Davis Bundi NtwigaUniversity of Nairobi, Nairobi, Kenya

23. Joseph NzabanitaUniversity of Rwanda, Kigali, [email protected]

24. Everlyne OderoMasinde Muliro University of Science and Technology, Kakamega, [email protected]

25. Rose Auma OdhiamboUniversity of Nairobi, Nairobi, [email protected]

26. Nelson OnyangoUniversity of Nairobi, Nairobi, [email protected]

CONTRIBUTED SPEAKERS 49

27. Idah OroweUniversity of Nairobi, Nairobi, [email protected]

28. Joseph A. M. OttienoUniversity of Nairobi, Nairobi, [email protected]

29. Andrea OtwandeJaramogi Oginga Odinga University of Science and Technology, Kisumu, [email protected]

30. Daniel PanAcademic Cardiology, Castle Hill Hospital, Cottingham, United Kingdom

31. Jolanta PielaszkiewiczLinnaeus University, Vaxjo, [email protected]

32. Mohamed Riad RemitaBadji Mokhtar University, Annaba, AlgeriaLaboratory of Probability and Statistics (LaPS), Badji Mokhtar University, Annaba, [email protected]

33. Bernardo Joao RotaEduardo Mondlane University, Maputo, [email protected]

34. Harouna SangareUniversite Gaston Berger, Saint-Louis, SenegalUniversite des Sciences Techniques et Technologies de Bamako (USTTB), Bamako, [email protected]

35. Rachel SargutaUniversity of Nairobi, Nairobi, [email protected]

36. Amadou SawadogoUniversity of Felix Houphouet-Boigny, Abidjan, Ivory [email protected]

37. Cheikh Tidiane SeckUniversite Alioune Diop, Bambey, [email protected]

38. Abdu Mohammed SeidBahir Dar University, Bahir Dar, [email protected]

39. Adison SilvaUniversity of Cape Verde, Praia, Cape VerdeNew University of Lisbon, Lisbon, Portugal

40. Martin SingullLinkoping university, Linkoping, [email protected]

41. Sandrine UnezezaUniversity of Rwanda, Kigali, Rwanda

42. Denise UwamariyaUniversity of Rwanda, Kigali, Rwanda

43. Dietrich von RosenSwedish University of Agricultural Sciences, Uppsala, SwedenLinkoping university, Linkoping, [email protected]

44. Tatjana von RosenStockholm University, Stockholm, [email protected]

45. Moses Wamalwa WakoliTechnical University of Kenya, Nairobi, [email protected]

46. Patrick G. O. WekeUniversity of Nairobi, Nairobi, [email protected]

Index

Afere, B. A., 17Alih, E., 17Asemota, G. N. O., 18Awichi, R. O., 18

Balakrishnan, N., 15Bale, E., 28Bamutaze, Y., 23Barhie, K. A., 19Basalirwa, C., 23Belhamra, T., 35Berhanie, T., 39Bodnar, T., 24

Chirove, F., 20

Dabye, A. S., 20Dossou-Gbete, S., 38

Fonseca, M., 39

Gueye, L., 20

Habyarimana, J. B., 40Hamidu, U. W., 26Hline, T., 34

Issaka, M. A., 20

Kagyera, D. K., 20Kamanu, T. K. K., 21

Laitila, T., 36Lindholm, M., 24Lo, G. S., 37, 38

Malouata, R. O., 22Malyarenko, A., 24Markiewicz, A., 16Mesquita, M. D. S., 23Mokgolele, M., 28Mugume, I., 23Muhinyuza, S., 24Murara, J.-P., 24Mutothya, N. M., 25Mwalilim, S., 31Mwaniki, J. I., 26

Nafiu, L. A., 26

Ndanguza, D., 42Ndiritu, J. M., 27Ngailo, T. J., 23Ngaruye, I., 27Ni, Y., 24Nthoiwa, G. P., 28Ntwali, D., 23Ntwiga, D. B., 29Nzabanita, J., 29

Odero, E. A., 30Odhiambo, R., 30Ogwang, B. A., 23Onyango, F., 30Onyango, N., 31Orowe, I., 32, 44, 45Ottieno, J. A. M., 25, 30, 32, 33, 37, 43Otwande, A., 33Ouknine, Y., 15

Pan, D., 34Pan, J., 15Pielaszkiewicz, J., 35Puntanen, S., 16

Remita, M. R., 35Reuder, J., 23Roininen, L., 39Rota, B. J., 36

Sangare, H., 37Sarguta, R. J., 37Sawadogo, A., 38Seck, C. T., 38Seid, M. A., 39Silva, A., 39Silvestrov, S., 24Singull, M., 27, 39, 40

Tumwine, F., 23Twinomuhangi, R., 23

Umunoza Gasana, E., 41Unezeza, U., 40Uwamariya, D., 41, 42

von Rosen, D., 27, 42, 43von Rosen, T., 43

51

52 INDEX

Waiswa, D., 23Wamalwa, M. W., 43Weke, P. G. O., 44, 45

Zwikiti, M., 28

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