Maximum lq-likelihood estimator of the heavy-tailed distribution parameter
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Kouider, Mohammed Tidha; Idiou, Nesrine; Toumi, Samia; Benatia, Fatah Article Maximum lq-likelihood estimator of the heavy-tailed distribution parameter Croatian Review of Economic, Business and Social Statistics (CREBSS) Provided in Cooperation with: Croatian Statistical Association (CSA), Zagreb Suggested Citation: Kouider, Mohammed Tidha; Idiou, Nesrine; Toumi, Samia; Benatia, Fatah (2024) : Maximum lq-likelihood estimator of the heavy-tailed distribution parameter, Croatian Review of Economic, Business and Social Statistics (CREBSS), ISSN 2459-5616, Croatian Statistical Association (CSA), Zagreb, Vol. 10, Iss. 2, pp. 29-48, https://doi.org/10.62366/crebss.2024.2.003 This Version is available at: https://hdl.handle.net/10419/323439 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/
Croatian Review of Economic, Business and Social Statistics 29 CREBSS 10(2):29–48 Maximum lq–likelihood estimator of the heavy–tailed distribution parameter Mohammed Ridha Kouider1,*, Nesrine Idiou 2, Samia Toumi 3and Fatah Benatia4 1,3,4 Mohamed Khider University of Biskra, Faculty of Exact Science and Natural and Life Science, Algeria Q 2Salah Boubnider University of Constantine 3, Faculty of Process Engineering, Algeria ARTICLE TYPE Preliminary communication ARTICLE INFO Received: September 7, 2024 Accepted: November 5, 2024 DOI: 10.62366/crebss.2024.2.003 JEL: C13, C46 SUMMARY Studying the extreme value theory (EVT) involves multiple main objectives, among them the estimation of the tail index parameter. Some estimation methods are used to estimate the tail index parameter like maximum likelihood estimation (MLE). Additionally, the Hill estimator is one type of maximum likelihood estimator, which is a more robust with a large sample than a small sample. This research proposes the construction of an alternative estimator for the parameter of the heavy–tailed distribution using the maximum lq–likelihood estimation (MLqE) approach in order to adapt the ML and Hill estimator with the small sample. Furthermore, the maximum lq–likelihood estimator asymptotic normality is established. Moreover, several simulation studies in order to compare the MLq estimator with the ML estimators are provided. In the excesses over high suitable threshold values the number of the largest observation kwill lead to an efficient estimate of the Hill estimator. For this, selection of kin the Hill estimator was investigated using the method of the quantile type 8 which is effective with the hydrology data. The performance of the Hill estimator and the lq–Hill estimator is subsequently compared by employing real relies with the distribution of hydrology data. KEYWORDS excesses over threshold, extreme value index, heavy–tailed distribution, maximum lq–likelihood estimator 1. Introduction Considering X1,X2, . . . , Xnof independently and identically distributed (iid) random variables (rv) defined over some probability space (Ω;A;P), with cumulative distribution function (cdf) F. We are interested in the probability that the maximum is not beyond a certain threshold x. This probability is given by P(max (X1,X2, . . . , Xn)≤x)=Fn(x). (1) ∗Corresponding author ©2024 Copyright of this article is retained by the author(s) This is an open access article under the CC BY–NC–ND 4.0 license
30 Kouider, Idiou, Toumi & Benatia As it is well known, when we are interested in the central part of a sample, the central limit theorem (CLT) giving the asymptotic law of the sum of observations which says that the sampling distribution of the mean will always be normally distributed as n→+∞. On the other hand, if we want to study the extreme values of this sample, the CLT presents only little of interest. Instead, we use a result establishing the asymptotic distribution of the maximum of the sample. This result is stated under EVT as demonstrated particularly by Gnedenko (1943). The EVT gives the conditions under which there exist sequences of normalizing constants an>0 and bn>0 such that lim n→+∞Fn(anx+bn)=Gγ(x). (2) Gγ(x)is so–called the extreme value distribution, defined by Gγ(x)=(exp −(1+γx)−1/γ, if γ=0 exp (−exp (−x)) , if γ=0(3) where Gγ(x)is a well–defined non–degenerate cdf. This law depends only on the parameter γ∈Rcalled the extreme value index or the tail index, or the shape parameter. According to the sign of γ, there are three domains of attraction that are defined from Gγ(x)depending on the tail index; among them is domain attraction of Fréchet. Also, it is referred to as heavy– tailed distribution. It contains laws whose survival function decreases as a power function like distributions of Pareto, Student, Cauchy, etc. So the heavy tailed limit distribution is the Fréchet distribution (Balkema and de Haan,1974), which is defined by Gγ>0(x)=e−x−1/γ. (4) As it is known and associated with EVT, the characterization of the domains of attraction makes extensive use of the notion of functions with regular variations which we define below. Let X1,X2, . . . , Xnbe a iid sequence of a non–negative rv Xover some probability space (Ω;A;P), with cdf F. We assume that the distribution tail F=1−Fis regularly varying at infinity, with index (−1/γ), notation: F∈RV(−1/γ). That is lim t→+∞ F(tx) F(t)=x−1/γ, for any x>0. (5) A distribution function Fbelongs to the domain attraction of Fréchet if and only if F∈RV(−1/γ). Hence, the tail behaves approximately as a power function x−1/γ. This implies that the distribution for the maximum has a one–to–one relationship with the shape parameter. Then, we will take heavy tailed to mean sub–exponential (definition below), but several definitions exist in the literature. For the convenience of the reader, we also sketch other common definitions and, where possible, relate them to one another. Sub-exponential distributions exhibit one of the general properties expected of heavy–tailed distributions on the level of aggregate losses, namely that the tail of the maximum determines the tail of the sum. All of the distributions considered here are sub–exponential. Let Fbe a cdf with support in [0; +∞[. Then Fis sub–exponential if, for all n≥2, lim x→+∞ Fn(x) F(x)=n, for any x>0. (6)
Maximum lq–likelihood estimator of the heavy–tailed distribution parameter 31 Then Fn(x)≈nF (x)as x→+∞. Subexponentiality implies another property that is sometimes taken as the definition of heavy tail, i.e. the tail decays more slowly than any exponential function. With the notation as above, the precise formulation is that for all t>0, lim t→+∞etx F(x):=∞. (7) An important subclass of sub–exponential distributions consists of regularly varying functions. For a regularly varying with tail index γ>0 , all moments of the associated rv higher than γ>0 will be unbounded (Embrechts et al.,1997). The parameter of interest is γ>0 is the tail index of F. Now for F∈RV(−1/γ)we take as F(x)=x−1/γ. Then we can check for Fis sub–exponential that lim t→+∞ lim x→+∞ 1 Fn(t) ∞ Zt Fn(x) xdx =1 F(t) ∞ Zt F(x) xdx :=γ. (8) Consider X1,n≤X2,n≤. . . ≤Xn,nthe order statistics of X1,X2, . . . , Xn. Let’s replace the distribution Fby its empirical version Fnand tby Xn−k,n. Thus, we find the Hill estimator (Hill,1975) defined by: b γH Xn−k,n:=1 Fn(Xn−k,n) ∞ Z Xn−k,n Fn(x)dx x. (9) The Hill estimator can only be used for distributions belonging to the Fréchet domain. The Hill estimator, which is a type of ML estimator, is the most common estimators for the tail index of heavy–tailed distributions; is probably the most studied estimator in the literature. As agreed that, Hill estimator and ML estimator are goods with the large sample is susceptible to be biased. But if we use very small k, both estimators have a large variance. In this article, we investigate a new class of parametric estimators based on the q–entropy function proposed by Havrda and Charvát (1967). It has been of considerable interest in different domains of application like physics, finance and biomedical sciences. As well, Altun and Smola (2006) have seen that the classical maximum entropy is dual of MLE. Ditto, Ferrari and Yang (2010) proposed a new parametric estimation method based on the q–entropy function, the MLqE where qis called the distortion parameter. Also, they have proven to be a very useful method when estimating high–dimensional parameters and small tail probabilities. This is important in many applications where the number of available observations is not great. They have shown that MLqE becomes the MLE with q=1. Since for large sample the ML and Hill estimators are at least as precise as any other estimators. However, for a moderate or small sample size the MLq estimator can offer dramatic improvement in mean squared error at the expense of a slight increase in bias. This paper has been organized as follows. In Section (1) it is presented as an introduction the asymptotic distribution of the maximum of the sample under the EVT and especially the heavy–tailed distribution or the Fréchet distribution. We also gave a definition about sub–exponential. As for later, we will focus on presenting the new estimate about the tail index of heavy–tailed distributions, i.e. MLq estimator. Next, with Section (2), we present the basic asymptotic normality of MLq estimator with their consistent for exponential families which we introduce in the same section. In Section (3), we present a simulation study with Pareto distribution to compare the MLqE with MLE. Also, the real data are utilized to illustrate the usefulness of the Fréchet distribution as the distribution of hydrology data. Finally, concluding notes are provided in Section (4).
32 Kouider, Idiou, Toumi & Benatia 1.1. Adaptive ML and Hill estimators for small sample In the domain of heavy–tailed distributions, the statistic of EVT translates into a semiparametric estimation problem. Indeed, if Fbelongs to the Fréchet domain, then Fis of the form x−1/γℓ(x)with ℓa slowly varying function i.e, lim t→+∞ ℓ(tx)/ℓ(x):=1. This paper concentrates on the distributions that have a regularly varying tail, F(x) x−1/γℓ(x):=1, as x→+∞,γ>0 (10) Note that Gγ>0(x)satisfies (10). Here (1/γ)is the index of regular variation, or the tail index. Then Fhas a parametric part x−1/γdepending only on γ, and a non–parametric part ℓ. For a real t>0 it’s clear that we deduce F(x):=F(t)x t−1/γ(11) Let X1,X2, . . . , Xnbe a sequence of iid rv from distribution function (df) Fand let X1,n≤ X2,n≤. . . ≤Xn,ndenote the order statistics correspondence. We denoting the number of absolute excesses over tby kfor rather the largest observations (Xn−k,n, . . . , Xn,n)where in the asymptotic setting k=knan intermediate sequence, that is, kn→∞and kn/n→0 as n→∞. Then, see Haan and Ferreira (2006) Lemma 3.4.1, the joint distribution of (Xn−k,n, . . . , Xn,n) for k=1, . . . , n−1 be the df given by Ft(x)=P(X≤x|X>t)=F(x)−F(t) 1−F(t)for x>t(12) Such that Ft(x)=1−Ft(x)we can rewrite (12) for x>tas Ft(x)=F(x) F(t)(13) Since F(x)is given by (11), also we can rewrite Ft(x)given in (13) for x>tby Ft(x)=x t−1/γ(14) It’s easy to checked that under (5) that Ft∈RV(−1/γ). The parameter of the Ft(x)can be estimated using standard methods such as the MLE. Another estimation method is the MLqE which based on q–order entropy. The q–order entropy which is provided by Havrda and Charvát (1967), has the function Lq(u)=(u1−q−1 1−qfor q<1 log ufor q=1(15) where uis probability density function(pdf) and qcalled the distortion parameter. The MLqE, which is proposed by Ferrari and Yang (2010), use the Lq(u)function instead of the log– likelihood function as in the MLE. Observed that when q=1, we have the MLq estimator approaches as the ML estimator approaches. The MLqE method reduces the effect of extreme observations on parameter estimates using q. The choice of qis another difficult problem in
Maximum lq–likelihood estimator of the heavy–tailed distribution parameter 33 MLqE estimation. In this research, we take q=1−1 kas given by Ferrari and Yang (2010) and we note that q→1 as k→∞. Then, the MLqE of γ>0 is given by b γ=arg max k ∑ i=1 Lq(ft(x)) (16) with Lq(ft(x)) =(ft(x)1−q−1 1−qfor 0 <q<1 log ufor q=1(17) where ft(x)is the pdf of Ft(x)given in (14). Thus defense for γ>0 by ft(x)=1 γ 1 tx t−1/γ−1(18) As is known, MLq estimator of γ>0 can be obtained by maximizing ∑k i=1Lq(ft(x)) with respect to the parameter γ>0. Then, for 0 <q≤1 we get ∂ ∂γ Lq(ft(x)) :=∂ ∂γ log (ft(x)) ft(x)1−q(19) Then, the equations from (19) are then given in term of the partial derivative respect to γby: ∂ ∂γ k ∑ i=1 Lq(ft(Xi)) := k ∑ i=1 1 γ2(log (Xi)−log (t)−γ)ft(Xi)1−q=0 (20) Next, we can define the estimator of γ>0 by b γMLq t= k ∑ i=1 wi(log (Xi)−log (t)) k ∑ i=1 wi with wi=ft(Xi)1−q(21) There more, if we take t=Xn−k,nwe find a new Hill estimator called lq–Hill estimator which is defined by b γHLq Xn−k,n= k ∑ i=1 wn−i+1,nlog Xn−i+1,n−log (Xn−k,n) k ∑ i=1 wn−i+1,n with wn−i+1,n=ftXn−i+1,n1−q(22) It is clear that if q=1 gives us the classic the MLE. We get ML estimator of γ>0 which is given by: b γML t=1 k k ∑ i=1 log (Xi)−log (t)(23) And by their equivalents in the order statistic, we find the formula of the Hill estimator with t=Xn−k,nby b γH Xn−k,n=1 k k ∑ i=1 log Xn−i+1,n−log (Xn−k,n)(24)
34 Kouider, Idiou, Toumi & Benatia Also, the Hill estimator was found to be very sensitive to the choice of index k. Because choosing the optimal value for the kindex will lead to an effective estimate of Hill estimator, the choice of kis another difficult problem and there are many researches on this. However, in this article, we will present in sub–section (3.2) a method that allows calculating the number kbased on the method of the quantile type 8 with the hydrology data. It is important to mention that Hill estimator is a consistent for the tail index and asymptotically normal with mean γand variance γ2/k. Hence, if one uses a very small k, the estimator has large variance, however, for very large k, the estimator is likely to be biased i.e asymptotically normal with mean 0 and variance γ2. This is why it is good with large sample also, ML estimator is effected with large sample. In this research, we have written this tow estimator with the MLqE. We can show that for q=1−1 kwe rewrite the estimators b γMLq tand b γHLq Xn−k,nas b γMLq t= k ∑ i=1 ft(Xi)1/k(log (Xi)−log (t)) k ∑ i=1 ft(Xi)1/k (25) and b γHLq Xn−k,n= k ∑ i=1 ftXn−i+1,n1/klog Xn−i+1,n−log (Xn−k,n) k ∑ i=1 ftXn−i+1,n1/k (26) Recall that when qis chosen correctly for small samples, the MLqE can trade bias for accuracy successfully. This leads to a significant decrease in the mean squared error. There are more for large sample if (k→∞)q=1−1 kor q→1 we focus on a necessary and sufficient condition, to ensure a proper asymptotic normality and efficiency of MLqE is established. This is what will be discussed in the next section. 2. Main result In this section, we discuss the basic asymptotic properties of the MLq estimator when the degree of distortion depends on the amount of information available in the sample. Such properties will be used later on to derive our main results. In the reminder of the paper, our analysis focuses on the distributions belonging to the exponential family. In particular, we consider pdf of Ft(x)given on (22) in the form ft(x)=exp 1 γb(x)−c(x)−A(γ)(27) For b(x)=log x tand c(x)=log (x),A(γ)=log (γ) In addition, we define ψγ(x)=1 γb(x)−c(x)−A(γ). So that ft(x)and log ft(x)with its derivative function respect to γexpressed respectively as ft(x)=exp ψγ(x)and ∂ ∂γ ft(x)=−1 γ2b(x)−1 γexp ψγ(x), (29)
Maximum lq–likelihood estimator of the heavy–tailed distribution parameter 35 and log ft(x)=ψγ(x)and ∂ ∂γ log ft(x)=−1 γ2b(x)−1 γ. (30) Until, for γ>0 we can check that A(γ)=log +∞ Zt exp 1 γb(x)−c(x)dx =log γ, (31) is the cumulative generating function (or log normalize) and differentiating ktimes gives ∂k ∂kγA(γ)=(−1)k−1(k−1)! γk, (32) Throughout the course of the discussion the true parameter will be denoted by γ>0 . Next, we explore consistency, which is a basic requirement for a good estimator. Let, for 0<q≤1 φk(γ)=1 k k ∑ i=1 ∂ ∂γ Lq(ft(Xi)) :=1 k k ∑ i=1−1 γ2b(x)−1 γe(1−q)1 γb(x)−c(x)−A(γ)(33) The MLqE is found by setting φk(γ)=0 and solving for γ. Since for q=1 and γ=0 in the above expression gives the usual MLE equation 1 k k ∑ i=1−1 b γMLq b(x)−1=0 (34) Theorem 2.1. Let X1,X2, . . . , Xnbe a sequence of iid rv from df F and let X1,n≤X2,n≤. . . ≤Xn,n denote the order statistics correspondence. Considering the largest observations (Xn−k,n, . . . , Xn,n) for k =1, . . . , n−1from the df Ft(x)and pdf ft(x)as (27). Then, for any MLq estimator of b γ=arg max ∑k i=1Lq(ft(x)) as k →∞we have that b γP →γ. with Lq(ft(x)) is given in (15) and q →1as k →∞. Proof. Define, φ(γ)=Eγh∂ ∂γ log ft(x)i. Then we can rewrite φ(γ)=Eγh−1 γ2b(x)−1 γi. Now, we want to show that for all φk−φ→0 as k→∞where φk−φ=1 k k ∑ i=1−1 γ2b(x)−1 γe(1−q)ψγ(x)−Eγ−1 γ2b(x)−1 γ(35) Then we have |φk−φ|= 1 k k ∑ i=1−1 γ2b(x)−1 γe(1−q)ψγ(x)−1+1 k k ∑ i=1−1 γ2b(x)−1 γ−Eγ−1 γ2b(x)−1 γ thus ≤ 1 k k ∑ i=1−1 γ2b(x)−1 γe(1−q)ψγ(x)−1+ 1 k k ∑ i=1−1 γ2b(x)−1 γ−Eγ−1 γ2b(x)−1 γ (36)
36 Kouider, Idiou, Toumi & Benatia By the law of large numbers we find 1 k k ∑ i=1−1 γ2b(x)−1 γ−Eγ−1 γ2b(x)−1 γ→0 (37) Then the inequality (35) becomes |φk−φ|= 1 k k ∑ i=1−1 γ2b(x)−1 γe(1−q)ψγ(x)−1(38) By the Hölder’s inequality, we can rewrite (38) as follows |φk−φ|≤v u u t1 k k ∑ i=1e(1−q)ψγ(x)−12v u u t1 k k ∑ i=1−1 γ2b(x)−1 γ2 (39) And under Jensen’s inequality we have |φk−φ|≤1 4 1 k k ∑ i=1e(1−q)ψγ(x)−121 k k ∑ i=1−1 γ2b(x)−1 γ2 (40) For the left side from the previous inequality and by the basic fact that (1+u)2≤e2ufor any real number uwe rewrite 1 γ2k k ∑ i=1−1 γb(x)−12 ≤1 γ2k k ∑ i=1 e2 γb(x)=1 γ2k k ∑ i=1x t−2 γ(41) Then we have 1 γ2k k ∑ i=1 e2 γb(x)=1 γ2k k ∑ i=1 Ft(x)2<∞(42) And for the right side of the inequality that under the law of large numbers we get 1 k k ∑ i=1e(1−q)ψγ(x)−12→Eγe(1−q)ψγ(x)−12(43) With q→1 as k→∞we have Eγe(1−q)ψγ(x)−12→0 Finally we obtained that φk→φas k→∞. As defined in theorem (2.1), the MLqE is a consistent estimator of γ. Although for a fixed q=1. The MLqE is clearly asymptotically biased; a clear improvement is obtained by letting the distortion parameter depends on the sample size. To obtain the asymptotic normality of MLqE, we shall discuss the reduction in terms of variance achieved by considering a slightly different target parameter for each γ>0 and k≥1. On particular, we consider b γMLq the value such that E∂Lq(ft(x)) ∂b γMLq =0 (44)
Maximum lq–likelihood estimator of the heavy–tailed distribution parameter 43 4. Concluding notes Heavy–tailed distributions would be a good alternative to the distributions that are used in economics, reliability, survival analysis and so on. The parameters of this distribution have been estimated using MLE method. Recently, ML and Hill estimators are the most estimators used to estimate the parameter of the tail behavior. For too large sample, ML and Hill estimators are likely to be good and robust methods should be used to estimate the shape parameter. However, ML and Hill estimators cannot be compatible with the small sample. In this paper, we have used the MLq estimation method to estimate the shape parameter of the heavy–tailed distribution for small sample. We have carried out to adapt the ML and Hill estimators in the case of small sample, b γMLq tand b γHLq Xn−k,nas in (25) and (26), respectively. Since b γMLq tand b γHLq Xn−k,nare both result from the MLqE, we establish the asymptotic normality of the MLq estimator of Heavy tailed distributions parameter γ>0. According to corollary (2.1) for very large k, the MLq estimator is likely to be biased. And if q=1, the MLqE becomes MLE and the normality asymptotic of MLq estimator becomes as ML estimator. Since the MLq estimator depends on the order statistical index kfor heavy tailed distributions, we used an approach for selecting kwhich is defined in (62) by using type 8 quantile estimator Q8(0.7) in (60) from the stable region of Hill plot, which will be a more flexible alternative for use in Hill’s estimator. This approach will be an approximation of the extreme value index at the tail end of the distribution of hydrology data. Appendix First, we calculate E∂Lq(ft(x)) ∂e γ2. Let’s consider E"∂Lq(ft(x)) ∂e γ2#=1 γe γ6t1 γ+2 e γe γ2qt−2q e γ ∞ Zt (log (Xi)−log (t)−e γ)2e−[2eα(1−q)+α]log xdx (66) with eα=1+1 e γ. Putting log (x)=u, then the previous equality (66) becomes E"∂Lq(ft(x)) ∂e γ2#=β ∞ Z log tu2−2u(log t+e γ)+(log t+e γ)2e−[2eα(1−q)+α−1]udu (67) where β=1 γe γ6t1 γ+2 e γe γ2qt−2q e γis constant with α=1+1 γ. Then, for e γ=q/(α−q)we have qeα=αwith e γ=1/ (eα−1). For e θ=eα(q−1)+α−1=eα−1then 2eα(1−q)+α−1= eα(2−q)−1. Consider this decomposition D(t)= ∞ Z log tu2−2u(log t+e γ)+(log t+e γ)2e−[eα(2−q)−1]udu =I1−I2+I3(68) where I1= ∞ Z log t u2e−[eα(2−q)−1]udu,
44 Kouider, Idiou, Toumi & Benatia I2= ∞ Z log t 2u(log t+e γ)e−[eα(2−q)−1]udu And I3= ∞ Z log t (log t+e γ)2e−[eα(2−q)−1]udu Beginning with I1=R∞ log tu2e−[eα(2−q)−1]udu, under integral by parts we have I1= ∞ Z log t u2e−[eα(2−q)−1]udu =−1 eα(2−q)−1 hu2e−[eα(2−q)−1]ui∞ log t−2 ∞ Z log t ue−[eα(2−q)−1]udu (69) Also with integral by parts we have ∞ Z log t ue−[eα(2−q)−1]udu =−1 eα(2−q)−1 "ue−[eα(2−q)−1]u+e−[eα(2−q)−1]u eα(2−q)−1#∞ log t (70) Substituting (26) into I1and since eα(2−q)−1>0 we get I1=e−[eα(2−q)−1]log t eα(2−q)−1(log t)2+2 eα(2−q)−1log t+1 eα(2−q)−1 (71) Then we go to I2=R∞ log t2u(log t+e γ)e−[eα(2−q)−1]udu, . By (26) and with eα(2−q)−1>0 we have I2=2(log t+e γ)e−[eα(2−q)−1]log t eα(2−q)−1log t+1 eα(2−q)−1(72) And for I3=R∞ log t(log t+e γ)2e−[eα(2−q)−1]udu, under-performing simple arithmetic operations, we get I3=(log t+e γ)2"−e−[eα(2−q)−1]u eα(2−q)−1#∞ log t =(log t+e γ)2e−[eα(2−q)−1]log t eα(2−q)−1(73) Under (71)-(73) and D(t)=I1−I2+I3we have (eα(2−q)−1)e[eα(2−q)−1]log tD(t)=1+(eαe γ(1−q))2 (eα(2−q)−1)2(74) Then it gives us that E"∂Lq(ft(x)) ∂e γ2#=βe−[eα(2−q)−1]log t eα(2−q)−1 1+(eαe γ(1−q))2 (eα(2−q)−1)2!(75) Since β=1 γe γ6t1 γ+2 e γe γ2qt−2q e γ
Maximum lq–likelihood estimator of the heavy–tailed distribution parameter 45 Next, for eαe γ=1+e γwe can find that E"∂Lq(ft(x)) ∂e γ2#=1 γt1 γe2(q−3)log e γ+2 e γ(1−q)−(eα(2−q)−1)log t eα(2−q)−1 1+((1+e γ) (1−q))2 (eα(2−q)−1)2! Then E"∂Lq(ft(x)) ∂e γ2#=1 γt1 γe2(q−3)log e γ−(q(eα−2)+1)log t 1+((1+e γ) (1−q))2 (eα(2−q)−1)3!(76) Next, we have to calculate E∂2Lq(ft(x)) ∂2e γ. Then, consider E" ∂2Lq(ft(x)) ∂2e γ!#=∂ ∂e γ ∞ Zt ∂Lq(ft(x)) ∂e γft(x)dx (77) Referring to equality (48), we conclude that E" ∂2Lq(ft(x)) ∂2e γ!#=∂ ∂e γ1 γt1 γe γq−3t1 e γ(1−q)1 e θ2−1 e θe−e θlog t =∂ ∂e γ1 γt1 γ1 e θ2−1 e θe−e θ+1 e γ(1−q)log t+(q−3)log e γ where e θ=eα(1−q)+α−1 with eα=1+1/e γimplies ∂e θ/∂e γ=−(1−q)/e γ2and ∂eα/∂e γ= −1/e γ2. Since ∂e−e θ+1 e γ(1−q)log t+(q−3)log e γ ∂e γ=(q−3) e γe−e θ+1 e γ(1−q)log t+(q−3)log e γ Therefore, we have E" ∂2Lq(ft(x)) ∂2e γ!#=1 γt1 γe−e θ+1 e γ(1−q)log t+(q−3)log e γ1−q e γ2e θ2−2(1−q) e γ2e θ3+q−2 e θ−q−3 e γe θ2 Under e θ=eα(1−q)+α−1=1/e γ. Then, we get E" ∂2Lq(ft(x)) ∂2e γ!#=1 γt1 γe−1 e γlog t+(q−3)log e γ((1−q) (1−2e γ)+e γ)(78) Now, we have to count the variance of √kb γMLq −e γ/σ→N(0; 1)as k→∞. With σ2= E∂Lq(ft(x)) ∂e γ2 Eh∂2Lq(ft(x)) ∂2e γi2(79) And under (76)-(78) we get σ2=1 γt−1 γe21 e γqlog t−(q(eα−2)+1)log t 1+((1+e γ)(1−q))2 (eα(2−q)−1)3 ((1−q) (1−2e γ)+e γ)2 (80)
46 Kouider, Idiou, Toumi & Benatia It’s easy to find that −(q(eα−2)+1)log t+2 e γqlog t=1 γlog t Since qeα=αwe find, γ=1/ (qeα−1). Then the variance becomes σ2=1 (qeα−1) 1+((e γ+1) (1−q))2 (eα(2−q)−1)3((1−q) (1−2e γ)+e γ)2. (81) with q=1 then eα=αand e γ=γthe variance σ2write as σ2=1 (α−1) 1 (α−1)3γ2(82) Since (α−1)=1/γwe have σ2=γ2. References Altun, Y. & Smola, A. (2006). Unifying divergence minimization and statistical inference via convex duality. In: Lugosi, G. & Simon, H. U. (Eds) Learning Theory, Conference Proceedings of the 19th Annual Conference on Learning Theory – COLT 2006, 139–153. Springer: Berlin. doi: 10.1007/11776420_13 Balkema, A. A. & de Haan, L. (1974). Residual life time at great age. The Annals of Probability, 2(5):792–804. doi: 10.1214/aop/1176996548 Boonradsamee, J., Bodhisuwan, W. & Jaroengeratikun, U. (2021). An estimation approach of classical tail index and an application to hydrology assessment. Proceedings of the World Congress on Engineering – WCE 2021, 19–24. https://www.iaeng.org/publicationWCE2021/WCE2021_pp19-24.pdf Davison, A. C. & Smith, R. L. (1990). Models for exceedances over high thresholds. Journal of the Royal Statistical Society, 52(3):393–425. doi: 10.1111/j.2517-6161.1990.tb01796.x Drees, H., Ferreira, A. & den Haan, L. (2004). On maximum likelihood estimation of the extreme value index. The Annals of Applied Probability, 14(3):1179–1201. doi: 10.1214/105051604000000279 Embrechts, P., Klüppelberg, C. & Mikosch, T. (1997). Modelling extremal events for insurance and finance. Springer: New York. doi: 10.1007/978-3-642-33483-2 Ferrari, D. & Yang, Y. (2010). Maximum lq–likelihood estimation. The Annals of Statistics, 38(2):753–783. doi: 10.1214/09-AOS687 Fisher, R. A. & Tippett, L. H. C. (1928). Limiting forms of the frequency distribution in the largest particle size and smallest member of a sample. Mathematical Proceedings of the Cambridge Philosophical Society, 24(2):180–190. doi: 10.1017/S0305004100015681 Gnedenko, B. (1943). Sur la distribution limite du terme maximum d’une série aléatoire. Annals of Mathematics, 44(3):423–453.doi: 10.2307/1968974 Haan, L. & Ferreira, A. (2006). Extreme value theory: An introduction. Springer: New York. doi: 10.1007/0-387-34471-3 Havrda, J. & Charvát, F. (1967). Quantification method of classification processes: Concept of structural a−entropy. Kybernetika, 1(3):30–35. https://www.kybernetika.cz/content1967/1/30/paper.pdf
Maximum lq–likelihood estimator of the heavy–tailed distribution parameter 47 Hill, B. M. (1975). A simple general approach to inference about the tail of a distribution, The Annals of Statistics, 3(5):1163–1174. doi: 10.1214/aos/1176343247 Hyndman, R. J. & Fan, Y. (1996). Sample quantiles in statistical packages. The American Statistician, 50(4):361–365.doi: 10.2307/2684934 Kouider, M. R. (2019). On maximum likelihood estimates for the shape parameter of the generalized Pareto distribution. Science Journal of Applied Mathematics and Statistics, 7(5):89– 94. doi: 10.11648/j.sjams.20190705.15 Kouider, M. R. & Benatia, F. (2023). Modified bisection algorithm in estimating the extreme value index. International Conference of Young Mathematicians, The Institute of Mathematics of the National Academy of Sciences of Ukraine. https://www.imath.kiev.uayoung/youngconf2023/Abstracts_2023/PS/Kouider-Benatia.pdf Kouider, M. R., Idiou, N. & Benatia, F. (2023a). Modified bisection algorithm in estimating the extreme value index under random censoring. TWMS Journal of Applied and Engineering Mathematics, 13(4):1408–1422. https://jaem.isikun.edu.tr/web/images/articlesvol.13.no.4/13.pdf Kouider, M. R., Idiou, N. & Benatia, F. (2023b). Adaptive estimators of the general Pareto distribution parameters under random censorship and application. Journal of Science and Arts, 23(2):395–412. doi: 10.46939/j.sci.arts-23.2-a07 Kouider, M. R., Idiou, N. & Benatia, F. (2024). Adaptive hazard estimator of the peak over threshold model under random censoring with application. Studies in Engineering and Exact Sciences, 5(2):1–24. doi: 10.54021/seesv5n2-243 Kouider, M. R., Kheireddine, S. & Benatia, F. (2024). The adaptive mean estimation of heavy tailed distribution under random censoring. Studies in Engineering and Exact Sciences, 5(2):1– 31. doi: 10.54021/seesv5n2-164 Pickands III, J. (1975). Statistical inference using extreme order statistics. The Annals of Statistics, 3(1):119–131. doi: 10.1214/aos/1176343003 Ramos, P. L., Louzada, F., Ramos, E. & Dey, S. (2018). The Fréchet distribution: Estimation and application an overview. Repository arXiv. doi: 10.48550/arXiv.1801.05327 Smith, R. L. (1987). Estimating tails of probability distributions. The annals of Statistics, 15(3):1174–1207. doi: 10.1214/aos/1176350499
48 Kouider, Idiou, Toumi & Benatia nista Procjenitelj najve´ce lq–vjerodostojnosti parametra distribucije s teškim repom VRSTA ˇ CLANKA Prethodno priop´cenje INFORMACIJE O ˇ CLANKU Primljeno: 7. rujna 2024. Prihva´ceno: 5. studenog 2024. DOI: 10.62366/crebss.2024.2.003 JEL: C13, C46 SAŽETAK Prouˇcavanje teorije ekstremnih vrijednosti (EVT) ukljuˇcuje nekoliko glavnih ciljeva, me ¯ du kojima je procjena parametra repnog indeksa. Razliˇcite metode procjene koriste se za procjenu ovog parametra, poput metode najve´ce vjerodostojnosti (MLE). Dodatno, Hillov procjenitelj je jedan tip procjenitelja najve´ce vjerodostojnosti koji je robusniji kod velikih uzoraka nego kod malih. Ovo istraživanje predlaže konstrukciju alternativnog procjenitelja za parametar distribucije s "teškim repom" koriste´ci pristup najve´ce lq–vjerodostojnosti (MLqE), kako bi se prilagodili MLE i Hillov procjenitelj za male uzorke. Nadalje, utvr ¯ dena je asimptotska normalnost procjenitelja najve´ce lq–vjerodostojnosti. Osim toga, provedene su simulacijske studije kako bi se usporedio MLq procjenitelj s MLE procjeniteljem. U sluˇcajevima prekoraˇcenja visokih razina pragova, prikladnih vrijednosti, broj najve´cih opažanja kvodi do efikasne procjene pomo´cu Hillovog procjenitelja. U tu svrhu, izbor k kod Hillovog procjenitelja istražen je metodom kvantila tipa 8, koja se pokazala uˇcinkovitom u analizi podataka iz hidrologije. Uˇcinkovitost Hillovog procjenitelja i lq–Hillovog procjenitelja zatim je uspore ¯ dena primjenom stvarnih podataka s distribucijom hidroloških vrijednosti. KLJU ˇ CNE RIJE ˇ CI prekoraˇcenja iznad razine praga, indeks ekstremnih vrijednosti, distribucija s teškim repom, procjenitelj najve´ce lq– vjerodostojnosti