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International Journal of Statistical Sciences

Volume 15, December 2015

Contents

D-Optimal Designs and Model Uncertainty in Mixture

Experiments-Nripes Kumar Mandal, Manisha Pal and Bikas Kumar

Sinha

1

An Efficient Product-type Exponential Estimator of Finite

Population Mean using Variable Transformation-K. B. Panda and

N. Sahoo

13

Optimisation Perspective in Managing Gender Centric

Prophesied Longevity-P. K. Tripathy, Sujata Sukla and Priyanka

Tripathy

21

Decision Support in a Credit Environment with Fuzzy Behaviour

of Cost-P. K. Tripathy and Sujata Sukla

37

Simulated Tests for Normality: A Comparative Study-Mezbahur

Rahman and Thevaraja Mayooran

55

Instructions for Authors 65-66

Panel of Reviewers in the Current Issue 67-68

Call for Papers 69

International Journal of Statistical Sciences

Editor-in-ChiefDr. Shahariar Huda, Professor, Dept. of Statistics and Operations ResearchFaculty of Science, Kuwait University, P.O. Box-5969, Safat-13060, Kuwait

Executive EditorDr. S. K. Bhattacharjee, Professor, Dept. of Statistics, University of Rajshahi, Rajshahi-6205, Bangladesh

Associate EditorsDr. Md. Golam Hossain, Professor, Dept. of Statistics, University of Rajshahi, Rajshahi-6205, BangladeshDr. Md. Sabiruzzaman, Dept. of Statistics, University of Rajshahi, Rajshahi-6205, BangladeshDr. Md. Mostafizur Rahman, Dept. of Statistics, University of Rajshahi, Rajshahi-6205, Bangladesh

SecretaryDr. Md. Nasim Mahmud, Dept. of Statistics, University of Rajshahi, Rajshahi-6205, Bangladesh

Editorial BoardAnwar H. Joarder, Dr., Prof.,School of BusinessUniversity of LiberalsArts Bangladesh, Dhanmondi,Dhaka 1209, Bangladesh

Beg, A.B.M. Rabiul Alam, Dr.,School of BusinessJames Cook UniversityTownsville, QLD 4811, Australia

Biswas, Atanu, Dr., Prof.,Applied Statistics UnitIndian Statistical InstituteKolkata-700108, India

Biswas, S., Dr., Senior Prof.,School of Insurance and ActuarialScience (ASIAS), Amity UniversityNoida-201303 (U.P.), India

El-Shaarawi, A.H., Dr., Prof.,Dept. of StatisticsMcMaster UniversityHamilton, Ontario, Canada

Imon, A.H.M.R., Dr., Prof.,Dept. of Mathematical SciencesBall State UniversityMuncie, IN 47306, USA

Islam, M. Nurul., Dr., Prof.,Dept. of StatisticsUniversity of RajshahiRajshahi-6205, Bangladesh

Karim, M. Rezaul, Dr., Prof.,Dept. of StatisticsUniversity of RajshahiRajshahi-6205, Bangladesh

Khan, S., Dr., Assoc. Prof.,Dept. of Math. & ComputingUniversity of Southern QueenslandQld 4350, Australia

Kibria, B.M.G., Dr., Prof.,Dept. of StatisticsFlorida International UniversityFL 33199, USA

Latif, Mahbub, Dr., Prof.,ISRT, University of DhakaDhaka-1000, Bangladesh

Rahman, M., Dr., Prof.,Dept. of StatisticsUniversity of JahangirnagarSavar, Dhaka-1342, Bangladesh

Rahman, M., Dr., Prof.,Dept. of StatisticsMinnesota State UniversityMankato, MN 56001, USA

Roy, M.K., Dr., Prof.,Pro-Vice ChancellorRanada Prasad Shaha UniversityNrayangonj, Bangladesh

Sen, K.P., Dr., Prof.,Dept. of StatisticsUniversity of DhakaDhaka-1000, Bangladesh

Shah, M.A., Dr., Prof.,Dept. of StatisticsUniversity of RajshahiRajshahi-6205, Bangladesh

Sinha, B.K., Dr., Prof.,Retired FacultyIndian Statistical InstituteKolkata-700108, India

Suzuki, K., Dr., Prof.,Dept. of Systems Eng.The University ofElectro-CommunicationsChofu, Tokyo, 182-8585, Japan

Verma, Med Ram, Dr.,Senior Scientist (Agri. Stat.)ICAR- Indian VeterinaryResearch Institute (IVRI)Izatnagar, BareillyUttar Pradesh, India 243 122

International Journal of Statistical Sciences ISSN 1683–5603

Vol. 15, 2015

c⃝ 2015 Dept. of Statistics, Univ. of Rajshahi, Bangladesh

Editorial note

It is a pleasure to see another issue of International Journal of Statistical Sciences(IJSS) go to the press. I would like to congratulate and thank all those who wereinvolved in the editorial and refereeing process. The labor put in is usually unpaid,voluntary and often not publicly appreciated.

This issue of IJSS consists of five articles. One of these is on Design of Experiments,one on Sampling, one on Applied Mathematical Demography, one on Applied De-cision Theory and one on Statistical Inference. There is quite a bit of diversity inthe areas covered and the articles are of reasonably good standard. Ideally, thereshould be more articles in an issue covering many more major areas of Statisticsand its applications. Bangladeshi statisticians, particularly those with connectionsto Rajshahi University, should be encouraged to publish in IJSS. The expatriateBangladeshi statistical community also should be tapped for getting more submis-sions of acceptable quality. The best way to encourage more quality submissionwould be to get the journal indexed in the major data bases. For example, inclusionin the ”Science Citation Index (SCI) Expanded” would give a tremendous boost tothe prestige of the journal. That should be an objective that we should strive for.

Finally, I thank the authors of the articles included in this issue and hope to seemore submissions from them in the future.

Prof. Shahariar HudaEditor-in-Chief, IJSS

International Journal

of

Statistical Sciences

Publisher:Department of StatisticsUniversity of Rajshahi

Rajshahi – 6205Bangladesh.

Published:Volume 15, December, 2015.

Copyright c⃝ 2015 by the Dept. of Statistics, Univ. of Rajshahi.All rights reserved.

Complementary Price:Individual: Tk. 500.00 (US $100.00)

Organizations: Tk. 1000.00 (US $200.00)

All communication should be addressed to the Executive Editor, IJSSDepartment of Statistics, University of Rajshahi, Rajshahi – 6205

Bangladesh. Web Site: http://www.statru.org/ijss/E-mail: ijss [email protected]: 88-0721-711122Fax: 88-0721-750194


Vol. 15, 2015, pp 1-12


D-Optimal Designs and Model Uncertainty in MixtureExperiments

Nripes Kumar MandalRetd. Professor

University of CalcuttaKolkata, India

Manisha PalDepartment of StatisticsUniversity of Calcutta

Kolkata, India

Bikas Kumar Sinha1

Retd. ProfessorIndian Statistical Institute

Kolkata, India

[Received March 17, 2016; Revised December 10, 2016; Accepted June 15, 2017;Published December 30, 2017]

Abstract

The problem of determining the optimal design to examine the degree of thepolynomial that well approximates the response function has been studiedby many authors. Stigler (1971) suggested optimal designs for polynomialregressions that enable efficient inferences to be made about the fitted modeland, at the same time, check its adequacy in defining the mean response. Inthis paper we consider a mixture experiment and investigate the optimaldesign that can estimate the parameters of a first degree mixture modeland also check the model adequacy as against the presence of one or moreinteraction terms.

Keywords and Phrases: Mixture experiment, model adequacy, D-optimality criterion, optimal design, power constraint.

AMS Classification: 62K05.

1Communicating author; e-mail: [email protected]

2 International Journal of Statistical Sciences, Vol. 15, 2015

1 Introduction

A standard design problem is concerned with choosing the values of the independentvariables so as to come up with the best experiment. To define a “best design” oran ‘optimum design’, several optimality criteria have been proposed and discussed bymany authors in the past several decades, (see, for example, Kiefer (1959, 1961, 1974),Karlin and Studden (1966), Atwood (1969)). However, these criteria have a seriousdrawback, namely they are model dependent. As such, if the assumed model is inad-equate in approximating the response function, they fail to detect this departure, nomatter how large the sample size is. One attempt to address this drawback has beenmade by Box and Draper (1959, 1963), who suggested that a design should minimizeV +B, where V is the error due to sampling variation and B is the error due to inade-quate modeling, and at the same time maximize the power of a goodness-of-fit test forsome class of alternative models. The main weakness of the approach is that it dependson the alternative model whose parameters are unknown. A further suggestion by Boxand Draper to overcome this difficulty was to minimize B alone. However, as Stigler(1971) observed, “minimum bias designs, while they are an important attempt to meetrealistically the problem of checking the representational adequacy of the model, mayoften be inappropriate, inefficient, or both”. This observation reduces the appeal ofthe approach of Karson et al. (1969), who proposed to choose the estimator by mini-mizing B, rather than by minimizing V.

Stigler (1971) suggested two optimality criteria, which “can be considered as compro-mises between the incompatible goals of inference about the regression function underan assumed model and of checking the model’s adequacy” (quote Stigler, 1971). Hecalled them restricted D- and G-optimality criteria and studied them for the problemof estimating a regression function which can be well-approximated by a polynomial.He imposed the condition that the independent variable lies in the interval [-1, 1],and illustrated the use of the criteria by assuming a first degree polynomial modelas against a second degree polynomial. His study yielded optimum designs whichwere superior to designs proposed by others, including the minimum bias designs. Analgorithm for constructing these designs was suggested by Mikulecka (1983). Usinga technique involving canonical moments, Studden (1982) investigated the problemof design construction under a generalization of Stigler’s criterion. Later, Lee (1987,1988) introduced several constrained optimality criteria and provided necessary andsufficient conditions for a design to be optimal.

In this paper, we adopt the approach of Stigler (1975) to suggest optimum designsin mixture experiments that would permit efficient inferences to be made about theassumed model while still allowing the model to be checked for adequacy. We assumea first degree mixture model and attempt to design an experiment which allows tocheck the competence of the model as against a quadratic mixture model, and alsotests hypotheses like βij = 0, with some specified degree of precision. The paper is

Mandal, Pal and Sinha: Uncertainty in Mixture Experiments 3

organized as follows. Section 2 discusses the problem and its perspectives. Optimaldesigns are studied in Section 3. Optimum designs in some specific situations arecomputed in Section 4. Finally, in Section 5, concluding remarks on the study aremade.

2 The problem and its perspectives

For the sake of completeness, we start with model adopted by Stigler (1971). Heconsidered the standard univariate regression set-up, where an experiment is conductedwith n fixed values of the independent variable x, x ∈ [−1, 1], and the response Y ismeasured. He assumed that the response function is sufficiently smooth over the rangeof interest and is adequately represented by an m-th degree polynomial

Pm : E(Y | x) = β0 + β1 + β2x2 + . . .+ βmxm,

where βi’s are unknown.

In order to find an optimum design which (i) checks for the adequacy of the fittedmodel Pm, (ii) enables to make reasonably efficient inferences concerning the model(parameters) if it is adequate, and (iii) does not depend on the unknown parameters,he proposed a restricted D-optimality criterion which maximizes | Mm(ξ) |, whereMm(ξ) is the information matrix of a design ξ when the fitted model is Pm, subjectto the constraint | Mm(ξ) |≤ C | Mm+1(ξ) | .

Stigler (1971) attempted to provide a justification for imposing this constraint. Thisrefers to the estimation of the ‘extra’ parameter βm+1 for which an expression for thevariance of the least square estimator βm+1 is given by Var (βm+1) = σ2 | Mm(ξ) |. | Mm+1(ξ) |−1 . Next he argues “minimize the generalized variance of the least

squares estimators β1, β2, . . . βm for the model Pm subject to the constraint thatVar(βm+1) ≤ σ2C.”

At this stage, it seems imperative to assume a prior knowledge about the error vari-ance σ2 as otherwise it is impractical to attach any meaning to the constraint σ2C.Henceforth, we will assume without any loss that σ2 = 1.

Stigler (1971) further indicated that in the event of finding an optimum design to testthe null hypothesis H0 : βm+1 = 0 against the alternative HA : βm+10, for a givenlevel of significance and the power at a stated alternative not below a specified value,the restricted D- optimality criterion can be made to meet the requirement.

Following Stigler (1971), we attempt to investigate optimum designs with the proper-ties (i) - (iii) above in the mixture set-up.


Consider a mixture experiment that is run under the assumption that the first degreemodel is adequate to approximate the response function:

E(Y | x) = ζ(1)x = Σ

iβixi, (1)

where x = (x1, x2, . . . , xq) denotes the mixing proportions defined in the experimentalregion

Ξ = (x1, x2, . . . , xq) | xi ≥ 0, i = 1, 2, . . . , q,Σxi = 1.

Several optimum designs have been suggested for estimation of βis using different opti-mality criteria. (Cf. Sinha, et al., 2014). However, these designs fail to check whetherthe assumed model provides an adequate fit to the true response function, no matterhow large the sample size is.

Suppose the experimenter suspects that there might be interaction between the firstand some of the other components and considers a quadratic model of the form

E(Y | x) = ζ(2)x = Σ

iβixi + Σ

2≤j≤sβ1jx1xj , (2)

where s ≤ q.

There exist optimum designs for testing the significance of the parameters β1js orestimating them. However, such designs will not be optimum for estimating the pa-rameters of (1) in case model (1) is true. Also, even if β1js are significant, these designswill not be optimal for making global inferences about the response function (2).

It would, therefore, be worthwhile to design a mixture experiment which allows effi-cient estimation of the parameters of model (1), and also tests hypotheses of the formβ1j = 0, j = 2, 3, . . . , s, 2 ≤ s ≤ q, with some specified degree of precision. FollowingStigler (1971), we define a criterion that minimizes the generalized variance of the leastsquares estimators of βis of the model (1) subject to the constraint that the minimumpower of the test for testing β1j = 0 for j = 2, 3, . . . ,s attains at least a specified valueC (in units of σ2, the error variance).

The choice of C reflects a compromise between two conflicting goals: precise inferenceabout β1js and precise inference about the parameters of model (1). While C should besmall to yield efficient designs for model (1), sufficiently large C will detect departuresfrom the model with a specified power of the test for β1j = 0, j ≥ 2. We note that forC = 0, the above criterion gives the D-optimal design for estimating the parameters ofmodel (1). On the other hand, when C attains its maximum value, we get the optimaldesign for testing β1j = 0, j ≥ 2.


3 Restricted D-optimal design for efficient estimation ofβi’s subject to testing β1j = 0 for 2 ≤ j ≤ s(s ≤ q) withspecified degree of precision

In consideration of the model fitting issue, the experimenter wishes to examine whetherthe interactions in model (2) influence the mean response or not, i.e. he wants to testthe null hypothesis

H0 : β12 = β13 = . . . = β1s = 0. (3)

Let us write the models (1) and (2) as

ζ(1)x = f ′1(x)β

(1)

ζ(2)x = f ′1(x)β

(1) + f ′2(x)β

(2),

where

f1(x) = (x1, x2, . . . , xq)′,f2(x) = (x1x2, x1x3, . . . , x1xs)

′,

β(1) = (β1β2, . . . , βq)′,β(2) = (β12, β13, . . . , β1s)

′.

The hypothesis (3) is tested using the classical F-test.

For any given design ξ, let M(ξ) denote the information matrix for β = (β(1)′ ,β(2)′)′,which may be partitioned as

M(ξ) =

M11 M12

M21 M22

, (4)

where Mij = Eξ[f i(x)f j(x)′], i, j = 1, 2.

M11 is the information matrix for β(1) in model (1). The power of the test for (3) isa non-decreasing function of the non-centrality parameter

δ = β(2)′ [Disp (β(2)

)]−1β(2) =1

σ2β(2)′M22,1β

(2),

where, σ2 is the error variance, M22,1 = M22−M21M−111 M12 andDisp(β

(2)) = σ2M−1

22,1.

Here under a non-null hypothesis, β(2) is unknown. Let us assume that β(2) is suchthat Ω = β(2) : β(2)′β(2) = D, for some D > 0 and known.


So, the statement “power of the test is at least equal to C for β(2) belonging to Ω′′ isequivalent to the statement” β(2)′M22,1β

(2) ≥ C0 for β(2) belonging to Ω′′, for somefunction C0 of C. It is tacitly assumed that the bounds are expressed in units of σ2.Now,

β(2)′M22,1β(2) ≥ C0, for all β

(2) ∈ Ω

⇔ minβ(2)∋β(2)β(2)=D

β(2)′M22,1β(2) ≥ C0

⇔ λmin(M22,1) ≥ C0D, (5)

where λmin denotes the minimum eigen value of M22,1.

WOLG, we can take D = 1.

Hence our problem reduces to finding a design ξ0 for the problem

maximize ϕ2(ξ) = log | M11(ξ) |

subject to ϕ1(ξ) = λmin(M22,1(ξ)) ≥ C0. (6)

Let, D(C0) denote the class of all designs ξ satisfying the restriction ϕ1(ξ) ≥ C0, and∆C0 be the set of all constrained optimal designs for a specific choice of C0, that is,∆C0 = ξ | ξ maximizes ϕ2(ξ) subject to the constraint ϕ1(ξ) ≥ C0. If there be achoice between designs in ∆C0 , then we select the design that does the best on theprimary criterion ϕ1(ξ). Arguing as in Stigler (1971), it is easy to show that D(C0) isa convex set and the set of restricted D-optimal designs ∆C0 forms a convex subset ofD(C0).

Let ξ1 be a restricted D-optimal design. Let us define ξP ∈ D(C0) such that ξP (x) =ξ1(Px), where P is a permutation matrix. The matrix P depends on the model andthe hypothesis considered as is evident from the examples considered below.

Example 3.1: Let s = 2 in (2). The model is, therefore,

E(Y | x) = ζ(2)x = Σiβixi + β12x1x2,

and the hypothesis to be tested is H0 : β12 = 0. In this case, the problem is invariantwith respect to x1 and x2, and with respect to x3, x4, . . . , xq and the permutationmatrix is

P =

0 1 0 0. . . 01 0 0 0. . . 0

0 Pq−2

,


where Pq−2 denotes a permutation of the matrix Iq−2.

Example 3.2: Suppose s = 3. In this case model (2) reduces to

E(Y | x) = ζ(2)x = Σiβixi + β12x1x2 + β13x1x3,

and the hypothesis to be tested is H0 : β12 = β13 = 0. Thus, the problem is invariantwith respect to x2 and x3, and with respect to x4, x5 . . . , xq, and the permutationmatrix is

P =

1 0 0 0. . . 0

0 0 1 0. . . 00 1 0 0. . . 0

0 0 Pq−3

,

where Pq−3 denotes a permutation of the matrix Iq−3.

In fact, for any general s, 3 ≤ s ≤ q, the problem is invariant with respect tox2, x3, . . . , xs, and with respect to xs+1, xs+2, . . . , xq, and the permutation matrix hasthe form

P =

1 0′ 0′

0 Ps−1 0

0 0 Pq−s

,

Then, ξP is also a restricted D-optimal design since both | M11 | and λmin(M−122,1) are

invariant with respect to the corresponding permutations.

Now, consider the design ξ∗ defined as ξ∗ = 1Q

Q

Σi=1

ξP1 , where Q denotes the number of

permutations leading to invariant null hypothesis. In example 3.1, Q = 2!.(q−2)!, whilein example 3.2, Q = 2!(q− 3)!. In the case of general s, 3 ≤ s ≤ q,Q = (s− 1)!(q− s)!.

Since the set of restricted D-optimal designs forms a convex subset of D(C0), ξ∗, which

is a permutation invariant design, is also a restricted D-optimal design.

Thus, we have the following theorem:


Theorem 3.1: There exists a permutation invariant restricted D-optimal design inD(C0).

The theorem simplifies the search for the optimal design by restricting to the class ofsymmetric designs that are permutation-invariant with respect to the invariant com-ponents under the null hypothesis. The search can be further reduced by the followingconsideration:

Lemma 3.1: Consider the model E(Y | x) = ζ(2)x = Σ

iβixi + Σ

1≤i<j≤qβijxixj ,x ∈

Ξ, and let (x1, x2, . . . xi, . . . xj , . . . , xq) be a support point of an arbitrary design ξ.If βij = 0 for some i < j, it is always possible to find two support points viz.(x1, x2, . . . , xi + xj , . . . , 0, . . . , xq) and (x1, x2, . . . , 0, . . . , xi + xj , . . . , xq) with weightsxi/(xi+xj) and xj/(xi+xj), respectively such that its information matrix dominatesthe information matrix of the single point design (x1, x2, . . . , xi, . . . xj , . . . , xq).

The proof is routine.

Hence, if an interaction effect, say βij , is absent in the model then an improved designcan be obtained by including support points in which xi and xj are not simultaneouslynon-zero. In our study we have, therefore, confined our search to the class of designsD1(C0)(⊂ D(C0)) with support points

(i) (1, 0, . . . , 0) and its permutations, and

(ii) points of the form (a, 0, . . . , 1 − a, . . . , 0) (with non-zero elements in the first andj-th positions) if β1j = 0. (7)

Remark 3.1: When β1j = 0, 3 ≤ j ≤ q, there is invariance between x1 and x2, andtherefore the support points of the optimum design are the extreme points of the sim-plex and the point (1/2, 1/2, 0, . . . , 0).

Remark 3.2: When β1j = 0, s + 1 ≤ j ≤ q, for some s ≥ 3, there is invariancebetween x2, . . . xs and between xs+1, xs+2, . . . , xq.

4 Optimum designs in specific situations

In this section we find the optimum design within D1(C0) for various forms of thealternative model (2).

4.1 Suppose the alternative model is ζ(2)x = Σ

iβixi + β12x1x2.


Here, there is invariance between components 1 and 2, and between componentsx3, x4, . . . , xq. The optimal design for the problem (5), therefore belongs to a class thatassigns mass α to the extreme points (1, 0, . . . , 0) and (0, 1, . . . , 0),mass γ to each of theremaining extreme points of the simplex and a mass β to the point (1/2, 1/2, 0, . . . , 0),where 2α+ (q − 2)γ + β = 1.

For any design ξ, we get

ϕ2(ξ) = αγq−2[α+1

2β] =

αγq−2

2[1− (q − 2)γ],

ϕ1(ξ) =α(1− 2α− (q − 2)γ)

8[1− (q − 2)γ]≤ 1

64.

For a given C0, the optimal value of α and γ can be obtained by solving the non-linearprogramming problem

maximize f(α, γ) = αγq−2[α+1

2β] =

αγq−2

2[1− (q − 2)γ],

subject toα(1− 2α− (q − 2)γ)

8[1− (q − 2)γ]≥ C0, α+ γ ≤ 1, 0 ≤ α ≤ 1

2, 0 ≤ γ ≤ 1

q − 2.

Since M22.1 ≤ 164 , C0 cannot exceed 1

64 .

Table 4.1 gives the optimal values of α and γ for some values of q(≥ 3) and C0.

Table 4.1: Optimum designs in D1 for some values of C0 in a q- component mixture

q C0 α β γ | M11 |3 0.002 0.3195 0.0337 0.3273 0.03517322848

0.006 0.2885 0.1152 0.3078 0.030735598590.010 0.2534 0.2338 0.2594 0.024339442990.015 0.2454 0.4697 0.0395 0.004659133855

4 0.002 0.2376 0.0344 0.2452 0.0036405163840.006 0.2114 0.1242 0.2264 0.00296614019330.010 0.2000 0.2668 0.1666 0.00185145186420.015 0.2426 0.4750 0.0198 4.59289961E-5

5 0.002 0.1885 0.0350 0.1960 2.92409516E-40.006 0.1681 0.1343 0.1765 2.17382987E-40.010 0.1825 0.2855 0.1165 9.38669829E-50.015 0.2407 0.4796 0.0130 2.53391735E-7

6 0.002 0.1551 0.0428 0.1617 1.8740489948E-50.006 0.1436 0.1512 0.1404 1.2233908896E-50.010 0.1753 0.2950 0.0886 3.490347345E-60.015 0.2454 0.4896 0.0049 7.149964739E-11


iβixi + Σ

2≤j≤qβ1jx1xj .


In this case, there being invariance among components 2, 3, . . . , q, we confine to theclass that assigns a mass α to the extreme point (1, 0, . . . , 0), mass γ to each of theremaining extreme points of the simplex and a mass β to each of the points of the form(a, 1−a, 0, . . . , 0), (a, 0, 1−a, 0, . . . , 0), . . . , (a, 0, . . . , 0, 1−a), where α+(q−1)(β+γ) =1. Clearly, the design is saturated.

Then, for any design ξ,

ϕ2(ξ) = [γ + (1− a)2β]q−2[αγ + (1− a)2β+ (q − 1)a2βγ] (8)

M22.1 =a2(1− a)2β

γ + (1− a)2β

[γIq−1 −

a2γ2

(q − 1)a2βγ + (γ + (1− a)2β)αβ1q−11

′q−1

].

The distinct eigen values of M22.1 are λ1 = a2(1−a)2βγγ+(1−a)2β

and λ2 = a2(1 − a)2βγα[α +

(q − 1)a2βγ + (1− a)2βα]−1. It is easy to check that λ2 is the minimum eigen value,whatever be the unknown parameters.

So, our problem is to maximize (8) with respect to a, α, β and γ subject to λ2 ≥C0, α+ (q − 1)(β + γ) = 1, 0 ≤ a, α ≤ 1, 0 ≤ β, γ ≤ 1/(q − 1).

Using Cauchy-Schwartz inequality, it is easy to check that λ2 ≤ 1/64(q − 1) = c0,say. Hence, C0 cannot exceed c0. However, this is a very crude bound and may not beattained by λ2.

Table 4.2 gives the optimal values of a, α, β and γ for some values of q(≥ 3) andC0.

Table 4.2: Optimum designs in D2 for some values of C0 in a q-component mixture

q C0 α β γ a | M11 |3 0.001 0.2382 0.0167 0.3642 0.4892 0.0123179745

0.002 0.2270 0.0353 0.3512 0.4782 0.0114015090.006 0.1996 0.1433 0.2569 0.4444 0.00677044400.007 0.2068 0.1868 0.2098 0.4507 0.0050268905

4 0.001 0.1844 0.0174 0.2545 0.4771 8.8529086E-40.002 0.1723 0.0386 0.2373 0.4547 7.4725697E-40.003 0.1666 0.0656 0.2122 0.4365 5.9183265E-40.005 0.2101 0.1466 0.1167 0.4607 1.7940228E-4

5 0.0005 0.1576 0.0091 0.2015 0.4773 5.8550308E-50.001 0.1496 0.0190 0.1936 0.4590 5.1783620E-50.002 0.1435 0.0443 0.1698 0.4279 3.6873693E-50.003 0.1600 0.0788 0.1312 0.4230 2.0033288E-5

6 0.0005 0.1462 0.0217 0.1490 0.4114 2.4108438E-60.001 0.1452 0.0322 0.1388 0.4072 1.9417063E-60.002 0.1716 0.0632 0.1025 0.4216 7.7929237E-70.003 0.2496 0.0969 0.0531 0.4951 7.3477238E-8


iβixi + Σ

2≤<j≤sβijxixj , 3 ≤ s ≤ q − 1.


Here there is invariance among components 2, 3, . . . , s, and among components s +1, s+2, . . . , q. So we confine to the class of designs that assign a mass α to the extremepoint (1, 0, . . . , 0), mass γ to each of the (s − 1) extreme points (0, 1, 0, . . . , 0, . . . , 0),(0, 0, 1, . . . , 0, . . . , 0), . . . , (0, 0, 0, . . . , 1, . . . , 0) and mass δ to each of the remaining(q − s) extreme points of the simplex, and a mass β to each of the (s − 1) pointsof the form (a, 1 − a, 0, . . . , 0), (a, 0, 1 − a, 0, . . . , 0), . . . , (a, 0, . . . , 1 − a, . . . , 0), whereα+ (s− 1)(β + γ) + (q − s)δ = 1.

For any design ξ we therefore have

ϕ2(ξ) = δq−s[γ + (1− a)2β]s−2[αγ + (1− a)2β+ (s− 1)a2βγ] (9)

M22.1 =a2(1− a)2β

γ + (1− a)2β

[γIs−1 −

a2γ2

(s− 1)a2βγ + (γ + (1− a)2β)αβ1s−11

′s−1

].

The expression for M22.1 is similar to that in Case 2, except that q is replaced by s.Hence, it is easy to find the minimum eigen value of M22.1.

The optimal values of a, α, β and γ for some values of q(≥ 3) and C0 are given in Table4.3.

Table 4.3: Optimum designs in D3 for some values of C0 and s = 3 in aq-component mixture

q C0 α β γ δ a | M11 |4 0.002 0.1796 0.0363 0.2782 0.1915 0.4730 8.9563062E-4

0.005 0.1679 0.1187 0.2180 0.1587 0.4444 5.4504091E-4

5 0.002 0.1483 0.0384 0.2286 0.1588 0.4672 5.6954921E-50.005 0.1536 0.1290 0.1712 0.1229 0.4469 2.7636170E-5

6 0.002 0.1346 0.0591 0.1784 0.1301 0.4460 2.6087455E-60.005 0.1821 0.1552 0.1297 0.0827 0.4650 6.9305479E-7

5 Conclusion

We discuss the necessary theoretical framework and computations for the study andspecification of D-optimal mixture designs that permit efficient inferences to be madeabout the assumed mixture model while still allowing the model to be checked foradequacy. We confine our study to the cases where the competence of a first degreemixture model is examined against possible presence of some quadratic terms.


References

[1] Atwood C. L. (1969). Optimal and efficient designs of experiments, Annals ofMathematical Statistics, 40, 1570 - 1602.

[2] Box, G. E. P. and Draper, N. R. (1959). A basis for the selection of a responsesurface design, Journal of the American Statistical Association, 54, 622 - 54.

[3] Box, G. E. P. and Draper, N. R. (1963). The choice of a second order rotatabledesign, Biometrika, 50, 335 - 52.

[4] Karlin, S. and Studden, W. J. (1966). Optimal Experimental Designs, The Annalsof Mathematical Statistics, 37, 783 - 815.

[5] Karson, M. J., Manson, A. R. and Hader, R. J. (1969). Minimum bias estimationand experimental design for response surfaces, Technometrics, 11, 461 - 75.

[6] Kiefer, J. (1959). Optimum experimental designs, Journal of the Royal StatisticalSociety, Ser. B, 21 (2), 273 - 319.

[7] Kiefer, J. (1961). Optimum Designs in Regression Problems, II, The Annals ofMathematical Statistics, 32, (March, 1961), 298 - 325.

[8] Kiefer, J. (1974). General Equivalence Theory for Optimal Designs, The Annalsof Statistics, 2, 849 - 879.

[9] Lee, C. M. S. (1987). Constrained optimal designs for regression models, Commu-nications in Statistics - Theory and Methods, 16, 765 - 783.

[10] Lee, C. M. S. (1988). Constrained optimal design, Journal of Statistical Planningand Inference, 18, 377 - 389.

[11] Mikulecka, J. (1983). On hybrid experimental design, Kybernetika, 19, 1 - 14.

[12] Sinha, B. K., Mandal, N. K., Pal, M. and Das, P. (2014). Optimal Mixture Ex-periments, Lecture series in Statistics, 1028 (Springer).

[13] Stigler, S. (1971). Optimal experimental design for polynomial regression, Journalof American Statistical Association, 66, 311 - 318.

[14] Studden, W. J. (1982). Some robust-type D-optimal designs in polynomial regres-sion, Journal of American Statistical Association, 77, 916 - 921.


Vol. 15, 2015, pp 13-19


An Efficient Product-type Exponential Estimator ofFinite Population Mean using Variable Transformation

K. B. PandaUtkal University

Vani Vihar, Odisha, IndiaEmail: [email protected]

N. SahooUtkal University

Vani Vihar, Odisha, IndiaEmail: [email protected]

[Received November 14, 2016; Revised February 5, 2017; Accepted June 10, 2017;Published December 30, 2017]

Abstract

This paper proposes an efficient product-type exponential estimator forestimating the finite population mean using variable transformation. Theexpressions of the bias and mean square error of the proposed estimator, tothe first order of approximation, are derived in general form. It has beenshown that the proposed estimator is more efficient than the usual unbiasedestimator, product estimator, product-type exponential estimator due toBahl and Tuteja [1] and product-type exponential estimator due to Tailorand Tailor [6]. An empirical study is carried out in support of theoreticalfindings.

Keywords and Phrases: Simple random sampling, product estima-tor, product-type exponential estimator, variable transformation, bias andmean square error.

AMS Classification: 62D05.

1 Introduction

In survey sampling, when the auxiliary variable x is negetively correlated with thestudy variable y and complete information on x is available, the product method of


estimation is followed to estimate the population mean (Y ) or the population total(Y ). The conventional product estimator is given as

yP = yx

X. (1)

With a view to improving estimates of the population mean (Y ) of the study variabley, Bahl and Tuteja [1] introduced the product-type exponential estimator as

yPe = yexp

(x−X

x+X

). (2)

Motivated by the works of Oyenka [3], Srivenkataramana [5] and Tailor and Sharma[7],we have used variable transformation to estimate the population mean. The obviousadvantage of variable transformation is the introduction of an additional auxiliary(transformed) variable without additional cost, since new auxiliary variable is a trans-formation of an already observed auxiliary variable.

Tailor and Tailor [6] proposed dual to Bahl and Tuteja product-type exponential esti-mator as

y∗Pe = yexp

(X − x∗

X + x∗

), (3)

where x∗ = NX−nxN−n , which is arrived at from the variable transformation given by

x∗i =NX−nxiN−n , i = 1, 2, . . . .., N.

2 Bias and mean square error of competing estimators

It is well known that mean per unit estimator y is an unbiased estimator of populationmean Y and its variance is given by

V (y) = MSE (y) = θY2C2y , (4)

where θ = N−nNn = 1−f

n , f = nN and C2

y =S2y

Y2 .

We also have, to O(n−1

),

B (yP ) = θY ρCyCx (5)

MSE (yP ) = θY2 (

C2y + C2

x + 2ρCyCx

)(6)

B (yPe) = θY

(1

2ρCyCx −

1

8C2x

)(7)

MSE (yPe) = θY2(C2y +

1

4C2x + ρCyCx

)(8)

Panda and Sahoo: An efficient product-type exponential estimator 15

B (y∗Pe) = θY

(3n2

8(N − n)2C2x +

n

2(N − n)ρCyCx

)(9)

and

MSE (y∗Pe) = θY2(C2y +

n2

4(N − n)2C2x +

n

(N − n)ρCyCx

), (10)

where ρ is the correlation coefficient between y and x assumed to be negative.

3 Proposed estimator

We propose the following product-type exponential estimator of population mean Y :

y∗∗Pe = y

[exp

(X − x∗

X + x∗

)(X

x∗

)a], (11)

where α is a scalar quantity and x∗ is as defined earlier.

To obtain the bias and mean square error of the proposed estimator y∗∗Pe, we definethe following quantities:

y = Y (1 + e0), x = X(1 + e1), such that E (e0) = E (e1) = 0, E(e20)

= θC2y ,

E(e21)= θC2

x, E (e0e1) = θρCyCx.

Expressing (11) in terms of e’s, we have

y∗∗Pe = Y

[1 + e0 +

(a+

1

2

)n

N − ne1 +

(a+

1

2

)n

N − ne0e1 +

(4a2 + 8a+ 3

8

)n2

(N − n)2 e

21

],

(12)

which gives rise to

B (y∗∗Pe) = θY

[(4a2 + 8a+ 3

8

)n2

(N − n)2C2x +

(a+

1

2

)n

N − nρCyCx

](13)

and

MSE (y∗∗Pe) = θY2

[C2y +

(a+

1

2

)2 n2

(N − n)2C2x + 2

(a+

1

2

)n

N − nρCyCx

]. (14)

The above expressions of bias and mean square error are, to the first order of approx-imation, i.e., to O(n−1).


4 Efficiency comparison

In this section, we have derived the conditions under which the proposed estimatory∗∗Pe is more efficient than the estimators y, yP , yPe and y∗Pe. From (4), (6), (8), (10)and (14), we find

MSE (y∗∗Pe) < MSE (y)

if(a+ 1

2

)n

N−n

[(a+ 1

2

)n

N−n + 2k]< 0, where k = ρ

Cy

Cx

i.e., if min

−1

2, −

(N − n

n

)2k − 1

2

< a < max

−1

2, −

(N − n

n

)2k − 1

2

.

(15)MSE (y∗∗Pe) < MSE (yP )

if(

a+ 12

)n

N−n − 1(

a+ 12

)n

N−n + 1 + 2k< 0

i.e., if min

2N − 3n

2n, (−2k − 1)

(N − n

n

)− 1

2

< a < max

2N − 3n

2n, (−2k − 1)

(N − n

n

)− 1

2

.

(16)

MSE (y∗∗Pe) < MSE (yPe)

if(

a+ 12

)n

N−n − 12

(a+ 1

2

)n

N−n + 12 + 2k

< 0

i.e., if min

N − 2n

2n, (−2k − 1

2)

(N − n

n

)− 1

2

< a < max

N − 2n

2n, (−2k − 1

2)

(N − n

n

)− 1

2

.

(17)

andMSE (y∗∗Pe) < MSE (y∗Pe)

if(

a+ 12

)− 1

2

[n

N−n

(a+ 1

2

)+ 1

2

+ 2k

]< 0

i.e., if min

0, −2k

(N − n

n

)− 1

< a < max

0, −2k

(N − n

n

)− 1

. (18)

5 Optimum choice of the scalar α

Differentiating (14) with respect to α and equating it to zero, we get the optimumvalue of α as

αopt. = −(N − n

n

)ρCy

Cx− 1

2= −

(N − n

n

)k − 1

2(19)


Substituting this value of α in (14), the optimum value of the MSE (y∗∗Pe) can be

MSE(y∗∗Pe)opt. = θY2C2y (1− ρ2). (20)

The expression in (20) is equal to the mean square error of linear regression estimator

ylr = y + β(X − x). Thus, the proposed estimator y∗∗Pe can be used as an alternativeto the usual linear regression estimator ylr when α is chosen optimally. It is aptto mention here that, for large samples, i.e., when n → N, the proposed estimatorapproaches the corresponding parameter implying thereby that the proposed estimatoris consistent.

6 Empirical Study

For the purpose of numerical illustration, we refer to Weisberg [8] wherein the givensample quantities have been considered as the corresponding population quantities.We consider the sample size n = 10.

The variables are

y: Rate (1973 accident rate per million vehicle miles)

x: SLIM = Speed limit =(in 1973, before the 55 mph limits)

and N = 39, n = 10, Y = 3.93, X = 55, Cy = 0.51, Cx = 0.11, ρ = −0.68.

Using the conditions which we have obtained in section 4, we calculate the range ofthe scalar α in which the proposed estimator y∗∗Pe is more efficient than the estimatorsy, yP , yPe and y∗Pe and present the same in Table 1.

Table 1: The range of α in which y∗∗Pe is more efficient than y, yP, yPe and y∗

Pe

Estimators y yP yPe y∗Pe

Range of α (-0.50, 17.78) (2.4, 16.33) (0.95, 16.33) (0, 17.28)

It may be noted here that the common range of the scalar α in which y∗∗Pe is moreefficient than y, yP , yPe and y∗Pe is (2.4, 16.33). To evaluate the performance of the y∗∗Peover the existing estimators, we have computed percent relative efficiencies (PREs) ofy∗∗Pe with respect to y, yP , yPe and y∗Pe in the common range (2.4, 16.33) of α usingrespective formulae given by

PRE (y∗∗Pe, y) =MSE (y)

MSE(y∗∗Pe

) × 100 (21)

PRE (y∗∗Pe, yP ) =MSE (yP )

MSE(y∗∗Pe

) × 100 (22)


PRE (y∗∗Pe, yPe) =MSE (yPe)

MSE(y∗∗Pe

) × 100 (23)

PRE (y∗∗Pe, y∗Pe) =

MSE (y∗Pe)

MSE(y∗∗Pe

) × 100 (24)

Table 2: PREs of y∗∗Pe with respect to the estimators y, yP, yPe and y∗

Pe

α PRE (y∗∗Pe, y) PRE (y∗∗Pe, yP ) PRE (y∗∗Pe, yPe) PRE (y∗∗Pe, y∗Pe)

2.4 132.76 100.00 114.89 126.22

2.85 138.29 104.17 119.67 131.48

3.15 141.90 106.89 122.80 134.92

4.50 158.04 119.05 136.77 150.26

5.25 166.31 125.28 143.93 158.13

6.35 176.43 132.90 152.69 167.75

7.80 184.72 139.15 159.86 175.63

8.64 185.99 140.10 160.96 176.84

9.37 185.07 139.40 160.16 175.96

10.25 181.14 136.45 156.76 172.22

11.50 171.57 129.24 148.48 163.12

12.75 158.46 119.36 137.13 150.66

13.85 145.35 109.49 125.79 138.20

14.55 136.83 103.07 118.41 130.10

15.30 127.76 96.24 110.56 121.47

16.33 115.68 87.14 100.17 109.99

It is observed from Table 2 that the performance of proposed estimator y∗∗Pe is betterthan the estimators y, yP , yPe and y∗Pe, the maximum gain in efficiency being attainedat the optimum value α = 8.64. It may also be noticed that in the event of an absolutedeviation to the extent of 70% from the optimum value of α, the superiority of theproposed estimator is maintained.

7 Conclusion

We have considered the problem of estimating the population mean of the study vari-able when the population mean of auxiliary variable is known using variable transfor-mation. The proposed estimator, to the first order of approximation, is more efficientthan the existing estimators. Theoretical findings have been demonstrated throughempirical investigation.


Acknowledgement

We thank the referee for his suggestions leading to greater clarity in the presentation.

References

[1] Bahl, S. and Tuteja, R. K. (1991). Ratio and product type exponential estimator,Information and Optimization Science, 12, 159-163.

[2] Cochran, W. G. (1977). Sampling Techniques, (New York, John Wiley).

[3] Oyenka, A. C. (2013). Dual to ratio estimators of population mean in post-stratified sampling using known value of some population parameters. GlobalJournal of Science Frontier Research, 13, 13-23.

[4] Sharma, B. and Tailor, R. (2010). A new ratio-cum-dual to ratio estimator of finitepopulation mean in simple random sampling. Global Journal of Science FrontierResearch, 10, 27-31.

[5] Singh, M. P. (1967). Ratio-cum-Productmethod of estimation, Metrika, 12, pp.34-42.

[6] Srivenkataramana, T. (1980). A Dual of ratio estimator in sample surveys.Biometrika, 67, 199-204.

[7] Tailor, R. and Tailor, R. (2012). Dual to ratioand product type exponential esti-mators of finite Population mean. Presented in National Conference on statisticalinference, at A.M.U., Aligarh, India, Aligarh-202002, Feb 11-12, 2012.

[8] Weisberg, S. (1980). Applied linear regression, (New York, John Wiley), 1980(p.179).


Vol. 15, 2015, pp 21-35


Optimisation Perspective in Managing Gender CentricProphesied Longevity

P. K. Tripathy and Sujata Sukla

P.G. Department of StatisticsUtkal University, Bhubaneswar 751004, India

Email:[email protected] & [email protected]

Priyanka Tripathy

Department of IT & ManagementRavenshaw University, Cuttack 751010, India

Email:[email protected]

[Received April 13, 2017; Revised May 11, 2017; Accepted June 15, 2017;Published December 30, 2017]

Abstract

Mortality transition leads to the change in the contribution of the agegroup which helps predominantly in enriching the life expectancy. Thiscontribution cannot simply be conceived from the mortality data as it mayalso vary gender wise. Hence, it is necessary to forecast the behaviourof the mortality transition of different age groups by gender towards thechange in longevity. Attempt has been constituted in the current study toforecast the age groups by sex which will contribute more in expansion of lifeexpectancy in future. For this purpose, the projected sex wise ASDR dataof developing country like India and its states of Kerala, Maharashtra andUttar Pradesh have been borrowed. The optimisation perspective is thenemployed to find out the most important age group which will contributemore in fortifying the longevity in future for both sex.

Keywords and Phrases: Mortality Transition, ASDR, Life table, LifeExpectancy.

AMS Classification: 91B28


1 Introduction

Life expectancy is a measure of how long a person belonging to a particular age groupis expected to survive. Improved health status throughout the developing countriesled to the enhancement in life expectancy in the last few decades. India has also keptpace in the improvement in health status over the country. As a result CDR, IMRand ASDR are lessened and the life expectancy has been ratcheted up considerably.

Many states in India are forging significant improvement in reinforcing the health sta-tus, which has led to the increment in life expectancy. But the role of all the agegroups in augmenting the longevity is not analogous. Game theory can be lucrativelyapplied to have erudition on this. Game theory was pioneered by Neumann and Mor-genstern [6] and later on explicated lucidly by Hillier and Liebermann[2] and Tripathy[11]. Bastian and Nair [8] proposed game theory approach to find out the age groupsplaying overriding role in enhancing life expectancy in India and some of its majorstates. Papers on life expectancy and mortality rate have been published by severalauthors. Tripathy and Pati [12] focused on that infant and child mortality that affectthe health of the persons in general and the health of women in particular, therebyaffecting the life expectancy. Dubey [10] projected the sex wise population and ASDRof India using Cohort-component and Lee-carter model. Sasson [4] adopted a multidi-mensional approach to life span inequality to study the trends in both life expectancyand life span variation by educational attainment in the United States. Li [5] exam-ined the application of a Poisson common factor model for the projection of mortality.Goldstein and Cassidy [7] evinced mathematically how varying the pace of senescenceinfluences life expectancy. Doblhammer [3] studied whether a woman’s reproductivehistory influences her life span. Mayhew and Smith [9] presented new methods forcomparing past improvements in life expectancy and also future prospects, using datafrom five developed and low-mortality countries. Poppel et. al. [1] investigated therole of urbanisation and plague on the changes in life expectancy amongst artists.

In the present study, a game theory prospective has been employed by using projectedASDR extracted from Dubey [10] and estimated life expectancy for both sexes inIndia and states of Kerala, Maharashtra and Uttar Pradesh, to find out the age groupaffecting mostly the life table in future. The objective of this paper is to make aninitiative for finding the age group which plays major role in enriching the longevityby gender.

2 Methodology

In a two person zero sum game, if maximum of row minima = minimum of columnmaxima, it does not possess a saddle point. In such a case to solve the game, theplayers must determine an optimal mixture of strategies to find a saddle point.

Tripathy, Sukla and Priyanka: Optimisation Perspective in Managing 23

There is no pure strategy (Nash equilibrium) in this game. If we play this game, weshould be unpredictable. That is, we should randomize (or mix) between strategies sothat we don’t get exploited.

The important observation is that if a player is using a mixed strategy at equilibriumthen both players should have the same expected payoff from the strategies they aremixing. We can easily find the mixed strategy (Nash equilibrium) in 2x2 games usingthis observation.

The optimal strategy mixture for each player may be determined by assigning to eachstrategy its probability of being selected. For the following two person zero sum game,

Player B

y1 y2

Player Ax1x2

[a11 a12a21 a22

]the player A plays his strategies x1 and x2 with probabilities p1 and q1 respectively,where

p1 =a22 − a21

(a11 + a22)− (a12 + a21)

and q1 = 1− p1 (because p1 + q1 = 1)

Likewise, the player B plays his strategies y1 and y2 with probabilities p2 and q2respectively, where

p2 =a22 − a12

(a11 + a22)− (a12 + a21)

and q2 = 1− p2 (because p2 + q2 = 1)

As per principle of dominance for easiness of solution, it is convenient to deal withsmaller payoff matrices. The size of the payoff matrix can be considerably reducedby using the principle of dominance, which includes, if each element in rth row isless than or equal to the corresponding element in any other row, say sth , then theplayer A will never choose rth strategy or in other words, rth strategy is said to bedominated by the sth strategy. If all the elements of a column, say Cr, are greaterthan or equal to the corresponding elements in any other column, say Cs, then thecolumn Cr is dominated by the column Cs. If the convex linear combination of somerows dominates the ith row, then the ith row will be deleted. If the ith row dominatesthe convex linear combination of some other rows, then one of the rows involving inthe combination may be deleted. Similar arguments follow for columns also. By us-ing dominance property, we reduce the size of the matrix to 2x2, the aforementioned


method is used to solve the game.

In case of 2×n or m×2 game, after applying the dominance property, if a 2x2 matrixis not obtained, graphical method can be employed to have such a matrix. In whichtwo axes of unit length apart are drawn and scale on each axis is marked. All the linesgiven in the matrix are drawn. Then the lowest point in the uppermost boundary,denoted by P, the interaction point of two strategies, helps in obtaining a 2x2 matrix.

3 Analysis

The methodology has been implemented to the gender wise data on projected ASDRand estimated Life Expectancy of India for 2016, 2021 & 2026.

Table 1: Sex wise Projected ASDR of India for the years 2016, 2021 & 2026

Age GroupProjected ASDR of India

2016Male

2016Female

2021Male

2021Female

2026Male

2026Female

0-45-910-1415-1920-2425-2930-3435-3940-4445-4950-5455-5960-6465-6970+

10.310.850.701.071.562.042.593.454.486.459.4314.0221.9035.6074.81

10.170.830.691.201.501.441.461.712.223.224.868.2613.8123.2461.05

8.050.670.600.961.421.912.433.264.175.998.6712.9020.0933.4170.68

8.490.680.601.081.331.281.281.511.982.904.357.5312.5621.5157.83

6.340.530.510.851.281.792.273.083.875.547.9411.8218.3731.2866.63

7.080.550.530.971.181.131.131.341.772.623.906.8711.4219.9054.79


Table 2: Sex wise Estimated life expectancy of India for 2016, 2021 & 2026

Age GroupEstimated Life Expectancy of India

2016Male

2016Female

2021Male

2021Female

2026Male

2026Female

0-45-910-1415-1920-2425-2930-3435-3940-4445-4950-5455-5960-6465-6970+

72.172.667.762.857.953.148.343.538.834.129.525.121.417.213.4

75.475.971.166.261.356.551.746.841.937.132.427.724.920.516.4

73.173.368.463.558.653.748.944.239.434.830.125.622.218.014.1

75.675.971.066.161.356.451.646.741.937.032.327.624.920.516.4

72.172.667.762.857.953.148.343.538.834.129.525.121.417.213.4

77.677.772.867.963.058.253.348.443.538.733.929.126.922.418.3

Taking into account the ASDR and Life expectancy of males and females in 2016, wehave the following tables.

Table 3: ASDR and Life expectancy for males of India in 2016

Age GroupMales of India 2016

Row MinimumProjectedASDR

Estimated LifeExpectancy

0-45-910-1415-1920-2425-2930-3435-3940-4445-4950-5455-5960-6465-6970+

10.310.850.701.071.562.042.593.454.486.459.4314.0221.9035.6074.81

72.172.667.762.857.953.148.343.538.834.129.525.121.417.213.4

10.310.850.701.071.562.042.593.454.486.459.4314.0221.417.213.4

ColumnMaximum

74.81 72.6


Table 4: The ASDR and Life expectancy of females of India in 2016

Age GroupFemales of India 2016

Row MinimumProjectedASDR


0-45-910-1415-1920-2425-2930-3435-3940-4445-4950-5455-5960-6465-6970+

10.170.830.691.201.501.441.461.712.223.224.868.2613.8123.2461.05

75.475.971.166.261.356.551.746.841.937.132.427.724.920.516.4

10.170.830.691.201.501.441.461.712.223.224.868.2613.8120.516.4

ColumnMaximum

61.05 75.9

It is evident from table 3 & table 4 that maximum of the Row Minima = minimum ofthe Column maxima i.e. 20.5 =61.05. So, they don’t possess a saddle point. Thereforedominance property is to be applied to reduce the size of the matrices. In table 3, theage groups 10-14 to 50-54 are dominated by the age group 0-4. In table 4 age groups10-14 to 55-59 are dominated by age group 0-4. The table 5 & table 6 represent theASDR and the Life expectancy after applying the dominance property.

Table 5: Reduced matrix after applying dominance property


ProjectedASDR


0-45-955-5960-6465-6970+

10.310.8514.0221.9035.6074.81

72.172.625.121.417.213.4




ProjectedASDR


0-45-960-6465-6970+

10.170.8313.8123.2461.05

75.475.924.920.516.4

Further 2x2 matrices have been formed by exerting graphical method.

Figure 1&2: Graphical representation of projectedASDR vis-a-vis estimated life expectancy

In Figure 1 & Figure 2, the lowest points P1 and P2 in the upper most boundariesare presented as the interaction of two strategies 0-4 and 70+. The two strategiescorresponding to the points P1 and P2 help in shortening the matrices into 2x2 forms.

Table 7: 2x2 matrix derived from Figure 1


ProjectedASDR


0-470+

10.3174.81

72.113.4




ProjectedASDR


0-470+

10.1761.05

75.416.4

Let p1 and q1 be the probabilities of males and p2 and q2 that of females for the agegroups of 0-4 and 70+ respectively. Then

p1 =13.4− 74.81

(10.13 + 13.4)− (72.1 + 74.81)= 0.4977

and q1 = 1− p1 = 1− 0.4977 = 0.5023, q1 > p1

p2 =16.4− 61.05

(10.17 + 16.4)− (75.4 + 61.05)= 0.4064

and q2 = 1− p2 = 1− 0.4064 = 0.5936, q2 > p2

For both the genders, the probability of the age group of 70+ is more exhibiting theimportance of the age group in expanding the life expectancy.

Applying the same method for the data of males and females of India for 2021 & 2026,the results obtained are presented in table 9.

Now the effectiveness of the proposed method has been studied by applying it for thedata of Kerala, Maharashtra and Uttar Pradesh.

The projected ASDR and estimated Life expectancy of males and females of Keralafor 2016, 2021 and 2026 are presented in the table 9 & table 10.


Table 9: Sex wise projected ASDR of Kerala for 2016, 2021 & 2026

Age GroupProjected ASDR of Kerala

2016Male

2016Female

2021Male

2021Female

2026Male

2026Female

0-45-910-1415-1920-2425-2930-3435-3940-4445-4950-5455-5960-6465-6970+

1.610.160.210.480.911.351.512.423.025.128.6012.4221.2529.9880.86

1.020.070.130.280.420.390.510.710.801.232.224.146.9312.3150.47

1.260.120.180.440.851.281.412.324.225.747.2511.7118.7629.3968.05

0.720.050.100.230.350.320.430.610.671.031.933.686.0810.8846.85

1.000.090.150.400.801.231.312.232.644.607.9911.3920.1627.7277.10

0.510.030.080.200.300.270.360.520.560.871.673.275.339.6243.50

Table 10: Sex wise estimated Life Expectancy of Kerala for 2016, 2021 & 2026

Age GroupEstimated Life Expectancy of Kerala

2016Male

2016female

2021Male

2021female

2026Male

2026Female

0-45-910-1415-1920-2425-2930-3435-3940-4445-4950-5455-5960-6465-6970+

72.871.967.062.057.152.247.342.537.732.928.223.620.716.512.4

80.479.674.669.664.659.754.749.844.839.934.930.128.924.319.8

73.172.367.362.357.452.547.642.737.933.128.423.920.916.812.7

81.880.975.970.966.061.156.151.146.241.336.331.530.525.821.3

73.572.667.662.657.752.847.943.038.233.428.724.121.317.112.9

83.482.477.472.567.562.557.652.647.742.737.832.932.227.522.9

Applying the above methodology for the sex wise data on ASDR and Life Expectancyof Kerala for 2016, it is observed that the maximum of the row minima = minimum ofthe Column maxima. Dominance property has been utilised and the table 11 & table12 are obtained.



Age GroupMales of Kerala 2016

ProjectedASDR


0-435-3940-4445-4950-5455-5960-6465-6970+

1.612.423.025.128.6012.4221.2529.9880.86

72.842.537.732.928.223.620.716.512.4


Age GroupFemales of Kerala 2016

ProjectedASDR


0-445-4950-5455-5960-6465-6970+

1.021.232.224.146.9312.3150.47

80.439.934.930.128.924.319.8

For further reduction of the above matrices to 2x2 form the graphical method has beenemployed.

The 2x2 matrices and the probabilities obtained from Figure 3 & Figure 4 are presentedbelow.


Age GroupMales of Kerala 2016

Males of Kerala2016

ProjectedASDR

0-470+

1.6180.86

72.812.4


Age GroupFemales of Kerala 2016

ProjectedASDR


0-470+

1.0250.47

80.419.8


Figure 3&4: Graphical representation of projectedASDR vis-a-vis estimated life expectancy

If p1 and q1 are defined as the probabilities of males and p2 and q2 that of females forthe age groups 0-4 and 70+ respectively, then

p1 =12.4− 80.86

(1.61 + 12.4)− (80.86 + 72.8)= 0.4902

and q1 = 1− p1 = 1− 0.4902 = 0.5098, q1 > p1

p2 =19.8− 50.47

(1.02 + 19.8)− (80.4 + 50.47)= 0.2787

and q2 = 1− p2 = 1− 0.2787 = 0.7213, q2 > p2

This implies that in Kerala for 2016, 70+ age group is important in enriching the lifeexpectancy.

Proceeding in the same way for 2021 and 2026, the result obtained is presented intable-19.

The following table 15, table 16, table 17 & table 18 represent the gender wise pro-jected ASDR and estimated life expectancy of Maharashtra and Uttar Pradesh for theyears 2016, 2021 & 2026.


Table 15: Sex wise Projected ASDR of Maharashtra for 2016, 2021 & 2026

Age GroupProjected ASDR of Maharashtra

2016Male

2016Female

2021Male

2021Female

2026Male

2026Female

0-45-910-1415-1920-2425-2930-3435-3940-4445-4950-5455-5960-6465-6970+

4.260.310.400.661.212.272.223.834.225.747.2511.7118.7629.3968.05

4.040.260.340.840.930.871.001.251.472.403.626.4411.7518.8655.54

3.050.210.320.561.092.272.053.763.905.266.4110.4416.7526.3962.68

2.730.160.250.690.750.690.811.021.171.973.015.4610.0616.2250.02

2.140.140.260.460.982.261.893.683.584.795.629.2414.8623.5657.46

1.800.100.190.560.590.530.650.820.921.602.484.588.5713.7944.70

Table 16: Sex wise Estimated Life expectancy of Maharashtra for 2016, 2021 & 2026

Age GroupEstimated Life Expectancy of Maharashtra

2016Male

2016Male

2021Male

2021Female

2026Male

2026Female

0-45-910-1415-1920-2425-2930-3435-3940-4445-4950-5455-5960-6465-6970+

77.977.572.667.662.757.852.948.043.138.333.428.626.722.218.0

74.273.868.863.958.954.149.344.639.935.230.626.022.918.714.7

73.172.367.362.357.452.547.642.737.933.128.423.920.916.812.7

79.979.374.469.464.559.654.749.844.939.935.130.328.724.319.9

77.076.371.466.461.456.651.846.942.337.632.928.325.821.517.4

82.281.576.571.666.661.756.851.846.942.037.132.331.226.722.4


Table 17: Sex wise Projected ASDR of Uttar Pradesh for 2016, 2021 & 2026

Age GroupProjected ASDR, Uttar Pradesh

Male2016

Female2016

Male2021

Female2021

Male2026

Female2026

0-45-910-1415-1920-2425-2930-3435-3940-4445-4950-5455-5960-6465-6970+

14.391.030.851.362.172.273.244.324.876.769.6614.0624.3638.3379.89

11.180.930.631.171.722.031.872.202.473.755.378.0513.0622.4063.12

10.990.770.721.252.102.173.194.274.496.158.6512.5222.4535.8476.28

7.460.660.480.911.341.671.501.832.033.204.456.6810.6819.0858.26

8.190.560.591.142.032.063.154.214.125.557.6611.0420.5533.3272.55

4.800.420.350.701.031.341.171.501.632.703.635.468.5716.0253.41

Table 18: Sex wise Estimated Life expectancy of Uttar Pradesh for 2016, 2021 &2026

Age GroupEstimated Life Expectancy, Uttar Pradesh

Male2016

Female2016

Male2021

Female2021

Male2026

Female2026

0-45-910-1415-1920-2425-2930-3435-3940-4445-4950-5455-5960-6465-6970+

70.671.766.961.957.152.447.642.938.233.629.024.520.416.412.5

74.875.570.765.760.956.151.346.541.736.932.127.524.420.015.8

71.772.367.462.557.652.848.143.438.734.029.424.921.116.9913.1

76.776.871.966.962.157.352.447.642.837.933.128.425.921.417.2

72.672.967.963.058.253.448.643.939.334.629.925.421.917.713.8

78.678.373.468.463.558.653.848.944.139.234.429.627.623.018.7

After applying the proposed methodology to the above data, the results are exhibitedin table 19.

The following table 19 explicitly focuses on the age groups playing dominant role inenriching the life expectancy.


Table-19: Predominant age groups affecting the life expectancyYearCountry/State

2016 2021 2026

Male Female Male Female Male FemaleIndia 70+ 70+ 70+ 70+ 70+ 70+Kerala 70+ 70+ 70+ 70+ 70+ 70+Maharashtra 70+ 70+ 70+ 70+ 70+ 70+Uttar Pradesh 0-4 70+ 0-4 70+ 70+ 70+

4 Conclusion

The propensity of life expectancy employing mortality data has been prevised effec-tively through game theory. Application of the method for sex wise data has evincedthat age group of 70+ plays dominant role in elevating longevity for all the three yearsin India. Similar consequences have been achieved for Kerala and Maharashtra. Butin Uttar Pradesh in 2016 & 2021, age group of 0-4 helps in augmenting life expectancyfor male while for female; it is 70+ age group which elevates the longevity. In 2026,for both sexes age group of 70+ helps in enriching the longevity. Hence it can beconceded that with the passage of time the old age mortality is becoming influentialin escalating the longevity. Erudition on the behaviour of mortality is essential fordecision maker for framing suitable policies as it affects the longevity. Therefore it isimperative to forecast the behaviour of mortality rate and life expectancy and thusprevise the age groups affecting the life expectancy for future planning. Optimisationperspective helps us in acquiring knowledge in this direction. For future research gametheory concept can be employed effectively for analysing other demographic problems.

References

[1] Frans Van Poppel, Dirk J. Van de kaa & Govert E. Bijwaard (2013). Life ex-pectancy of artists in the low countries from the fifteenth to the twentieth century.Population Studies, 67:3, 275-292.

[2] Frederick S. Hillier & Gerald J. Liebermann (2001). Introduction to OperationsResearch, (7th edition), Mc Graw- Hill.

[3] Gabriele Doblhammer (2000). Productive history and mortality later in life: Acomparative study of England and Wales and Austria. Population Studies.169-176, 54.

[4] Issac Sasson (2016).Trends in life expectancy and life span variation by educationalattainment: United States, 1990-2010. Demography, 53(2): 269-293.


[5] Jackie Li (2013). A Poisson common factor model for projecting mortality andlife expectancy jointly for females and males. Population Studies, Vol.67, No. 1,111-126.

[6] John Von Neumann & Oskar Morgenstern (2004). Theory of Games and EconomicBehavior, (60th Anniversary edition), Princeton.

[7] Joshua R. Goldstein and Thomas Cassidy (2012). How slowing senescence trans-lates into longer life expectancy. Population Studies, Vol. 66, No. 1, pp. 29-37.

[8] K. S. Bastian & Mohanachandran Nair (2009). Mortality changes and lifeexpectancy-an application of Operations Research. Demography India, Vol. 38,No. 1, pp. 103-116.

[9] Les Mayhew & David Smith (2015). On the decomposition of life expectancy andlimits of limits. Population Studies. Vol. 69, No. 1, 73-89.

[10] Manisha Dubey (2016). Trends and prospects of mortality by age and sexing India:1991-2030. International Institute of Population Sciences (IIPS).

[11] P. K. Tripathy (2007). Operations Research: Methods and Practice, (2nd edition),Kalyani Publishers, New Delhi.

[12] P. K. Tripathy and A Pati (2003). Bio-social relationship in fertility transitionfor promoting family planning programme in India. Population Stabilisation andDevelopment. Edited Book of International Conference on Population and De-velopment, Editor-B. Mishra, Published by- Council of Cultural Relations andCultural Growths & Supported by UNFPI & PFI, India, pp. 780-788.


Vol. 15, 2015, pp 37-54


Decision Support in a Credit Environment with FuzzyBehaviour of Cost

P. K. Tripathy and Sujata SuklaP.G. Department of Statistics

Utkal University, Bhubaneswar 751004, IndiaEmail:[email protected], [email protected]

[Received February 9, 2017; Revised May 17, 2017; Accepted October 22, 2017;Published December 30, 2017]

Abstract

In our present study demand is assumed to be proportional to the time which isdependent on permissible trade credit. Necessary and sufficient conditions havebeen discussed to frame up permissible trade credit period and purchase quan-tity. The default risk emerging in sales revenue is incorporated in the objectiveof profit maximization. Both crisp and fuzzy models have been proposed todetermine the optimal solution. Ordering cost, purchase cost, holding cost andselling price are considered as triangular as well as trapezoidal fuzzy numbers.Defuzzification of the seller’s annual profit has been carried out by graded meanintegration method and signed distance method. The eminence of fuzzy modelover the crisp model in exalting profit, reducing credit period and achievingoptimal solution, has been avowed through numerical examples.

Keywords and Phrases: Credit Period, Default Risk, Fuzzy Number, De-fuzzification.

AMS Classification: 90B05

1 Introduction

To meet the dynamic pace of the retail sector, the seller being the decision maker,offers the credit period to settle the account which woos the buyers and enhances themarket demand. This also evolves default risk for the suppliers. Most of the modelsdiscussed in the retail market situations have been studied in the crisp environment.However, in the real world, especially for new products, the relevant precise informa-tion is not possible to get due to lack of historical data. Moreover, in today’s highly


competitive market useful statistical data are not available. Thus fuzzy theory ratherthan crisp theory is well suited to this type of supply chain.

The traditional inventory model tacitly assumes that as soon as the buyers purchasethe item, they have to pay for it. But in real situation it is not always practicable tomake payments at the time of purchasing. Therefore, the offer of credit period hasbeen introduced, which has changed the entire market scenario. The offer of creditperiod is beneficial for the sellers as well as for the buyers, as it enhances the marketdemand for that product without offering discount and favors the buyers to receiveitems with lesser price as they will pay later.

Goyal (1985) first introduced the economic order quantity model by allowing permissi-ble delay in payment. Shah et al. (2015) developed an economic order quantity modelto find out the optimal credit period and purchase quantity for the seller. Jaggi et al.(2008) introduced retailer’s optimal replenishment decisions with credit linked demandunder permissible delay in payments. Tripathy and Pradhan (2011) elaborated on anintegrated partial backlogging inventory model having weibull demand and variabledeterioration rate with the effect of trade credit. Shah et al. (2014) developed an op-timal pricing and ordering policies for inventory system under trended demand whenthe supplier offers a credit period to the retailer and the retailer also gives a creditperiod to his customers. Chung and Haung (2003) developed an optimal cycle time foreconomic order quantity model under permissible delay in payments. Chang (2004)carried out an economic order quantity model with deteriorating items under inflationwhen supplier credits are linked to order quantity.

In the above cases, it has been assumed that the inventory parameters are crisp orprecise or probabilistic but in reality they may deviate a little from their actual valuewithout following any probability distribution. To deal with such type of uncertaintyin inventory parameters, the notion of fuzziness has been initialized by several au-thors (Zimmermann (2001) and Lee (2005)). This model has also been maneuveredby various researchers. Tripathy et al. (2011) established a fuzzy economic orderquantity model with reliability where the unit cost depends on demand. Dutta andKumar (2012) developed a fuzzy inventory model without shortage where holding costand ordering cost were taken as trapezoidal fuzzy numbers. Tripathy and Pattnaik(2009) focused on optimal disposal mechanism by considering the system cost as fuzzyunder flexibility and reliability criteria. Kundu and Goswami (2003) introduced anEPQ inventory model involving fuzzy demand rate and deterioration rate. Yao et al.(2000) established a fuzzy inventory model without backorder where order quantityand total demand were treated as fuzzy numbers. Tripathy and Behera (2016) formu-lated a fuzzy inventory model for time deteriorating items using penalty cost under thecondition of infinite production rate. Dutta and Kumar (2013) explored an optimalordering policy for an inventory model for deteriorating items without shortage where

Tripathy and Sukla: Decision Support in a Credit Environment 39

demand rate, ordering cost and holding cost were taken as fuzzy in nature. Jaggi et al.(2012) introduced a fuzzy inventory model for deteriorating items with time varyingdemand and shortages. Mahata and Goswami (2007) elaborated on an economic orderquantity model under the condition of permissible delay in payments in fuzzy sense.Shah et al. (2012) proposed an economic order quantity model under the condition ofpermissible delay in payments in fuzzy sense.

In the study developed by Shah et al. (2015), demand is assumed to be proportionalto the time and is dependent on permissible trade credit. Necessary and sufficient con-ditions have been discussed to frame up permissible trade credit period and purchasequantity. The default risk emerging in sales revenue is incorporated in the objective ofprofit maximization. Here the optimal solution is obtained only in crisp environment.The present study elaborates the above model in fuzzy environment and comparesthe result obtained in crisp and fuzzy environment to prove the superiority of theproposed model over the crisp model in achieving the optimal solution. Here orderingcost, purchase cost, holding cost and selling price are considered as triangular as wellas trapezoidal fuzzy numbers. Defuzzification of the Seller’s annual Profit has beenaccomplished by graded mean integration method and signed distance method. Nu-merical examples have been cited as a proof of the validation of the model. Sensitivityanalysis has also been carried out to prove the credibility of the proposed model overcrisp model.

2 Notations and Assumptions

The following notations and assumptions are used to develop the model.

2.1 Notations

(i) A and A-ordering cost and fuzzy ordering cost per order

(ii) C and C-purchase cost and fuzzy purchase cost per unit

(iii) P and P -selling price and fuzzy selling price per unit

(iv) h and h-holding cost and fuzzy holding cost per unit per annum

(v) M - credit period offered by the seller to his buyers (a decision variable)

(vi) R(M,T )-time and credit period dependent annual demand rate

(vii) I(t)-inventory level at any instant of time t, 0 ≤ t ≤ T

(viii) T -cycle time (a decision variable)

(ix) Q-seller’s purchase quantity


(x) π(M, T) and π (M,T )-the total average profit of the seller per unit time in crispand fuzzy environment.

(xi) π(M,T )SD and π(M,T )GM -defuzzified profit by using signed distance and gradedmean integration method

2.2 Assumptions

(i) The seller deals with single item, for which replenishment rate is infinite.

(ii) Shortages are not permitted. Lead time is zero or negligible.

(iii) In global market, seller keeps selling price constant to bind his retailers.

(iv) Trade credit is similar to price discount. Demand rate is considered to be functionof time and credit period

R (M, t) = a (1 + bt)Mβ (1)

where a>0 is scale demand, 0 ≤ b < 1 denotes rate of change of demand withrespect to time and β>0 is constant.

(v) For seller, default risk increases when longer credit period offered to the buyer.

Here, the rate of default risk giving the credit period M is assumed to be

F (M) = 1−M−γ (2)

where γ>0 is a constant.

3 Mathematical Model

3.1 Crisp Model

The seller’s inventory is depleting due to increasing demand and offer of credit period.The rate of change of inventory is governed by the differential equation

dI(t)

dt= −R(M, t) (3)

with I (T ) = 0. The solution of differential equation (3) is

I (t) = aMβ(T − t+1

2b(T 2 − t2)) (4)

Initially, the seller has Q units in inventory system i.e.

Q = I (0) = aMβ(T +1

2bT 2) (5)

The relevant costs per cycle for the seller are


• Net revenue after default risk: SR=P∫ To R (M, t) dt(1− F (M))

• Purchase cost ; PC= CQ=CaMβ(T + 12bT

2)

• Ordering cost ; OC=A

• Holding cost ; HC= haMβ(T2

2 + 13bT

3)

Hence, the seller’s annual profit per unit time is

π(M,T ) =1

T(SR− PC −OC −HC)

=1

T

(aPM−γ+β(T +

1

2bT 2)− CaMβ(T +

1

2bT 2)− haβ(

T 2

2+

1

3bT 3)−A

)(6)

For maximizing annual profit per unit time with respect to credit period and cycletime, the necessary and sufficient conditions are

∂π(M,T )∂M = 0, ∂π(M,T )

∂T = 0 and∣∣∣∣∣ ∂2π(M,T )∂M2

∂2π(M,T )∂M ∂T

∂π(M,T )∂T ∂M

∂π(M,T )∂T 2

∣∣∣∣∣ > 0 (7)

3.2 Fuzzy Model

Due to uncertainty in the environment, we assume some of the parameters of the in-ventory system like A, C, h and P may change within certain limits. Here we haveconsidered h, A, C and P are trapezoidal as well as triangular fuzzy number.

The fuzzy total profit is given by

π(M,T ) =1

T

(aPM−γ+β(T +

1

2bT 2)− CaMβ(T +

1

2bT 2)− haβ(

T 2

2+

1

3bT 3)A

)(8)

The total profit π(M,T ) has been defuzzified by employing graded mean integrationmethod and signed distance method.

When P = (P1, P2, P3, P4,), C = (C1,C2,C3,C4), h = (h1, h2, h3, h4 ) and A=(A1, A2, A3, A4)are trapezoidal fuzzy numbers, by using signed distance method for defuzzification,we have

πSD(M,T ) =1

4[πSD1(M,T ) + πSD2(M,T ) + πSD3(M,T ) + πSD4(M,T )] (9)


where

πSD1(M,T ) =1

T

(aP1M

−γ+β(T +1

2bT 2)− aC1M

β(T +1

2bT 2)− h1aβ(

T 2

2+

1

3bT 3)−A1

)

πSD2(M,T ) =1

T

(aP2M

−γ+β(T +1

2bT 2)− aC2M

β(T +1

2bT 2)− h2aβ(

T 2

2+

1

3bT 3)−A2

)πSD3(M,T ) =

1

T

(aP3M

−γ+β(T +1

2bT 2)− aC3M

β(T +1

2bT 2)− h3aβ(

T 2

2+

1

3bT 3)−A1

)πSD4(M,T ) =

1

T

(aP4M

−γ+β(T +1

2bT 2)− aC4M

β(T +1

2bT 2)− h4aβ(

T 2

2+

1

3bT 3)−A4

)Using graded mean integration method, the total profit is given by

πGM (M,T ) =1

6[πGM1(M,T ) + 2πGM2(M,T ) + 2πGM3(M,T ) + πGM4(M,T )] (10)

where

πGM1(M,T ) =1

T

(aP1M

−γ+β(T +1

2bT 2)− aC1M

β(T +1

2bT 2)− h1aβ(

T 2

2+

1

3bT 3)−A1

)

πGM2(M,T ) =1

T

(aP2M

−γ+β(T +1

2bT 2)− aC2M

β(T +1

2bT 2)− h2aβ(

T 2

2+

1

3bT 3)−A2

)πGM3(M,T ) =

1

T

(aP3M

−γ+β(T +1

2bT 2)− aC3M

β(T +1

2bT 2)− h3aβ(

T 2

2+

1

3bT 3)−A1

)πGM4(M,T ) =

1

T

(aP4M

−γ+β(T +1

2bT 2)− aC4M

β(T +1

2bT 2)− h4aβ(

T 2

2+

1

3bT 3)−A4

)When P = (P1 ,P2 ,P3), C = (C1 ,C2 ,C3 ), h = (h1 ,h2 ,h3 ) and A=(A1, A2, A3)

are triangular fuzzy numbers, by using signed distance method for defuzzification, wehave

πSD(M,T ) =1

4[πSD1(M,T ) + 2πSD2(M,T ) + πSD3(M,T )] (11)

where

πSD1(M,T ) =1

T

(aP1M

−γ+β(T +1

2bT 2)− aC1M

β(T +1

2bT 2)− h1aβ(

T 2

2+

1

3bT 3)−A1

)

πSD2(M,T ) =1

T

(aP2M

−γ+β(T +1

2bT 2)− aC2M

β(T +1

2bT 2)− h2aβ(

T 2

2+

1

3bT 3)−A2

)πSD3(M,T ) =

1

T

(aP3M

−γ+β(T +1

2bT 2)− aC3M

β(T +1

2bT 2)− h3aβ(

T 2

2+

1

3bT 3)−A1

)


Using graded mean integration method, the total profit is given by

πGM (M,T ) =1

6[πGM1(M,T ) + 4πGM2(M,T ) + πGM3(M,T )] (12)

where

πGM1(M,T ) =1

T

(aP1M

−γ+β(T +1

2bT 2)− aC1M

β(T +1

2bT 2)− h1aβ(

T 2

2+

1

3bT 3)−A1

)

πGM2(M,T ) =1

T

(aP2M

−γ+β(T +1

2bT 2)− aC2M

β(T +1

2bT 2)− h2aβ(

T 2

2+

1

3bT 3)−A2

)πGM3(M,T ) =

1

T

(aP3M

−γ+β(T +1

2bT 2)− aC3M

β(T +1

2bT 2)− h3aβ(

T 2

2+

1

3bT 3)−A1

)The equations (9), (10), (11) and (12) satisfy the necessary and sufficient conditionspresented in the equation (7).

4 Numerical Example

4.1 Example-1

Crisp ModelLet A=$80 per order, a=1250 units, b=50%, h=$6 per unit, C=$10 per unit, P=$16per unit, β=6, γ=3. The solution of the crisp model is: credit period M= 0.890417years, cycle time T = 0.430106 years, purchase quantity Q =296.757 units and totalprofit π(M,T ) =$7632.70.

Fuzzy Model

Case-IA= (76, 80, 83), C=(5, 10, 13), h=(3, 6, 8)and P=(14, 16, 19) are considered astriangular fuzzy numbers.Scenario-1:On applying signed distance method for defuzzification: M= 0.917546 years,T= 0.398403 years, Q =326.764 units and πSD(M,T ) =$8296.1.Scenario-2:On applying graded mean integration method for defuzzification: M= 0.908367 years,T= 0.408794 years, Q =316.404 units and πGM (M,T ) =$8238.04.

Case-IIA= (75, 78, 82, 84), C= (7, 9, 11, 13), h= (3, 5, 7, 9) and P= (15, 18, 20, 23) areconsidered as trapezoidal fuzzy numbers.


Scenario-1:

On applying signed distance method for defuzzification: M= 0.877927 years, T=0.382267years, Q =239.699 units and πSD(M,T ) =$7552.22.

Scenario-2:

On applying graded mean integration method for defuzzification: M=0.880744 years,T=0.387836 years, Q =248.226 units and πGM (M,T ) =$7600.86.

4.2 Sensitivity Analysis

Table 1: Sensitivity analysis for Crisp model in example-1ChangingParameters

Change(%)

CreditPeriod (M)

CycleTime (T)

Total Profitp(M,T )

A

+40%+20%-20%-40%

0.8839200.8870230.8942000.898535

0.5087910.4710140.3850210.333988

7564.537597.197671.967716.47

C

+40%+20%-20%-40%

0.7370190.8197490.9685721.071830

1.8068300.7967790.2626970.163525

5767.586505.519378.712327.5

h

+40%+20%-20%-40%

0.8965770.8951840.867360*

0.2567640.3124550.887910*

7358.217477.927883.64*

a

+40%+20%-20%-40%

0.8959770.8935370.8862230.880132

0.3640230.3928870.4807210.555427

10766.49198.126071.034514.66

b

+40%+20%-20%-40%

0.8244950.8711130.8991430.904016

1.2906000.6639830.3283710.272541

8074.617806.617502.967396.53

β

+40%+20%-20%-40%

0.9898850.953084**

0.1982960.255105**

6693.046931.10**

γ

+40%+20%-20%-40%

**0.9549301.037730

**0.2367820.157251

**6839.476776.64

P

+40%+20%-20%-40%

1.0076400.9525720.8183780.731317

0.30833200.35898800.53765400.7207440

15168.111080.34825.402658.61

*indicates infeasible solution


The following figures exhibit the effect of different system parameters on credit period,cycle time and total average profit.

Fig.1: Sensitivity analysis for credit period

Fig.2: Sensitivity analysis for cycle Time

Fig.3: Sensitivity analysis for Total Profit

It is evident from the fig.1 that the credit period increases with ascent in selling priceand holding cost but it decreases with rise in purchase cost and rate of change indemand. For other parameters it remains constant.

Slightly opposite behavior of fig.1 is attained by fig.2, which demonstrates that cycletime increases with incline in purchase cost and rate of change in demand and declineswith increase in selling price and holding cost.


It is elucidated from fig.3 that the total profit increases when selling price and scaledemand increase and declines when unit purchase cost and trade credit elasticityincrease.

4.3 Comparative Analysis

Table 2: Fuzzy model when A, h, P and C are triangular fuzzy numbersMethod employed M T Q π(M,T )

Graded Mean IntegrationMethod

0.908367 0.408794 316.404 8238.04

Signed Distance Method 0.917546 0.398403 326.764 8478.49

Table 3: Fuzzy model when A, h, P and C are trapezoidal fuzzy numbersMethod employed M T Q π(M,T )


0.948950 0.362350 360.710 10846.5


Table 4: Fuzzy model when h, P and C are triangular fuzzy numbersMethod employed M T Q π(M,T )


0.926269 0.170704 140.516 8294.52


Table 5: Fuzzy model when h, P and C are trapezoidal fuzzy numbersMethod employed M T Q π(M,T )


0.968538 0.150249 160.855 11105.6


4.4 Example-2

Crisp ModelLet A=$100 per unit, a=1500 units, b=60%, h=$8 per unit, C=$12 per unit, P=$18per unit, β=9, γ=4. The solution of the crisp model is: credit period M= 0.932546,cycle time T =0.297535 years, purchase quantity Q =319.729 units and total profitπ(M,T ) =$8882.5.


Fuzzy Model

Case-I

A=(95, 100, 103), C=(9, 12, 16), h=(3, 8, 10)and P=(16, 18, 21) are considered astriangular fuzzy numbers.

Scenario-1:On applying signed distance method for defuzzification: M=0.929376 years, T=0.358125years, Q=307.702 units and πSD(M,T ) =$9064.34.

Scenario-2:

On applying graded mean integration method for defuzzification: M=0.93057 years,T=0.334404 years, Q=288.818 units and πGM (M,T )=$9001.24.Case-II

A=(95, 97, 105, 108), C= (9, 10, 13, 15), h=(5, 7, 9, 12) and P=(16, 19, 21, 23) areconsidered as trapezoidal fuzzy numbers.

Scenario-1:On applying signed distance method for defuzzification: M=0.962322 years, T=0.24744years, Q=282.191 units and πSD(M,T )=$11263.6.

Scenario-2:

On applying graded mean integration method for defuzzification: M=0.965224 years,T=0.246037 years, Q=288.186 units and πGM (M,T )=$11484.1.

4.5 Sensitivity Analysis

Table 6: Sensitivity analysis for Crisp model in example-2ChangingParameters

Change (%) CreditPeriod (M)

CycleTime(T)

Total Profitπ(M,T )

A

+40%+20%-20%-40%

0.9284750.9304230.9349030.937589

0.3527720.3262500.2659090.230156

8759.478818.388953.509034.14

C

+40%+20%-20%-40%

0.8429700.8852710.9902631.067410

0.7094720.4523650.1961550.124317

6054.137189.1311589.716443.7

h

+40%+20%-20%-40%

0.9336940.9336030.9280810.902590

0.2028670.2370260.4447261.175870

8540.678694.549139.999639.17

a

+40%+20%-20%-40%

0.9360060.9344910.9299210.926089

0.2511910.2714250.3330640.385509

12581.310729.37041.765212.33

b

+40%+20%-20%-40%

0.9181150.9271400.9359940.938364

0.4905260.3690240.2524120.221605

9237.209041.788746.868627.49

β

+40%+20%-20%-40%

0.9919490.969049**

0.1696290.208922**

7813.158102.63**

γ

+40%+20%-20%-40%

*0.8771460.9696581.02094

*0.8787040.1968450.138028

*10378.77995.097872.86

P

+40%+20%-20%-40%

1.0220700.9803750.8759590.805643

0.2034630.2420630.3837920.535846

19389.613575.35260.442649.90

*indicates infeasible solution


The following figures exhibit the effect of different system parameters on credit period,cycle time and total average profit.

Fig.4: Sensitivity analysis for credit period

Fig.5: Sensitivity analysis for cycle Time

Fig.6: Sensitivity analysis for Total Profit

It is stipulated by fig.4 that the credit period increases with increase in selling priceand holding cost and decreases with increase in purchase cost and rate of change indemand. For other parameters it remains constant.

Referring to fig.5, it can be observed that it behaves oppositely to fig.4. Here thecycle time increases with increase in purchase cost, scale demand and rate of change


in demand and decreases with increase in selling price, holding cost.

As exhibited by fig.6 that the total profit increases when selling price and scale demandincrease and declines when purchase cost and trade credit elasticity increase.

4.6 Comparative Analysis

Table 7: Fuzzy model when A, h, P and C are triangular fuzzy numbersMethod employed M T Q π(M,T )

Graded Mean Integra-tion Method

0.930570 0.334404 288.818 9001.24


Table 8: Fuzzy model when A, h, P and C are trapezoidal fuzzy numbersMethod employed M T Q π(M,T )


0.965224 0.246037 288.186 11484.1


Table 9: Fuzzy model when h, P and C are triangular fuzzy numbersMethod employed M T Q π(M,T )


0.944255 0.139128 129.738 9351.07


Table 10: Fuzzy model when h, P and C are trapezoidal fuzzy numbersMethod employed M T Q π(M,T )


0.977354 0.0992367 124.730 11974.5


Behavior of profit, credit period, cycle time and purchase quantity in both the exampleshas been presented in the following figures.

Fig.7: Behaviour of profit in example-1


Fig.8: Behaviour of credit period in example-1

Fig.9: Behaviour of cycle time in example-1

Fig.10: Behaviour of purchase quantity in example-1

Fig.11: Behaviour of profit in example-2


Fig.12: Behaviour of credit period in example-2

Fig.13: Behaviour of cycle time in example-2

Fig.14: Behaviour of purchase quantity in example-2

5 Result & Discussion

Though utmost profit is attained by trapezoidal fuzzy number in both the examples(fig. 7&11), by treating four parameters like holding cost, unit purchase cost, orderingcost and selling price and three parameters like holding cost, unit purchase cost andselling price, as fuzzy, it cannot be adjudged as ideal one owing to possessing of lengthycredit period but shorter cycle time (fig. 8,9,12 and 13).

The next highest profit is achieved through triangular fuzzy number in both the il-


lustrations (fig. 7&11), considering four parameters and three parameters as fuzzy, asstated earlier. It can be treated as ideal one as compared to trapezoidal fuzzy numberand crisp method, as it possesses shorter credit period and lengthy cycle time. Nowcomparing the two methods, signed distance and graded mean integration, of defuzzi-fication, both can be deemed as equally effective in achieving our goal as they acquirelittle difference. From figures 7,8,11 & 12, it is elucidated that the profit is acceleratedwith longer credit period.

6 Conclusion

The determination of optimal credit period and purchase quantity is inevitable forthe seller. But attainability of maximum profit is not easily accessible as it is swayedby many constraints. The present paper quests on enriching the profit of the sellerunder the constraint of default risk. The proposed model is adorned with two differ-ent fuzzy numbers like triangular and trapezoidal. For defuzzification, signed distanceand graded mean integration method have been expended. Sensitivity analysis hasbeen accomplished both for crisp and fuzzy model. Induction of sensitivity analysisfor fuzzy model, considering four parameters (holding cost, unit purchase cost, or-dering cost and selling price) and three parameters (holding cost, unit purchase costand selling price) as fuzzy, enables us to toughen the conclusion up. Result achievedelucidates the importance of fuzzy model over the crisp model in ratcheting up theprofit, reducing the credit period and obtaining optimal purchase quantity. Opting forfour parameters (holding cost, unit purchase cost, ordering cost and selling price) astriangular fuzzy number can be regarded as more productive in reinforcing our goal.Furthermore, both signed distance and graded mean integration methods of defuzzi-fication can be treated as equally effective in enabling us in maximizing the profit,minimizing credit period and attaining optimal purchase quantity.

The proposed model can be extended by introducing partial credit period, shortagesand deterioration of the items.

References

[1] Chandra K. Jaggi, Sarla Pareek, Anuj Sharma, Nidhi (2012). Fuzzy inventorymodel for deteriorating items with time varying demand and shortages. AmericanJournal of Operational Research, 2(6), 81-92.

[2] C. K. Jaggi, S. K. Goyal and S.K. Goel (2008). Retailer’s optimal replenishmentdecisions with credit linked demand under permissible delay in payments. Eur. J.Operations Research, 190(1), 130-135.


[3] C. T. Chang (2004). An EOQ model with deteriorating items under inflationwhen supplier credits linked to order quantity. International Journal of ProductionEconomics, 88(3), 307-316.

[4] D. Dutta and P. Kumar (2012). Fuzzy inventory model without shortage usingtrapezoidal fuzzy number with sensitivity analysis. IOSR Journal of Mathematics,4(3), 32-37.

[5] D. Dutta and P. Kumar (2013). Optimal ordering policy for an inventory modelfor deteriorating items without shortages by considering fuzziness in demand rate,ordering cost and holding cost. International Journal of Advanced Innovation andResearch, 2(3), 320-325.

[6] G. C. Mahata and A. Goswami (2007). An EOQ model for deteriorating itemsunder trade credit financing in the fuzzy sense. Production Planning & Control,18(8), 618-692.

[7] H. J. Zimmermann (2001). Fuzzy set theory-and its applications. Springer Science+ Business Media, LLC, 4th edition.

[8] J. S. Yao, S. C. Chang and J. S. Su (2000). Fuzzy inventory without backorder forfuzzy order quantity and fuzzy total demand quantity. Computer and OperationsResearch, 27, 935-962.

[9] K. J. Chung, Y. F. Haung (2003). The optimal cycle time for EPQ inventorymodel under permissible delay in payments. International Journal of ProductionEconomics, 84(3), 307-318.

[10] Kwang Hyung Lee (2005). First course in fuzzy theory and applications, Springerpublication.

[11] N. H. Shah, S. Pareek and Isha Sangal. (2012). EOQ in fuzzy environment andtrade credit. International J. of Industrial Engineering and Computations, 3(2),133-144.

[12] N.H. Shah, Dushyant kumar, G. Patel, Digesh kumar, B. Shah (2014). Optimalpricing and ordering policies for inventory system with two level trade creditsunder price sensitive trended demand. International Journal of Applied and Com-putational Mathematics, 1(1), 101-110.

[13] N. H. Shah, Digesh kumar, B. Shah, Dushyant kumar, G. Patel (2015). Op-timal credit period and purchase quantity for credit dependent trended demand.Opsearch, 52(1),101-107.

[14] P. K. Tripathy and M. Pattnaik (2009). Optimal disposal mechanism with fuzzysystem cost under flexibility and reliability criteria in non-random optimisationenvironment. Applied Mathematical Sciences, Vol.3, No.37, 1823-1847.


[15] P. K. Tripathy, P. Tripathy, M. Pattnaik (2011). A fuzzy EOQ model with relia-bility and demand depended unit cost. Int. J. Contemp. Math. Science, Vol.6, No.30, 1467-1482.

[16] P. K. Tripathy, S. Pradhan (2011). An integrated partial backlogging inventorymodel having weibull demand and variable deterioration rate with the effect oftrade credit. International Journal of Science and Engineering Research, vol.2, 4.

[17] P. K. Tripathy and N. P. Behera (2016). Fuzzy EOQ model for time-deterioratingitems using penalty cost. American Journal of Operational Research, 6,(1): 1-8.

[18] S. K. De Kundu, P. K. Goswami, A. (2003). An EPQ inventory model involvingfuzzy demand rate and fuzzy deterioration rate. Journal of Applied Mathematicsand Computing, 12(1-2), 251-260.

[19] S. K. Goyal (1985). Economic Order Quantity under conditions of permissibledelay in payments. Journal of Operations Research Society, 36(4), 335-338.


Vol. 15, 2015, pp 55-64


Simulated Tests for Normality: A Comparative Study

Mezbahur RahmanDepartment of Mathematics and Statistics

Minnesota State UniversityMankato, MN 56001, USA

Thevaraja MayooranDepartment of Mathematics and Statistics

Minnesota State UniversityMankato, MN 56001, USA

[Received May 10, 2017; Revised July 20, 2017; Accepted August 14, 2017;Published December 30, 2017]

Abstract

The subject of assessing whether a data set is from a specific distribution hasreceived a good deal of attention. This topic is critically important for the nor-mal distribution. Often the distributions of the test statistics are intractable.Here we consider simulation based distributions for several commonly usednormality test statistics, such as, Anderson-Darling A2 test, Chi-square test,Shapiro-Wilk W test, Shapiro-Francia W ′ test, D’Agostino-Pearson test, andJarque-Bera test. Practitioners are used to with the Chi-square test because allother tests are dependent on specialized tables and/or software. Here, we givealgorithms, how those specialized tables can be generated and then the respec-tive tests can be implemented without much difficulty. A power comparison isalso performed using simulation.

Keywords and Phrases: Central moments; Kurtosis; Legendre polynomials;Monte-Carlo simulation; Normal score; Skewness.

AMS Classification: Primary 62J02; Secondary 62J20.

1 Introduction

The subject of assessing whether a data set is from a specific distribution has receiveda good deal of attention. This topic is critically important for normal distributions.


Recently, Wah (2011), Rahman and Wu (2013a), and Rahman and Wu (2013b) com-pared several normality tests. Here we compare six different commonly used paramet-ric goodness of fit tests for normality through simulation. In this study, we considersimulation based distributions for several commonly used normality test statistics, suchas, Anderson-Darling A2 test, Chi-square test, Shapiro-Wilk W test, Shapiro-FranciaW ′ test, D’Agostino-Pearson test, and Jarque-Bera test. A brief motivation for thisstudy is given in Section 2. Descriptions of all the seven (previously mentioned sixplus traditional Chi-square test) tests considered are given in Section 3. In Section 4,a power comparison is presented. A brief conclusion based on the simulation resultsis given in Section 5. In Appendix, the simulation results are provided.

2 Motivation

The most commonly used method to check for normality is the Normal Probabil-ity Plot. The most commonly used method to test for normality is the Chi-squaregoodness-of-fit test, which is very simple to comprehend and very easy to implement.Some softwares are developed to implement other specialized tests but the accessibil-ity of these softwares is limited to the general practitioners. Most tests rely on tableswhich are also not easily accessible. Here we give brief descriptions of the tests underconsideration, implementation of algorithms, and finally a comparison using simula-tion. In addition, we also consider traditional Chi-square goodness-of-fit test alongwith simulation based Chi-square test.

3 Tests for Normality

3.1 Anderson-Darling Test

A distribution function test is suggested by Anderson and Darling (1952). TheAnderson-Darling A2 statistic is computed as

A2 = −n− 1

n

n∑i=1

(2i− 1) lnΦi + ln(1− Φi) , (1)

where Φi’s are the normal cumulative distribution function (CDF) value for the ith

ordered data point. Large sample approximations of the percentiles were given byAnderson and Darling (1952) based on simulation and then through polynomial fit-tings with respect to the sample size. Due to computational developments, now onecan use simulation to generate the percentiles of the A2 statistic for the sample sizeunder consideration and the null hypothesis that the data is from the standard normaldistribution. Then a p-value can be obtained for the observed A2 statistic for the dataat hand.

Rahman and Mayooran: Simulated Tests for Normality 57

3.2 Chi-square Test

Pearson’s chi-squared test is used to assess two types of comparison: tests of goodnessof fit and tests of independence. In this paper, we use it to establish whether ornot an observed frequency distribution differs from a hypothetical distribution. Thetest-statistic is defined as

χ2 =

g∑i=1

(Oi − Ei)2

Ei, (2)

where g is the number of groups, Oi is an observed frequency while Ei is the expectedfrequency under the null hypothesis. Then χ2 follows approximate Chi-square dis-tribution with g − k − 1 degrees of freedom, where k is the number of parametersestimated for the distribution under consideration. A rule can be maintained that theexpected frequencies are at least 5. For moderate to small samples, the highest num-ber of groups possible should be considered. For large samples, the highest number ofgroups might over smooth in complying with Chi-square approximation. To examinesuch a behavior, different number of groups can be considered while comparisons aremade. In literature, number of classes in the Chi-square test is analyzed by Dahiyaand Gurland (1971), Hamdan (1963), Mann and Wald (1942), and Williams (1950).Here we consider expected frequencies as 5 throughout our simulations.

Due to computational developments, now one can use simulation to generate the per-centiles of the χ2 statistic for the sample size under consideration and the null hy-pothesis that the data is from the standard normal distribution. Then a p-value canbe obtained for the observed χ2 statistic for the data at hand. Here we denote suchstatistic as χ2

s, where s stands for simulation.

3.3 Shapiro-Wilk W Test

Let (X1, X2, · · · , Xn) be a random sample to be tested for departure from normality,based on the ordered sample X(1) < X(2) < · · · < X(n) , and let mn×1 denote thevector of expected values Vn×n variance-covariance matrix of the standard normalorder statistics. Shapiro and Wilk (1965) suggested the following test.Define

W =

(n∑

i=1

aiX(i)

)2

n∑i=1

(Xi − X)2, (3)

where X is the sample mean and an×1 = m′V−1

(m′V−1V−1m)1/2. Note that W equals the

square of the standard product-moment correlation coefficient between the X(i) and


ai, and therefore measures the straightness of the normal probability plot of the X(i);small values of W indicate non-normality. In literature, ai’s are tabulated for limitedsample sizes and some approximations are provided for larger samples. Here, we sug-gest obtaining ai’s using simulation and then creating a simulation of the W statisticdistribution under the null hypothesis for the sample size needed prior to implementingthe test.

3.4 Shapiro-Francia W ′ Test

W ′ =

(n∑

i=1

miX(i)

)2

n∑i=1

m2i ×

n∑i=1

(Xi − X

)2 , (4)

a modified W statistic proposed by Shapiro and Francia (1972).Sarkadi (1975) dis-cussed about consistency of this test. Implementation of W ′ is easier and gainedpopularity as it requires only the means of the order statistics of the standard nor-mal variates unlike W which also requires covariance matrix of the order statistics ofthe standard normal variates. Harter (1961) provided the approximate means of theorder statistics of the standard normal variates. Parrish (1992) provided the moreaccurate means of the order statistics of the standard normal variates using LegendrePolynomials, which are accurate up to thirty two decimal places. Following Parrish(1992), Rahman and Pearson (2000) showed that by simulation the means of the orderstatistics of the standard normal variates can be approximated pretty accurately upto about eight decimal places which serves the purpose in most cases.

Rahman and Pearson (2000) showed that through exclusive Monte-Carlo simulation,that is, incorporating approximate expected values, the W ′ percentiles can be com-puted pretty accurately, while Rahman and Ali (1999) provided revised W ′ percentilesusing Parrish (1992) expected values.

Hence, before implementing W ′ test one can easily simulate the means of the orderstatistics of the standard normal variates and W ′ percentiles and then compute thep-value for the data at hand. Similar can be done for the W statistic.

3.5 D’Agostino-Pearson Test

Pearson (1895) suggested that the following sample estimates could be used to describenonnormal distributions and used as the bases for tests of normality. Let (X1, · · · , Xn)


denote a sample of n observations. Mj ’s are the j-th sample central moments, and Xis the sample mean.

√b1 and b2 are defined as

√b1 =

M3√M3

2

=

1

n

n∑i=1

(Xi −X)3

(1

n

n∑i=1

(Xi −X)2

)3/2, (5)

and

b2 =M4

M22

=

1

n

n∑i=1

(Xi −X)4(1

n

n∑i=1

(Xi −X)2

)2 , (6)

where Mj =1n

∑ni=1(Xi −X)j .

D’Agostino et al. (1990) have done separate tests based on√b1 and b2. They indicate

how these two can be used in conjunction with normal probability plotting. They havealso provided the procedures on how to calculate the normal approximations of the teststatisticsX(

√b1) andX(b2) when the sample sizes are large enough (n > 8 and n ≥ 20,

respectively), where X(√b1) is the standard normal score for the respective percentile

position for√b1 value, and X(b2) is similarly defined as X(

√b1). In this paper, we will

obtain the estimates of these two statistics X(√b1) and X(b2) by simulations. The

test proposed by D’Agostino and Pearson (1973) combines√b1 and b2 for an omnibus

test. The test statistic is

DPC = X2(√b1) +X2(b2). (7)

Under the normal null hypothesis, DPC follows a Chi-square distribution with 2 de-grees of freedom. In literature, simulated

√b1 and b2 values, and their tables are given

for a wide range of sample sizes.

At this computational age, one can easily simulate percentiles for√b1 and b2 under

normality assumption prior to implementing the DPC test. Computational steps:

• Step 1: Simulate samples of size n from N(0, 1), compute√b1 and b2, store the

values.

• Step 2: Compute√b1 and b2 for the data, obtain percentile positions of

√b1 and

b2 in the stored respective empirical distributions in Step 1, compute DPC in(7) and then obtain the upper tail p-value by using the Chi-square distributionwith 2 degrees of freedom.


Alternatively, instead of using the Chi-square table, one can use the simulated empiri-cal distribution to determine the critical values or the p-values in making the decisions.Let us call such test DPS. Computational steps:

• Step 1: Simulate samples of size n from N(0, 1), compute√b1 and b2, store the

values.

• Step 2: Separately, take samples of size n from N(0, 1), compute√b1 and b2,

obtain the percentile positions of√b1 and b2 in the stored respective empirical

distributions in Step 1, compute DPC in (7) and store.

• Step 3: Compute√b1 and b2 for the data, obtain percentile positions of

√b1 and

b2 in the stored respective empirical distributions in Step 1, compute DPC in(7) and then obtain the upper tail p-value by using the empirical distribution inStep 2.

Rahman and Wu (2013b) showed that performance of DPS is better than DPC, hencewe will consider DPS in this study.

3.6 Jarque-Bera Test

Jarque and Bera (1987) suggested a moment based statistic

JBC =n

6

(b1 +

1

4(b2 − 3)2

), (8)

where b1 and b2 are defined above.

For large samples, JBC follows a Chi-square distribution with 2 degrees of freedom.Computational steps:

• Compute b1 and b2 for the data, compute JBC in (8) and then obtain the uppertail p-value by using the Chi-square distribution with 2 degrees of freedom.

Alternatively, instead of using the Chi-square table one can use the simulated empiricaldistribution to determine the critical values or the p-values in making the decisions.Let us call such test JBS. Computational steps:

• Step 1: Simulate samples of size n from N(0, 1), compute b1 and b2, computeJBC in (8) and store.

• Step 2: Compute b1 and b2 for the data, compute JBC in (8) and then obtainthe upper tail p-value by using the empirical distribution in Step 1.

Here, we will consider JBS in our comparisons.


Figure 1: Different Distributions

4 Power Comparison

All seven tests mentioned in Section 3 are compared using simulation. Data weregenerated from N(0, 1) to investigate the null distributions. Then data were gener-ated from some non-normal distributions, such as, Uniform(0, 1), Exponential(1),a mixture 1

4N(0, 1) + 34N(32 ,

13

), t7, Gamma(4, 5), and χ2

4 to compare the powers ofthe tests. All the alternative distributions are shown in Figure 1. Samples are con-sidered of sizes n = 20, 30, 50, and 100. In all simulations 10, 000 replications wereconsidered. Proportions of rejections (p-values when the null distribution is consid-ered) were computed and their percentiles are given in the tables in the Appendix.Levels of significances are considered 1%, 5% and 10%.

Test Abbreviation

D’Agostino-Pearson Test DPSJarque-Bera Test JBSAnderson-Darling Test ADSShapiro-Wilk Test SWSShapiro-Francia Test SFSχ2 Simulation Test C2Sχ2 Table Test C2T

We notice that when data are generated from the standard normal distribution, thatis, the null hypothesis is true, powers are the level of significances, all powers are close


to the respective level of significances, except for C2T for smaller sample sizes. ForUniform(0, 1) alternatives, JBS has poor performance compared to all others. DPShas the best performance for larger samples.

5 Concluding Remarks

In general, the further the alternative distributions are away from the normal distri-bution, the more powerful the tests are. The Chi-square test is the least powerful testfor the sample sizes considered in this study. For very large samples, the Chi-squaretest might outperform others but might not be noticeable. All other tests are morepowerful but more computational difficulty. At this computational age, the researchersshould overcome the hesitance and use one of the other tests other than the Chi-squaretest. All other tests are equally computational burdensome but in this study as weexperienced that they could be performed with some computational background andwithout much difficulty. If one has to use the Chi-square test, should use the numberof groups as high as possible at least for the samples of sizes below one hundred.

6 Acknowledgment

We would like to thank the referees for their constructive comments which improvedthe presentation of he paper significantly.

References

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[2] D’Agostino, R. B., A. Belanger, and R. B. D’Agostino, Jr. (1990). A suggestionfor using powerful and informative tests of normality. The American Statistician,44(4), 316-321.

[3] D’Agostino, R. B. and E. S. Pearson (1973). Testing for departures from normal-ity. I. Fuller empirical results for the distribution of b2 and

√b1. Biometrika, 60,

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[6] Harter, H. L. (1961). Expected values of normal order statistics. Biometrika, 48(1and 2), 151-159.

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[8] Mann, H. B. and A. Wald(1942). On the Choice of the Number of Class Inter-vals in the Application of Chi-Square Test. Annals of Mathematical Statistics,13(1942), 306-317.

[9] Parrish, R. S. (1992). Computing expected values of normal order statistics. Com-munications in Statistics — Simulation and Computation, 21(1), 57-70.

[10] Pearson, K. I. (1895). Contributions to the mathematical theory of evolution: Skewvariation of homogeneous material. Philosophical Transactions of the Royal Soci-ety of London, 186, 343-412.

[11] Rahman, M. and M. M. Ali (1999). Quantiles for Shapiro-Francia W ′ statistic.Journal of the Korean Data & Information Science Society, 10(1), 1-10.

[12] Rahman, M. and L. M. Pearson (2000). Shapiro-Francia W ′ statistic using exclu-sive Monte-Carlo simulation. Journal of the Korean Data & Information ScienceSociety, 11(2), 139-155.

[13] Rahman, M. and H. Wu (2013a). Tests for Normality: A Comparative Study. FarEast Journal of Mathematical Sciences (FJMS), 75(1), 143-164.

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[16] Shapiro, S. S. and R. S. Francia (1972). An approximate analysis of variance testfor normality, Journal of the American Statistical Association, 67(337), 215-216.

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[19] Williams, G. A., Jr. (1950). On the Choice of the Number and Width of Classesfor the Chi-Square Test of Goodness-of-Fit. Journal of the American StatisticalAssociation, 45(March 1950), 77-86.


Appendix

Rejection Proportionsα = 0.01 α = 0.05 α = 0.10

n = 20 n = 30 n = 50 n = 100 n = 20 n = 30 n = 50 n = 100 n = 20 n = 30 n = 50 n = 100

TS Normal(0, 1),√β1 = 0, β2 = 3

DPS 0.0194 0.0181 0.0192 0.0155 0.0575 0.0592 0.0534 0.0513 0.1038 0.0948 0.1011 0.0955JBS 0.0095 0.0094 0.0103 0.0116 0.0504 0.0536 0.0519 0.0502 0.1072 0.0962 0.1011 0.1017ADS 0.0128 0.0094 0.0096 0.0082 0.0516 0.0540 0.0442 0.0502 0.1014 0.1009 0.0967 0.0931SWS 0.0107 0.0107 0.0075 0.0092 0.0516 0.0560 0.0467 0.0520 0.1042 0.0999 0.1025 0.0961SFS 0.0102 0.0118 0.0102 0.0103 0.0501 0.0538 0.0470 0.0541 0.1021 0.1015 0.0975 0.1042C2S 0.0116 0.0099 0.0074 0.0091 0.0541 0.0524 0.0489 0.0519 0.1130 0.1103 0.1041 0.0944C2T 0.0199 0.0121 0.0100 0.0091 0.1085 0.0659 0.0524 0.0536 0.2475 0.1346 0.1041 0.1057

Uniform(0, 1),√

β1 = 0, β2 = 1DPS 0.0508 0.1752 0.4211 0.9712 0.1905 0.3896 0.7634 0.9953 0.2995 0.5171 0.8698 0.9983JBS 0.0001 0.0000 0.0000 0.0000 0.0214 0.0172 0.0806 0.2888 0.1442 0.1800 0.2760 0.5390ADS 0.0432 0.0888 0.2549 0.8041 0.1742 0.2989 0.5566 0.9521 0.2898 0.4406 0.7227 0.9785SWS 0.0162 0.0453 0.9300 0.9347 0.0424 0.4627 0.1572 0.4579 0.1858 0.3706 0.2976 1.0000SFS 0.0062 0.0273 0.0638 0.7368 0.0462 0.3490 0.2766 0.9261 0.0828 0.3789 0.7115 0.9978C2S 0.0243 0.0204 0.0810 0.6093 0.1079 0.1135 0.1968 0.7801 0.1791 0.1943 0.2815 0.8481C2T 0.0380 0.0211 0.0818 0.5989 0.1759 0.1135 0.1968 0.7807 0.3925 0.2372 0.2896 0.8501

Exponential(1),√


NormalMixture : 14N(0, 1) + 3

4N

(32, 13

),√

β1 = 5.4819, β2 = 5.5438


t-distribution with 7 degrees of freedom,√


Gamma(4, 5),√β1 = 1, β2 = 4.5


χ24,

√β1 = 1.41, β2 = 6



Vol. 15, 2015, pp 65-66


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