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ARFIMA processes and outliers: a weighted likelihood approachClaudio Agostinellia; Luisa Bisagliab

a Dipartimento di Statistica, Università Ca' Foscari and School for Advanced Studies in Venice,Venezia, Italy b Dipartimento di Scienze Statistiche, Università di Padova, Padova, Italy

Online publication date: 06 September 2010

To cite this Article Agostinelli, Claudio and Bisaglia, Luisa(2010) 'ARFIMA processes and outliers: a weighted likelihoodapproach', Journal of Applied Statistics, 37: 9, 1569 — 1584To link to this Article: DOI: 10.1080/02664760903093609URL: http://dx.doi.org/10.1080/02664760903093609

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Journal of Applied StatisticsVol. 37, No. 9, September 2010, 1569–1584

ARFIMA processes and outliers: a weightedlikelihood approach

Claudio Agostinellia* and Luisa Bisagliab

aDipartimento di Statistica, Università Ca’ Foscari and School for Advanced Studies in Venice, 30121Venezia, Italy; bDipartimento di Scienze Statistiche, Università di Padova, 35121 Padova, Italy

(Received 6 October 2008; final version received 4 June 2009)

In this paper, we consider the problem of robust estimation of the fractional parameter, d, in long memoryautoregressive fractionally integrated moving average processes, when two types of outliers, i.e. additiveand innovation, are taken into account without knowing their number, position or intensity. The proposedmethod is a weighted likelihood estimation (WLE) approach for which needed definitions and algorithmare given. By an extensive Monte Carlo simulation study, we compare the performance of the WLE methodwith the performance of both the approximated maximum likelihood estimation (MLE) and the robust M-estimator proposed by Beran (Statistics for Long-Memory Processes, Chapman & Hall, London, 1994). Wefind that robustness against the two types of considered outliers can be achieved without loss of efficiency.Moreover, as a byproduct of the procedure, we can classify the suspicious observations in different kindsof outliers. Finally, we apply the proposed methodology to the Nile River annual minima time series.

Keywords: ARFIMA processes; outliers; robust estimation; weighted likelihood

1. Introduction

Long memory processes have recently become very popular in the analysis of many empiricaltime series as, for instance, economic time series [2], financial intra-day series [11] and hydro-logical time series (of which the Nile River minima of Hurst [15] is a classical example). Eventhough these data appear to satisfy the assumption of stationarity, perhaps after some differencingtransformations, they typically exhibit a dependence between distant observations. The sampleautocorrelations of these series decrease to zero as a power function rather than exponentially andthe spectral density diverges as the frequencies tend to zero. The most popular class of modelsfor this particular behavior of the autocorrelation and spectral density functions are the autore-gressive fractionally integrated moving average models (ARFIMA) introduced by Granger andJoyeux [12] and Hosking [13]. This class of models generalizes the usual ARIMA(p, d, q), theintegration parameter d being allowed to assume any real number. More specifically, the form of

*Corresponding author. Email: claudio@unive.it

ISSN 0266-4763 print/ISSN 1360-0532 online© 2010 Taylor & FrancisDOI: 10.1080/02664760903093609http://www.informaworld.com

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1570 C. Agostinelli and L. Bisaglia

ARFIMA(p, d, q) models is

(1 − φ1B − · · · − φpBp)(1 − B)d(Xt − μ) = (1 + θ1B + · · · θqBq)εt , (1)

where εt ∼ N(0, σ 2) is a white noise (w.n.), B is the backward operator, i.e. Bxt = xt−1, andthe real coefficients φi and θ i are such that the AR and MA polynomials, of degrees p andq, respectively, have no common roots and all the roots lie outside of the unit circle. Finally,(1 − B)d = ∑∞

j=0 πjBj , with π j =�(j − d)/[�(j + 1)�( − d)], �( · ) is the gamma function, and

d ∈ ( − 1/2, 1/2) is called long memory or fractional integration parameter. In the following, wewill concentrate on ARFIMA(p, d, q) processes with d ∈ (0, 1/2). In fact, for this range of valuesthe process is stationary, invertible and with long range dependence. Moreover, we will assumefor simplicity and without loss of generality that μ= 0. Many authors have studied the problemof estimating the parameters of an ARFIMA(p, d, q) process by exact or approximate Gaussianmaximum likelihood methods (see [4], for a good review on these arguments). All these methodshave the drawback of being very sensitive to deviations from the Gaussian model. In particular,occasional outliers can force the estimate of the long memory parameter d to be close to zero,even in the presence of strong long memory in the data. Some contributions in the literature (see,for instance [17]) show that outliers have several adverse effects on time series analysis.

In the case of long memory processes, the outliers’ effect on parameters estimates persistslonger than for short memory processes. Thus, neglecting outliers makes the resultant inferenceunreliable or even invalid with respect to model selection, estimation and forecasting. Amongothers, Granger [10] shows the effects of the presence of outliers in time series. In most cases, thesimple remotion of the substantial outliers from the data before carrying out the analysis does notappear to be the best solution. Beran [5] proposes an approximated maximum likelihood estimatorbased on the autoregressive representation of the process and he proves that it belongs to a class ofM-estimators. By choosing an appropriate ψ − function, the robustness against isolated additiveoutliers (AO) can be achieved, while keeping the high efficiency under the ideal model. Theproblem of more general outliers scheme remains open. Our aim here is to introduce a procedurefor efficient and robust estimation of the unknown parameters ofARFIMA(p, d, q) processes whentwo types of outliers, i.e. additive and innovation, are taken into account without knowing theirnumber, position or intensity. Our method is based on the weighted likelihood [19]. Moreover,we are able to detect the outlier observations. In the next section, the necessary notation and themethod are introduced. In Section 3, the results of an extensive Monte Carlo study are reported.Section 4 presents a sensitivity analysis on real data. Finally, Section 5 concludes.

2. Weighted likelihood and robust estimation for ARFIMA models

In this section, we describe the methodology we developed based on weighted likelihood and onan outliers classification procedure. So, first we define, the weighted estimating equations in thecase of ARFIMA(p, d, q) processes. Then, we obtain the derivatives involved in the estimatingequations. Finally, we describe the outliers classification procedures and derive the algorithm.

2.1 Weighted estimating equations

Let Xt be a causal and invertible ARFIMA(p, d, q) process, with d ∈ ( − 1/2, 1/2). This process hasan AR(∞) and an MA(∞) representation [7]; then, if we knew the infinite past of Xt (t ≤T ), we

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Journal of Applied Statistics 1571

could construct the sequence of i.i.d. innovations εt (t ≤T ) by

εt = Xt +∞∑

j=1

ajXt−j , (2)

where the parameters aj are such that

(z)

∞∑j=0

ajzj = (z)

∞∑j=0

πjzj , | z |≤ 1. (3)

It is straightforward to see that in the case of the simpleARFIMA(0, d, 0) process the parametersaj are equal to the π j defined in Section 1, while for the more general ARFIMA(p, d, q) processwe have a0 = 1 and

aj = πj −min(j,p)∑

i=1

φiπj−i −min(j,q)∑

i=1

θiaj−i , j = 1, 2, . . . , (4)

(see [6] for details), hence aj = aj(η) is a function of the unknown parameters vector η= (d, φ1, . . . ,φp, θ1, . . . , θq). Then, if Xt is Gaussian the contribution of a single observation to the log-likelihoodfunction of ε1, . . . , εT is given by

LT (εt ; σ 2) = −1

2log 2π − 1

2log σ 2 − 1

2

(ε2t

σ 2

). (5)

and the log-likelihood is

LT =T∑

t=1

LT (εt ; σ 2). (6)

Since we can observe only a finite number of past values of the series, we can assume thatXt = 0 for t ≤ 0 so that we can estimate the innovations as

et (η) = Xt +t−1∑j=1

aj (η)Xt−j . (7)

The conditional log-likelihood function can be defined by replacing εt by et(η), and an approx-imate maximum likelihood estimator of the parameters ξ = (σ 2, η) is obtained by maximizingthe conditional log-likelihood function with respect to ξ (for details about this approximate MLEmethod for ARFIMA processes see [4,5]). Beran [5] proves that a generalization of Equation (6)leads to a class of M-estimators for which a central limit theorem holds. By choosing an appropri-ate Huber ψ-function [14], robustness against isolated AO can be achieved, while keeping highefficiency under the ideal model. Our approach here is different. We will introduce an algorithmbased on weighted likelihood modified by an outliers classification procedure. This method pro-vides an estimation of the unknown parameters ξ that is robust in the presence of contaminationin a general setting and that has no loss of efficiency under the correctly specified model. Weanticipate that a side effect of our procedure is a classification of the suspicious observations indifferent kinds of outliers. We define the conditional weighted likelihood estimating equation as

T∑t=1

w(et (η); σ 2, FT (η))u(et (η); σ 2) = 0, (8)

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1572 C. Agostinelli and L. Bisaglia

where u(et(η);σ 2) is the score function that can be obtained as

u(et (η); σ 2) = ∂

∂ξLT (et (η); σ 2), (9)

and w( · ) is the weight function which depends on the unknown parameters and on the empiricalcumulative distribution function, FT (η). The weight function is a measure of the agreementbetween the parametric model of the errors and the estimated distribution of the actual residualsand it is defined according to a minimum disparity measure. In particular, following Markatouet al. [19], since the εt are from a white noise process, we define the Pearson residual, δt , for theapproximated residuals et from the non-parametric kernel density estimator

f ∗(et (η); FT (η)) =∫

k(et (η); r, g) dFT (r; η) (10)

and the smoothed model

m∗(et (η); σ 2) =∫

k(et (η); r, g) dM(r; σ 2), (11)

where FT (η) is the empirical cumulative distribution function based on the residuals et(η), M(σ 2)is the model cumulative distribution (in general Gaussian), k(et(η);r, g) is a kernel density and gis its bandwidth. Now, by the definition of the Pearson residuals and weight function, we have,respectively,

δt = δ(et (η); M(σ 2), FT (η))

= f ∗(et (η), FT (η)) − m∗(et (η); σ 2)

m∗(et (η); σ 2)

and

w(et (η); M(σ 2), FT (η)) min

{1,

[A(δt ) + 1]+δt + 1

}= w(δt ),

where [ · ]+ indicates the positive part and A( · ) is the residual adjustment function (RAF) ofLindsay [18] that operates on Pearson residuals as the Huber ψ-function operates on the structuralresiduals. When A(δ) = δ the weights are equal to one and this corresponds to the maximumlikelihood estimation (MLE) method. An example of RAF is A(δ) = 2{(δ + 1)1/2 − 1} namelyHellinger RAF that we will use in our simulations and example. For further details about theweight function see Agostinelli [1] and the literature herein cited.

Hereafter, we give the details necessary to solve the estimating equations (Equation (8)) involv-ing the partial derivatives (Equation (9)). For sake of simplicity, we assume that μ is known andequal to zero. Beran [5] proves that the asymptotic result does not change if μ is substituted by aconsistent estimate. Notice that maximizing the conditional log-likelihood function is equivalentto minimizing the conditional sum of the squared residuals

S(η) =T∑

t=1

e2t (η)

with respect to η. From Equation (7) the log-likelihood function for a single observation is

proportional to(∑t−1

j=0 aj (η)Xt−j

)2and the score function becomes

u(et (η); σ 2) =⎛⎝ t−1∑

j=0

aj (η)Xt−j

⎞⎠ t−1∑

j=0

∂aj (η)

∂ηi

Xt−j t = 1, · · · , T .

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Journal of Applied Statistics 1573

This leads to a system of (p + q + 1) nonlinear conditional weighted likelihood estimatingequations of the form

T∑t=1

w(et (η); σ 2, FT (η))

⎛⎝ t−1∑

j=0

aj (η)Xt−j

⎞⎠ t−1∑

j=0

∂aj (η)

∂ηi

Xt−j = 0.

The derivative of aj with respect to d is easily computed using Equation (4) and the fact that

πj (d) = �(j − d)

�(j + 1)�(−d)

by the following recursive formula:

∂aj (η)

∂d= ∂πj

∂d−

min(j,p)∑i=1

φi

∂πj−i

∂d−

min(j,q)∑i=1

θi

∂aj−i

∂d

∂aj (η)

∂φi

= πj−i −min(j,q)∑

i=1

θi

∂aj−i

∂φi

, i = 1, 2, . . . , p

∂aj (η)

∂θi

= −aj−i − θi · ∂aj−i

∂θi

−min(j,q)∑l=1,l �=i

θl

∂aj−l

∂θi

, i = 1, 2, . . . , q.

Thus, all we need are the derivatives of π j(d) with respect to d. Using the fact that

∂ log �(x)

∂x= �(x)

with �( · ) the digamma function, we have

∂�(x)

∂x= �(x)�(x)

and then, after some algebra, it is possible to see that

∂πj (d)

∂d= ∂

∂d

�(j − d)

�(j + 1)�(−d)= �(j − d)

�(j + 1)·(

d�(j − d) − d�(−d)

�(1 − d)

). (12)

So, for example, to estimate an ARFIMA(0, d, 0) process, by weighted likelihood equations weneed only formula (12).

2.2 Outliers classification procedure

In this section, we review the definition of outliers in time series and we present the outliersclassification procedure used in our algorithm.

In the literature (see [3,8]), two different types of contamination are often defined, namely AOand innovation outliers (IO). In this setting, any contaminated time series zt can be represented asa function of the uncontaminated time series xt (following an ARFIMA model) and a componenttaking into account the outliers generating process (OGP):

zt =n∑

j=1

hjvj (B)ξt (t∗j ) + xt , (13)

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1574 C. Agostinelli and L. Bisaglia

where

• vj(B) = 1 for an AO,• vj(B) = [(B)/(B, d)] for an IO,

at time t = t∗j , hj represents the magnitude of the outliers, ξ t(t∗) = 1 if t = t∗, and = 0 otherwise,(B, d) = (1 −φ1B −· · · −φpBp)(1 − B)d , (B) = (1 + θ1B +· · · θqBq) and n is the number ofoutliers.

Note that while an additive outlier will affect only the corresponding observation, the effect ofan innovation outlier will be propagated according to (B)/(B, d).

Since we can have realizations only from zt when we apply the ARFIMA model xt to theobserved time series we have the following observed residuals rt as function of the error εt

rt = (B, d)

(B)zt = εt +

n∑j=1

hj vj (B)ξt (t∗j ), (14)

where

• vj (B) = (B, d)/(B) for an AO,• vj (B) = 1 for an IO,

We can conclude that an IO affects only one residual while the effect of an AO at time t will bepropagated to the subsequent residuals (at time t + 1, . . . ) according to the data generating process.Since the weighted log-likelihood estimating equations (Equation (8)) are defined on residuals itwill be robust against the presence of innovation outliers while it may be not robust in the presenceof AO. This behavior is very well-known in classical M-estimation (see, for instance [20]).

It is important to note that the OGP cannot be identified uniquely by the observed time series,since it is always possible to find a different OGP (maybe with infinite elements) that gener-ates the same observed time series. Hence, a classification procedure that correctly identifiesthe OGP cannot be constructed [1]. Here our goal is to provide a procedure that bounds theeffects of AO by selecting the most suitable outliers pattern to describe the actual contamination.The selection will be performed by introducing an optimal criterion based on the sum of theweights.

The procedure is as follows. Let (η, σ 2) be an initial estimate of the true parameters, et =et (η) the estimated residuals and for simplicity let w(t) = w(et ; σ 2, F (η)) be the weight of eachobservation. Let wl be a threshold level such that 0 ≤wl ≤ 1 and define O as the set that containsthe observations for which w(t) ≤ wl and let dO be its dimension. We focus our attention onlyon the observations in this set. Each observation in O might be either an additive outlier or not(i.e., an innovation outlier or a good observation). All the possible combinations are 2dO . Foreach combination, namely O(i), i = 1, · · · , 2dO , we consider a new time series, x

(i)t , and the

corresponding new set of residuals, e(i)t , evaluated recursively as

x(i)t = −

t−1∑j=1

aj (η)x(i)t−j , (15)

where

x(i)t−j =

⎧⎨⎩

xt−j t − j /∈ O(i)

x(i)t−j t − j ∈ O(i)

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Journal of Applied Statistics 1575

and

e(i)t = x

(i)t +

t−1∑j=1

aj (η)x(i)t−j .

For the set of residuals e(i)t (t /∈ O(i)) we consider the problem of estimating the associated

weights wO(i) (t) = w(e(i)t ; σ 2, F (η)) where F (η) is based on e

(i)t . Then we calculate the average

weights function

w(i) =∑

t /∈O(i) wO(i) (t)

T, i = 1, · · · , 2dO , (16)

where we assign weight zero to each observation in O(i). We choose the set O(i�) such thatw(i�) = maxi w

(i).Some remarks on the introduced algorithm follow. The observations in O(i) but not in O(i�)

could be classified as innovation outliers. Using wl = 1 means looking for a possible additiveoutlier in the whole time series, and this gives the best results. If so, dO = T . This can makethe classification step infeasible because the number of all patterns, 2T , becomes huge (even formoderate sample size). Hence, in practice, wl < 1. In particular, suitable values should be in therange 0.4–0.6 and in our work we have chosen wl = 0.5.

Without contamination the proposed procedure is asymptotically first-order efficient regardlessof the wl choice since the weights will converge uniformly to 1 and no observations will beclassified as AO.

In our work, the search for the optimal set of AO is implemented using genetic-like algorithm[9]. From the O we extract a fixed number of subsets to create the initial population. We alsoinclude all the best sets used in the previous iteration if these are subsets of O. For each ofthe sets in the population we calculate the average weight function (16). To get an offspring Awe sample two elements from the the population (with probability proportional to their averageweight function). We build a set B based on the union of the sampled sets plus a small fixednumber of elements from O. We finally sample from B using a Bernoulli scheme to have theoffspring. Often the population size could be small, for our simulations we fix the size to 6 andwe include in B two elements.

2.3 A robust estimation algorithm

In practice, given a time series, x = (x1, . . . , xT ):

(1) obtain the initial values using an approximate maximum likelihood approach (like for instancethat of [5]), {η(0), σ

2(0)} and the vector of residuals e(0);

(2) run the classification procedure described in the previous section and obtain a set of AO O(0)

and the corresponding modified time series x(0) and e(0) (if the set is empty then x(0) = x ande(0) = e(0));

(3) use η(0), σ 2(0) as starting values and x(0) as the data set to obtain a solution from the weighted

likelihood estimating equations re-weighting algorithm, say, η(1), σ 2(1);

(4) run the classification procedure using η(1), σ2(1) and the original time series x, and obtain a set

of AO O(1) and the corresponding modified time series x(1) and e(1);(5) if the new set of AO O(1) is the same as O(0) the convergence is achieved;(6) otherwise, we iterate steps 3–5 until convergence.

With this procedure, we bound the influence of any additive outlier on the future residuals andwe have a robust and more efficient estimation of the unknown parameters.

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1576 C. Agostinelli and L. Bisaglia

3. Monte Carlo study and results

This section reports some Monte Carlo experiments designed to put in evidence the robust proper-ties in finite samples of the weighted likelihood method we have proposed above. In addition, wecompare the performance of our method with the performance of the approximate maximum like-lihood approach and the robust M-estimator of Beran [5] both when no contamination is involvedand in the presence of outliers in the samples. Moreover, these experiments are conducted toinvestigate the effects of AO for different contamination levels and sample sizes. The functionsthat we use are written in R language [16] and are available upon request from the correspondingauthor. In this simulation we use the Hellinger RAF and the bandwidth needed in Equations (10)and (11) is iteratively calculated proportional to the estimate of the innovation variance by a factorof 0.0031 (for details on how to choose this value see [19]).

We use time series generated by ARFIMA(0, d, 0) models with parameter d = 0.1, 0.2, 0.3, 0.4,0.45 and normal error component with zero mean and variance σ 2 = 1. Our results are based on500 independent replications of series with different sample sizes T = 100, 250, 500 and 1000.We have generated each time series of length T 0 =T + 500 and then we have discarded the first500 observations to avoid initialization effects. Moreover, we have conducted two different exper-iments: (i) we have considered a symmetric contamination, with heavy tails generated accordingto a t2 (Table 1); (ii) we have considered also an asymmetric contamination generated accordingto a N(8, 1) (Tables 2–4). This second set of experiments has the aim of testing the goodness of thedifferent methodologies in a more difficult situation. In the experiments we present, all the out-liers are generated as AO and the contamination levels we consider are γ = 0%, 5%, 10%, 15%. Inpractice, we generate a time series, xt , from an ARFIMA(0, d, 0) process with innovation variance1 together with the realizations, et , of the innovation process. Then we build the contaminatedtime series as

yt = xt + α(t)Vt ,

where V t =ηt − et and ηt is an independent realization of a t2 (in the case of symmetric con-tamination) or a N(8, 1) (in the case of asymmetric contamination) distributed random variable;α(t) is an indicator function such that α(t) = 1 for every t in the sample of size [γT ] sampledwithout replacement from the set of index {1, 2, . . . , T}. We use this contamination scheme to havea fixed number of contaminated observations in each time series. For each simulated series andcontamination levels, the long memory parameter d, the innovations variance σ 2 and the mean μ

are estimated in three ways: (i) we use the approximate maximum likelihood estimator [5] (L inthe tables); (ii) we use the robust M-estimator of Beran [5] with his suggested ψ-function (M inthe tables); (iii) we use our method (W in the tables).

Results for the long memory parameter are reported in Table 1 for the symmetric contaminationand in Table 2 for the asymmetric contamination. Results for the mean parameter (Table 3) andfor the variance parameter (Table 4) are reported only for the asymmetric case. For each entry wereport the mean and the standard error (in round brackets) of the Monte Carlo replications. Thecomplete results for the symmetric case are available upon request by the authors.

In the case of no contamination (ε = 0%), we have an empirical evaluation of the efficiencyof the proposed method with respect to approximate maximum likelihood in finite samples. Wecan see that with regard to the estimation of the long memory parameter d and the mean μ allthe proposed methodologies are equivalent both in terms of bias and standard error for all samplesizes. With regard to σ 2 (Table 4) all methods work well but the standard error of the robustM-estimator of Beran is between 1.4–1.8 times the standard error of the weighted likelihoodmethod.

For the symmetric contaminated case, the maximum likelihood method performs worse withthe increasing contamination level, also for large sample sizes. With regard to the estimation of

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Journal of Applied Statistics 1577

Table 1. Monte Carlo results for the estimation of long memory parameter, d, in case of symmetriccontamination.

True d

T ε Type 0.1 0.2 0.3 0.4 0.45

Long memory parameter estimation (d) Symmetric contamination100 0% L 0.103 (0.061) 0.165 (0.074) 0.244 (0.082) 0.341 (0.076) 0.368 (0.080)

M 0.105 (0.066) 0.175 (0.083) 0.262 (0.091) 0.358 (0.085) 0.373 (0.085)W 0.106 (0.065) 0.171 (0.079) 0.257 (0.090) 0.370 (0.093) 0.407 (0.105)

5% L 0.094 (0.059) 0.145 (0.073) 0.213 (0.087) 0.304 (0.094) 0.334 (0.092)M 0.101 (0.062) 0.163 (0.081) 0.248 (0.090) 0.345 (0.091) 0.365 (0.089)W 0.099 (0.062) 0.155 (0.077) 0.236 (0.092) 0.348 (0.096) 0.384 (0.105)

10% L 0.093 (0.054) 0.131 (0.074) 0.196 (0.082) 0.275 (0.091) 0.300 (0.099)M 0.104 (0.062) 0.157 (0.085) 0.234 (0.089) 0.330 (0.090) 0.350 (0.097)W 0.096 (0.058) 0.147 (0.077) 0.220 (0.090) 0.322 (0.094) 0.360 (0.103)

15% L 0.088 (0.051) 0.121 (0.070) 0.177 (0.082) 0.245 (0.100) 0.278 (0.100)M 0.100 (0.061) 0.146 (0.078) 0.221 (0.091) 0.314 (0.099) 0.339 (0.097)W 0.090 (0.058) 0.134 (0.074) 0.207 (0.086) 0.301 (0.101) 0.337 (0.108)

250 0% L 0.090 (0.043) 0.179 (0.052) 0.276 (0.053) 0.375 (0.048) 0.415 (0.045)M 0.094 (0.044) 0.185 (0.055) 0.283 (0.056) 0.386 (0.054) 0.423 (0.048)W 0.090 (0.044) 0.182 (0.054) 0.282 (0.056) 0.388 (0.053) 0.436 (0.056)

5% L 0.080 (0.042) 0.156 (0.056) 0.240 (0.063) 0.329 (0.071) 0.370 (0.067)M 0.089 (0.043) 0.173 (0.056) 0.269 (0.057) 0.371 (0.055) 0.411 (0.051)W 0.084 (0.041) 0.166 (0.056) 0.260 (0.057) 0.364 (0.055) 0.413 (0.059)

10% L 0.075 (0.040) 0.131 (0.056) 0.204 (0.072) 0.288 (0.078) 0.333 (0.080)M 0.084 (0.044) 0.161 (0.056) 0.254 (0.062) 0.354 (0.059) 0.399 (0.056)W 0.078 (0.040) 0.151 (0.054) 0.240 (0.060) 0.338 (0.057) 0.390 (0.061)

15% L 0.064 (0.036) 0.113 (0.052) 0.181 (0.067) 0.258 (0.078) 0.294 (0.085)M 0.080 (0.042) 0.152 (0.057) 0.241 (0.061) 0.341 (0.066) 0.385 (0.060)W 0.070 (0.038) 0.136 (0.057) 0.221 (0.059) 0.316 (0.062) 0.363 (0.062)

500 0% L 0.090 (0.034) 0.190 (0.037) 0.288 (0.036) 0.386 (0.033) 0.432 (0.031)M 0.092 (0.034) 0.192 (0.039) 0.290 (0.038) 0.393 (0.037) 0.437 (0.034)W 0.091 (0.034) 0.191 (0.038) 0.290 (0.037) 0.393 (0.035) 0.444 (0.037)

5% L 0.076 (0.035) 0.160 (0.044) 0.245 (0.053) 0.335 (0.058) 0.377 (0.063)M 0.085 (0.035) 0.181 (0.040) 0.277 (0.040) 0.379 (0.039) 0.425 (0.037)W 0.080 (0.034) 0.173 (0.039) 0.269 (0.039) 0.367 (0.036) 0.419 (0.038)

10% L 0.067 (0.033) 0.136 (0.046) 0.213 (0.056) 0.289 (0.068) 0.334 (0.071)M 0.080 (0.034) 0.169 (0.040) 0.264 (0.041) 0.363 (0.040) 0.414 (0.040)W 0.071 (0.031) 0.154 (0.040) 0.247 (0.041) 0.344 (0.038) 0.395 (0.041)

15% L 0.060 (0.030) 0.117 (0.043) 0.184 (0.054) 0.257 (0.067) 0.303 (0.071)M 0.075 (0.034) 0.158 (0.040) 0.250 (0.042) 0.351 (0.042) 0.402 (0.044)W 0.065 (0.032) 0.139 (0.039) 0.226 (0.044) 0.322 (0.041) 0.372 (0.043)

1000 0% L 0.095 (0.025) 0.196 (0.026) 0.294 (0.026) 0.392 (0.025) 0.440 (0.023)M 0.096 (0.026) 0.196 (0.026) 0.294 (0.027) 0.395 (0.027) 0.445 (0.026)W 0.095 (0.025) 0.196 (0.026) 0.294 (0.026) 0.394 (0.025) 0.447 (0.026)

5% L 0.076 (0.027) 0.158 (0.041) 0.246 (0.044) 0.332 (0.053) 0.384 (0.056)M 0.088 (0.027) 0.184 (0.027) 0.281 (0.028) 0.380 (0.028) 0.433 (0.028)W 0.083 (0.025) 0.176 (0.027) 0.272 (0.027) 0.368 (0.028) 0.421 (0.028)

10% L 0.064 (0.027) 0.132 (0.038) 0.208 (0.050) 0.293 (0.061) 0.335 (0.060)M 0.081 (0.028) 0.173 (0.027) 0.267 (0.028) 0.367 (0.028) 0.418 (0.030)W 0.071 (0.026) 0.157 (0.029) 0.250 (0.028) 0.346 (0.028) 0.395 (0.029)

15% L 0.056 (0.025) 0.117 (0.037) 0.183 (0.050) 0.254 (0.060) 0.304 (0.063)M 0.076 (0.025) 0.163 (0.027) 0.255 (0.029) 0.353 (0.030) 0.406 (0.031)W 0.063 (0.025) 0.142 (0.029) 0.231 (0.030) 0.322 (0.030) 0.373 (0.030)

Notes: L is maximum likelihood estimator, M is M-estimator and W is weighted likelihood estimator. T is the samplesize, ε is the level of contamination. In round parenthesis the Monte Carlo standard deviation.

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1578 C. Agostinelli and L. Bisaglia

Table 2. Monte Carlo results for the estimation of long memory parameter, d, in case of asymmetriccontamination.

True d

T ε Type 0.1 0.2 0.3 0.4 0.45

Long memory parameter estimation (d) Asymmetric contamination100 0% L 0.103 (0.061) 0.165 (0.074) 0.244 (0.082) 0.341 (0.076) 0.368 (0.080)

M 0.105 (0.066) 0.175 (0.083) 0.262 (0.091) 0.358 (0.085) 0.373 (0.085)W 0.106 (0.065) 0.171 (0.079) 0.257 (0.090) 0.370 (0.093) 0.407 (0.105)

5% L 0.075 (0.052) 0.082 (0.060) 0.098 (0.062) 0.125 (0.068) 0.146 (0.072)M 0.079 (0.056) 0.129 (0.083) 0.202 (0.108) 0.294 (0.123) 0.316 (0.115)W 0.064 (0.043) 0.103 (0.069) 0.183 (0.102) 0.308 (0.130) 0.358 (0.148)

10% L 0.071 (0.049) 0.075 (0.051) 0.083 (0.050) 0.105 (0.063) 0.113 (0.069)M 0.118 (0.146) 0.179 (0.163) 0.221 (0.158) 0.266 (0.142) 0.246 (0.150)W 0.052 (0.034) 0.081 (0.055) 0.143 (0.107) 0.269 (0.151) 0.333 (0.186)

15% L 0.060 (0.041) 0.073 (0.051) 0.078 (0.051) 0.084 (0.056) 0.096 (0.059)M 0.368 (0.144) 0.360 (0.137) 0.328 (0.132) 0.290 (0.124) 0.262 (0.130)W 0.047 (0.036) 0.074 (0.059) 0.146 (0.117) 0.258 (0.177) 0.275 (0.214)

250 0% L 0.090 (0.043) 0.179 (0.052) 0.276 (0.053) 0.375 (0.048) 0.415 (0.045)M 0.094 (0.044) 0.185 (0.055) 0.283 (0.056) 0.386 (0.054) 0.423 (0.048)W 0.090 (0.044) 0.182 (0.054) 0.282 (0.056) 0.388 (0.053) 0.436 (0.056)

5% L 0.053 (0.034) 0.070 (0.041) 0.101 (0.047) 0.147 (0.053) 0.178 (0.053)M 0.058 (0.035) 0.117 (0.059) 0.223 (0.089) 0.357 (0.085) 0.396 (0.071)W 0.053 (0.028) 0.104 (0.050) 0.187 (0.077) 0.324 (0.093) 0.387 (0.091)

10% L 0.048 (0.033) 0.056 (0.034) 0.073 (0.039) 0.098 (0.046) 0.119 (0.049)M 0.096 (0.148) 0.184 (0.180) 0.264 (0.157) 0.260 (0.136) 0.239 (0.132)W 0.042 (0.027) 0.081 (0.051) 0.183 (0.104) 0.308 (0.117) 0.345 (0.130)

15% L 0.044 (0.023) 0.052 (0.032) 0.063 (0.036) 0.081 (0.041) 0.099 (0.047)M 0.405 (0.059) 0.360 (0.080) 0.316 (0.085) 0.236 (0.099) 0.200 (0.103)W 0.045 (0.032) 0.090 (0.068) 0.179 (0.098) 0.272 (0.118) 0.295 (0.145)

500 0% L 0.090 (0.034) 0.190 (0.037) 0.288 (0.036) 0.386 (0.033) 0.432 (0.031)M 0.092 (0.034) 0.192 (0.039) 0.290 (0.038) 0.393 (0.037) 0.437 (0.034)W 0.091 (0.034) 0.191 (0.038) 0.290 (0.037) 0.393 (0.035) 0.444 (0.037)

5% L 0.041 (0.025) 0.063 (0.031) 0.101 (0.038) 0.158 (0.039) 0.189 (0.040)M 0.046 (0.025) 0.108 (0.046) 0.231 (0.066) 0.388 (0.067) 0.434 (0.050)W 0.048 (0.022) 0.104 (0.043) 0.212 (0.072) 0.349 (0.069) 0.400 (0.075)

10% L 0.040 (0.022) 0.047 (0.025) 0.068 (0.032) 0.106 (0.038) 0.136 (0.040)M 0.036 (0.019) 0.220 (0.196) 0.295 (0.151) 0.207 (0.123) 0.174 (0.119)W 0.043 (0.025) 0.093 (0.057) 0.210 (0.089) 0.328 (0.088) 0.363 (0.099)

15% L 0.035 (0.018) 0.043 (0.022) 0.056 (0.029) 0.086 (0.034) 0.110 (0.038)M 0.394 (0.031) 0.357 (0.044) 0.306 (0.068) 0.216 (0.085) 0.160 (0.090)W 0.052 (0.036) 0.115 (0.065) 0.208 (0.081) 0.287 (0.077) 0.307 (0.107)

1000 0% L 0.095 (0.025) 0.196 (0.026) 0.294 (0.026) 0.392 (0.025) 0.440 (0.023)M 0.096 (0.026) 0.196 (0.026) 0.294 (0.027) 0.395 (0.027) 0.445 (0.026)W 0.095 (0.025) 0.196 (0.026) 0.294 (0.026) 0.394 (0.025) 0.447 (0.026)

5% L 0.035 (0.017) 0.063 (0.024) 0.109 (0.026) 0.165 (0.027) 0.201 (0.028)M 0.039 (0.018) 0.111 (0.035) 0.237 (0.048) 0.409 (0.052) 0.459 (0.033)W 0.049 (0.02) 0.115 (0.044) 0.229 (0.064) 0.369 (0.050) 0.410 (0.063)

10% L 0.031 (0.015) 0.041 (0.019) 0.072 (0.025) 0.114 (0.028) 0.148 (0.028)M 0.019 (0.001) 0.265 (0.209) 0.293 (0.148) 0.179 (0.116) 0.079 (0.078)W 0.055 (0.029) 0.118 (0.055) 0.232 (0.070) 0.338 (0.066) 0.366 (0.084)

15% L 0.029 (0.015) 0.038 (0.018) 0.054 (0.022) 0.090 (0.028) 0.120 (0.030)M 0.382 (0.016) 0.352 (0.015) 0.308 (0.044) 0.197 (0.084) 0.140 (0.077)W 0.059 (0.033) 0.132 (0.056) 0.222 (0.060) 0.298 (0.059) 0.328 (0.072)

Notes: L is maximum likelihood estimator, M is M-estimator and W is weighted likelihood estimator. T is the samplesize, ε is the level of contamination. In round parenthesis the Monte Carlo standard deviation.

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Journal of Applied Statistics 1579

Table 3. Monte Carlo results for the estimation of mean parameter, μ, in case of asymmetric contamination.

True d

T ε Type 0.1 0.2 0.3 0.4 0.45

Mean estimation (μ) Asymmetric contamination100 0% L 0.002 (0.160) − 0.006 (0.265) 0.026 (0.424) − 0.013 (0.877) − 0.062 (1.462)

M − 0.005 (0.170) − 0.001 (0.272) 0.017 (0.434) − 0.011 (0.880) − 0.033 (1.436)W − 0.001 (0.161) − 0.004 (0.262) 0.025 (0.424) − 0.014 (0.879) − 0.062 (1.462)

5% L 0.396 (0.160) 0.387 (0.259) 0.435 (0.427) 0.374 (0.896) 0.370 (1.463)M 0.047 (0.180) 0.062 (0.285) 0.097 (0.438) 0.060 (0.892) − 0.001 (1.470)W − 0.001 (0.166) − 0.008 (0.273) 0.028 (0.428) − 0.007 (0.898) − 0.067 (1.463)

10% L 0.798 (0.158) 0.791 (0.260) 0.840 (0.415) 0.796 (0.854) 0.682 (1.469)M 0.113 (0.176) 0.137 (0.289) 0.180 (0.442) 0.174 (0.897) 0.037 (1.449)W 0.001 (0.167) − 0.008 (0.267) 0.036 (0.432) − 0.009 (0.888) − 0.032 (1.435)

15% L 1.208 (0.146) 1.183 (0.257) 1.246 (0.417) 1.208 (0.870) 1.097 (1.400)M 0.206 (0.198) 0.235 (0.272) 0.246 (0.430) 0.247 (0.881) 0.178 (1.486)W − 0.013 (0.152) − 0.010 (0.269) 0.028 (0.446) − 0.012 (0.895) − 0.003 (1.479)

250 0% L − 0.001 (0.111) 0.000 (0.201) 0.010 (0.366) − 0.009 (0.799) − 0.043 (1.366)M − 0.004 (0.122) 0.002 (0.207) 0.003 (0.364) − 0.014 (0.796) − 0.017 (1.365)W − 0.001 (0.111) 0.000 (0.200) 0.010 (0.366) − 0.009 (0.799) − 0.042 (1.367)

5% L 0.375 (0.118) 0.385 (0.203) 0.391 (0.369) 0.371 (0.799) 0.341 (1.368)M 0.059 (0.126) 0.071 (0.210) 0.071 (0.363) 0.070 (0.805) 0.070 (1.376)W 0.003 (0.111) − 0.007 (0.203) 0.005 (0.363) 0.001 (0.794) − 0.033 (1.365)

10% L 0.811 (0.106) 0.811 (0.204) 0.817 (0.356) 0.799 (0.796) 0.762 (1.367)M 0.140 (0.136) 0.142 (0.201) 0.171 (0.373) 0.183 (0.764) − 0.010 (1.434)W 0.006 (0.113) 0.003 (0.202) 0.006 (0.358) − 0.008 (0.802) − 0.066 (1.356)

15% L 1.210 (0.100) 1.220 (0.199) 1.208 (0.365) 1.216 (0.812) 1.211 (1.363)M 0.222 (0.129) 0.238 (0.214) 0.250 (0.369) 0.231 (0.789) 0.217 (1.374)W − 0.003 (0.113) 0.010 (0.205) 0.009 (0.364) − 0.003 (0.809) 0.022 (1.353)

500 0% L 0.000 (0.082) 0.008 (0.159) − 0.001 (0.311) 0.020 (0.751) − 0.026 (1.322)M − 0.003 (0.090) 0.009 (0.163) − 0.004 (0.311) 0.018 (0.753) − 0.020 (1.304)W 0.000 (0.082) 0.008 (0.158) − 0.001 (0.311) 0.019 (0.751) − 0.026 (1.322)

5% L 0.399 (0.079) 0.406 (0.158) 0.398 (0.309) 0.421 (0.748) 0.373 (1.321)M 0.059 (0.094) 0.078 (0.163) 0.068 (0.311) 0.116 (0.749) 0.103 (1.27)W − 0.001 (0.084) 0.010 (0.160) − 0.001 (0.308) 0.005 (0.747) 0.008 (1.303)

10% L 0.807 (0.076) 0.819 (0.152) 0.800 (0.304) 0.818 (0.749) 0.774 (1.323)M 0.086 (0.087) 0.146 (0.167) 0.160 (0.325) 0.207 (0.682) 0.339 (1.328)W 0.001 (0.082) 0.010 (0.160) 0.002 (0.312) 0.027 (0.750) − 0.003 (1.316)

15% L 1.196 (0.078) 1.214 (0.159) 1.209 (0.306) 1.223 (0.743) 1.179 (1.325)M 0.222 (0.092) 0.242 (0.168) 0.242 (0.315) 0.277 (0.727) 0.202 (1.327)W 0.000 (0.084) 0.011 (0.162) 0.004 (0.308) 0.021 (0.736) 0.020 (1.304)

1000 0% L 0.003 (0.063) 0.005 (0.129) − 0.007 (0.271) − 0.006 (0.701) − 0.060 (1.302)M 0.002 (0.068) 0.005 (0.133) − 0.006 (0.272) − 0.007 (0.703) − 0.065 (1.305)W 0.003 (0.063) 0.005 (0.129) − 0.007 (0.271) − 0.006 (0.701) − 0.060 (1.302)

5% L 0.400 (0.062) 0.404 (0.129) 0.393 (0.269) 0.395 (0.701) 0.340 (1.301)M 0.067 (0.070) 0.073 (0.133) 0.067 (0.272) 0.092 (0.701) 0.097 (1.233)W 0.003 (0.065) 0.005 (0.130) − 0.003 (0.272) − 0.008 (0.701) − 0.055 (1.301)

10% L 0.802 (0.062) 0.805 (0.128) 0.793 (0.269) 0.795 (0.699) 0.740 (1.301)M 0.140 (0.054) 0.139 (0.145) 0.140 (0.273) 0.165 (0.690) 0.083 (1.612)W 0.002 (0.065) 0.008 (0.131) − 0.004 (0.271) 0.001 (0.687) − 0.078 (1.304)

15% L 1.204 (0.059) 1.210 (0.126) 1.192 (0.266) 1.195 (0.697) 1.140 (1.301)M 0.228 (0.071) 0.237 (0.134) 0.241 (0.272) 0.266 (0.703) 0.155 (1.278)W 0.002 (0.065) 0.008 (0.130) − 0.004 (0.267) 0.005 (0.693) − 0.031 (1.300)

Notes: L is Maximum likelihood estimator, M is M-estimator and W is weighted likelihood estimator. T is the samplesize, ε is the level of contamination. In round parenthesis the Monte Carlo standard deviation.

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1580 C. Agostinelli and L. Bisaglia

Table 4. Monte Carlo results for the estimation of innovation variance parameter, σ 2, in case of asymmetriccontamination.

True d

T ε Type 0.1 0.2 0.3 0.4 0.45

Variance estimation (σ 2) Asymmetric contamination100 0% L 0.996(0.147) 1.003(0.136) 0.991 (0.139) 0.993 (0.147) 0.997 (0.146)

M 1.045 (0.246) 1.043 (0.245) 1.023 (0.248) 1.044 (0.260) 1.044 (0.245)W 0.983 (0.146) 0.989 (0.136) 0.975 (0.141) 0.976 (0.148) 0.978 (0.146)

5% L 4.068 (0.381) 4.075 (0.389) 4.144 (0.413) 4.300 (0.450) 4.342 (0.451)M 1.096 (0.276) 1.134 (0.290) 1.136 (0.281) 1.158 (0.311) 1.156 (0.312)W 0.983 (0.145) 1.007 (0.144) 1.000 (0.151) 1.014 (0.162) 1.013 (0.165)

10% L 6.824 (0.497) 6.858 (0.510) 6.914 (0.546) 7.047 (0.558) 7.209 (0.627)M 1.221 (0.339) 1.273 (0.368) 1.330 (0.364) 1.398 (0.381) 1.370 (0.398)W 0.992 (0.154) 1.019 (0.157) 1.034 (0.163) 1.063 (0.205) 1.074 (0.222)

15% L 9.219 (0.565) 9.271 (0.637) 9.332 (0.621) 9.496 (0.673) 9.647 (0.713)M 1.774 (0.611) 1.774 (0.601) 1.802 (0.587) 1.872 (0.603) 1.899 (0.569)W 0.998 (0.148) 1.018 (0.163) 1.037 (0.173) 1.104 (0.313) 1.149 (0.275)

250 0% L 1.000 (0.091) 1.004 (0.087) 0.993 (0.089) 0.995 (0.088) 1.000 (0.089)M 1.061 (0.158) 1.056 (0.161) 1.041 (0.158) 1.053 (0.155) 1.057 (0.155)W 0.990 (0.091) 0.995 (0.088) 0.983 (0.090) 0.985 (0.088) 0.989 (0.088)

5% L 3.937 (0.227) 3.988 (0.246) 4.052 (0.258) 4.146 (0.294) 4.220 (0.281)M 1.108 (0.164) 1.151 (0.176) 1.177 (0.192) 1.180 (0.184) 1.183 (0.185)W 0.999 (0.097) 1.021 (0.089) 1.031 (0.098) 1.028 (0.104) 1.038 (0.106)

10% L 6.771 (0.307) 6.841 (0.319) 6.930 (0.351) 7.095 (0.366) 7.176 (0.397)M 1.204 (0.192) 1.278 (0.229) 1.399 (0.247) 1.470 (0.247) 1.537 (0.272)W 1.014 (0.097) 1.038 (0.098) 1.042 (0.113) 1.079 (0.148) 1.117 (0.157)

15% L 9.262 (0.386) 9.297 (0.406) 9.440 (0.406) 9.620 (0.474) 9.736 (0.474)M 2.061 (0.393) 1.998 (0.426) 2.086 (0.419) 2.126 (0.432) 2.186 (0.416)W 1.011 (0.097) 1.036 (0.105) 1.066 (0.137) 1.132 (0.185) 1.224 (0.256)

500 0% L 1.000 (0.064) 1.004 (0.062) 0.996 (0.064) 0.998 (0.065) 1.002 (0.063)M 1.060 (0.110) 1.060 (0.113) 1.049 (0.114) 1.055 (0.113) 1.066 (0.109)W 0.994 (0.063) 0.997 (0.063) 0.990 (0.064) 0.991 (0.065) 0.995 (0.063)

5% L 4.067 (0.171) 4.091 (0.164) 4.165 (0.192) 4.281 (0.202) 4.359 (0.205)M 1.103 (0.124) 1.164 (0.136) 1.214 (0.134) 1.196 (0.138) 1.185 (0.132)W 1.005 (0.067) 1.029 (0.067) 1.037 (0.077) 1.035 (0.088) 1.052 (0.089)

10% L 6.802 (0.224) 6.840 (0.221) 6.917 (0.246) 7.082 (0.253) 7.183 (0.296)M 1.175 (0.109) 1.352 (0.196) 1.444 (0.194) 1.530 (0.204) 1.615 (0.255)W 1.012 (0.067) 1.042 (0.072) 1.052 (0.093) 1.081 (0.123) 1.133 (0.153)

15% L 9.193 (0.286) 9.240 (0.283) 9.362 (0.298) 9.514 (0.309) 9.680 (0.331)M 2.093 (0.276) 2.100 (0.297) 2.127 (0.311) 2.188 (0.316) 2.232 (0.298)W 1.008 (0.071) 1.037 (0.081) 1.064 (0.094) 1.136 (0.129) 1.241 (0.201)

1000 0% L 0.999 (0.045) 1.000 (0.045) 0.995 (0.045) 0.998 (0.047) 0.999 (0.047)M 1.061 (0.074) 1.055 (0.080) 1.053 (0.080) 1.055 (0.081) 1.060 (0.080)W 0.995 (0.045) 0.996 (0.046) 0.991 (0.045) 0.994 (0.047) 0.995 (0.047)

5% L 4.043 (0.120) 4.096 (0.120) 4.166 (0.133) 4.294 (0.150) 4.365 (0.150)M 1.101 (0.081) 1.166 (0.093) 1.221 (0.100) 1.187 (0.101) 1.174 (0.092)W 1.007 (0.048) 1.029 (0.049) 1.035 (0.059) 1.027 (0.069) 1.051 (0.079)

10% L 6.786 (0.162) 6.833 (0.165) 6.918 (0.176) 7.101 (0.201) 7.210 (0.195)M 1.196 (0.050) 1.325 (0.140) 1.444 (0.141) 1.582 (0.158) 1.635 (0.124)W 1.007 (0.049) 1.032 (0.055) 1.049 (0.070) 1.078 (0.096) 1.137 (0.138)

15% L 9.188 (0.180) 9.243 (0.188) 9.350 (0.201) 9.542 (0.242) 9.681 (0.245)M 2.128 (0.196) 2.124 (0.207) 2.187 (0.235) 2.250 (0.264) 2.252 (0.244)W 1.006 (0.052) 1.033 (0.058) 1.062 (0.075) 1.139 (0.111) 1.221 (0.160)

Notes: L is maximum likelihood estimator, M is M-estimator and W is weighted likelihood estimator. T is the samplesize, ε is the level of contamination. In round parenthesis the Monte Carlo standard deviation.

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Journal of Applied Statistics 1581

σ 2, this method has a bias that increases both with the contamination level and the sample size.So, in the following the behavior of this estimator will not be commented upon. With regard tothe estimation of d and μ the robust M-estimator of Beran and weighted likelihood method areequivalent and work well, especially with increasing sample sizes, for all contamination levels. Forthe estimation of the innovation variance both techniques are not biased but weighted likelihoodgains significant improvements in terms of its standard errors. This is particularly important forthe inference analysis.

For the asymmetric contaminated case (Tables 2–4), the maximum likelihood method heav-ily under-estimates the long memory parameter d and it is heavily biased for both mean andvariance parameters for all considered sample sizes, especially when the contamination levelincreases.

The robust M-estimator of Beran and the weighted likelihood method perform somewhat sim-ilar for 5% contamination level, but for higher contamination the weighted likelihood methodoutperforms the M-estimator for the estimation of μ and σ 2 both in terms of bias and standarderror. With respect to the estimation of d the weighted likelihood method under-estimates thevalue of d as the contamination level increases, while the robust M-estimator of Beran has anestimate of d that decreases as the true value of d increases. This behavior is accentuated as thesample size increases and the level of contamination increases from 10% to 15%. Thus, it seemswise to discourage its use in the case of a suspect contamination greater than 10%.

4. Example: the time series of Nile River annual

We have applied the procedure introduced in Section 2.3 to the first 663 observations from thetime series of annual minimum water levels of the Nile River, measured at the Roda Gorgebetween 622 and 1984 A.D. [21]. We consider only this period since for subsequent years thereare missing values. For this data set several analysis has suggested that an ARFIMA model withonly the fractional parameter will be adequate. Running the approximate maximum likelihoodand the conditional weighted likelihood we have obtained the results reported in Table 5. Theestimated values are very similar and our method gives weights close to one to every obser-vations but 6, 25, 39, 98, 188, 257, 258, 342, 479 with weights 0.350, 0.073, 0.658, 0.650,0.007, 0.589, 0.251, 0.793, 0.203, respectively, and the observation 25 is classified as additiveoutlier.

To study the stability of these results, we perform a sensitivity analysis by moving the obser-vation 188 (that with minimum weight) within the interval (6, 21) (its original value is 14.66). Asshown in Figure 1 our method classifies as additive outlier the observation 188 when its valueis outside the interval (7.587, 16.423). Out of this interval the effect of the observation in theestimation is negligible, that is, we have bound its influence. The stability of our method withrespect to the MLE is particularly evident for σ 2, confirming the results observed in Monte Carloexperiments.

Table 5. The approximate maximum likelihood, approximate M andthe approximate weighted likelihood results for the time series ofNile River annual.

MLE M WLE

d 0.394 0.435 0.416Mean 11.484 11.480 11.465σ 2 0.489 0.305 0.424

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1582 C. Agostinelli and L. Bisaglia

6 8 10 12 14 16 18

0.38

0.40

0.42

0.44

0.46

0.48

x188

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mlewleBeran M

6 8 10 12 14 16 18

0.30

0.35

0.40

0.45

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x188

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mlewleBeran M

6 8 10 12 14 16 18

0.0

0.2

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1.0

wle

x188

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ght

6 8 10 12 14 16 18

wle

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addi

tive

FALS

ET

RU

E

Figure 1. The result of the sensitivity analysis performed on the Nile River annual time series movingobservation 188 within the interval (6, 21) (its original value is 14.66).

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Journal of Applied Statistics 1583

5. Conclusions

In the literature so far developed, no much attention has been devoted to the problem of robustestimation of long memory ARFIMA processes when outliers are present in the time series.In this paper, we propose a robust estimation procedure for strictly stationary long memoryARFIMA(p, d, q) processes. The proposed estimator is a weighted likelihood estimator definedby a weighted score function, which is implied under a strict white noise assumption for the errorterm. Our procedure is able, simultaneously, (i) to detect suspicious observations, (ii) to classifythe suspicious observations in different kind of outliers (additive or innovation) and (iii) to obtainrobust estimate of the parameters without loss of efficiency.

By an extensive Monte Carlo exercise, we find that our approach seems to be a promisingmethod for robust estimation/inference in long memory time series. Since the qualities of ourprocedure have been shown mainly by mean of simulations, it would be interesting to completeour results with a formal discussion about the precise robustness properties of the proposedestimator. Fulfilling this task is beyond the aims of this paper and it is left for future research.

Moreover, it still remains to investigate the behavior of our method when forecasting futurevalues or computing prediction intervals for ARFIMA(p, d, q) processes in presence of outliers.The latter is perhaps the main objective in time series analysis and several empirical studies revealthat classical models tend to provide poor forecasts in the presence of outliers. Further work aboutthis issue is ongoing.

Acknowledgements

We thank two anonymous referees for their insightful comments.

References

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