Exact

Exact inference and optimal invariant estimation for the tail coefficient of symmetricα-stable∗distributionsJean-Marie Dufour†C.R.D.E.and D´e partment de Sciences EconomiqueUniversit´e de Montr´e al,Montr´e al,Qu´e bec H3C3J7,CanadaJeong-Ryeol Kurz-Kim‡Deutsche BundesbankWilhelm-Epstein-Straße14,60431Frankfurt am Main,GermanyAbstractBecause of its simplicity the Hill estimator(Hill,1975)is the most popu-lar one for estimation of the stability parameter ofα-stable distributions.Itsconfidence interval forfinite samples,however,can be given only based on theasymptotic distribution,which can be far different from thefinite sample dis-tributions.In this paper,using the Monte-Carlo method we provide the Hillestimator with exact confidence intervals forfinite samples.By performingconfidence intervals forfinite samples,we use the so-called Hodges-Lehmannestimator(Hodges and Lehmann,1963).A powerful advantage of the Hodges-Lehmann estimator is that the truncation parameter for tail,k/n(which de-pends on unknownα)do not need to be assumed as known.It reveals alsothat the Hodges-Lehmann estimate is more efficient than the Hill estimate inthe sense of mean square error.Furthermore,we alsofind that median is theoptimal centering for the Hill estimator.JEL Classification: C1;C15;C22;G0;E0∗The views expressed in this paper are those of the authors,and not necessarily those of the Deutsche Bundesbank.Research support from the Alexander-von-Humboldt Foundation is gratefully acknowledged by the two authors†e-mail:jean.marie.dufour@umontreal.ca‡e-mail:jeong-ryeol.kurz-kim@bundesbank.deKeywords: α-stable distribution;Hill estimator;Monte-Carlo test;exact confidence interval;finite-sample;Hodges-Lehmann estimator.1IntroductionSince the influential works of Mandelbrot(1963),stable Paretian(shortly,α-stable) distributions have often been considered to be a more realistic distribution for high frequency variables,such asfinancial data than the normal distribution,because asset returns,for example,are typically heavy–tailed and excessively peaked around zero—phenomena that can be captured byα-stable distributions withα<2.Statistical inferences for estimations and hypothesis tests under theα-stable distributional assumption depends crucially onα,see DuMouchel(1973a,1975)and Mittnik et al.(1998).Therefore,one of the most important tasks for using theα-stable distribution is tofind exact confidence intervals forfinite samples for estimated α.To discriminateα<2fromα=2would be one example how important it is to know what the exact confidence interval is.However,the confidence interval of all the estimators usually used can be only given by the asymptotic distribution which can be far different from that forfinite samples.The followings are the usually used estimators forα:the most popular estimation forαis the Hill estimator(Hill, 1975)which is a simple nonparametric estimator based on order statistic.Pickands (Pickands,1975)and Dekkers,Einmal and Haan estimator(Dekkers,Einmal and de Haan,1989)are some variations of the Hill estimator.For a roughly checking,the quantile estimation of McCulloch(1986)may be also used.Some modifications are also considered by many authors:Mittnik and Rachev(1998)modify the Pickands estimator using high order approximations and Huisman et al.(2001)propose a weighted Hill estimator taking the trade-offof bias and variance of the Hill estimator into account.One more drawback of the Hill estimator and its modified estimators is that the truncation parameter,ratio of the observations in tails to the sample size,must be known(which depends again on unknownα).In this paper,using Monte Carlo(MC)method we propose an exact test method and at the same time an estimation1 for the stability parameter ofα-stable distri-butions.This is because one can construct an exact confidence interval forfinite samples in the estimation procedures,or rather an estimate in the test procedure. Our MC method improves the Hill estimation in three ways.First,it provides exact confidence intervals forfinite samples.Second,the truncation parameter for tail, k/n do not need to be assumed as known for our estimator.Third,the Monte Carlo estimate is more efficient than the Hill estimate in the sense of mean square error.By doing this,we additionally show that the Hill estimator based on median-relocated data is more efficient than that based on mean-relocated.The rest of the paper is organized as follows.Section2gives a short summary of 1An estimation procedure which is based on MC method is termed Hodges-Lehmann estimation in the literature.See Hodges and Lehmann(1963)for the basic idea.α-stable distributions and Hill estimator. The procedures of estimation and test are described as well. In Section 3, the MC method based estimation and test procedure are explained. In Section 4 various simulation studies demonstrate i) the power of the MC method based test, ii) compare confidence intervals for finite samples of the Hodges-Lehmann estimate with those of the Hill estimate, and iii) efficiency of the two estimates. An empirical application is given in Section 5. Section 6 summarizes the paper and contain some concluding remarks.2 Framework2.1 A brief summary on α-stable distributionsA r.v. X is said to be stable if for any positive numbers A andB , there is a positived number C and a real number D such that AX 1 + BX 2 = CX + D, where X 1 d d and X 2 are independent r.v.’s with X i = X, i =1, 2; and “ = ” denotes equality indistribution. Moreover, C =(A α + B α)1/α for some α ∈ (0, 2], where the exponent α is called index of stability. When 0 <α < 2, the tails of the distribution are thicker than those of the normal distribution. The tails become thicker as α decreases such that moments of order α or higher do not exist. A stable r.v., X , with index α is called α-stable. The α-stable distributions are described by four parameters denoted by S (α, β, µ, σ). Although the α-stable laws are absolutely continuous, their densities can be expressed only by complicated special function except some special cases.2 Therefore, the logarithm of the characteristic function, φ(t ), of the α-stable distribution is the best way of characterizing all members of this family and given as ∞ ln φ(t )=ln e ist d P(S < s )= −σα|t |α[1 − iβ sign(t )tan πα =1,2 ]+ i µt, for α −∞ −σ|t |[1 + iβ π 2sign(t )ln |t |]+ i µt, for α =1, The shape of the α-stable distribution is determined by the stability parameter α.For α =2 t he α-stable distribution reduces to the normal distribution, (i.e. S (2, 0,σ,µ) ≡ N (µ, 2σ2)), the only member of the α-stable family with finite vari-ance. If α< 2, moments of order α or higher do not exist and the tails of the distri-bution become thicker i.e., the magnitude and frequency of outliers (from the view-point of the Gaussian) increase as α decreases. Skewness is governed by β ∈ [−1, 1]. When β = 0, the distribution is symmetric. The location and scale of the α-stable distributions are denoted by µ and σ. The standardized version of the α-stable distribution is given by S ((x − µ)/σ; α, β, 1, 0).2The Gaussian (α = 2) the symmetric Cauchy (α =1,β = 0) and the L´e vy distribution (α =0.5,β = 1) are the special cases whose densities are expressible via elementary functions.Assume that random sequence {X i }∞ is in the domain of attraction 3 (DA) ofi =1 an α-stable law with index α ∈ (0,2). This is equivalent to saying that X 1,X 2,··· are i.i.d. r.v.s and that constants a n > 0and b n ∈ R exist, such thatd b −1(S n −a n ) −→ S (α), (1)n d where S n = X 1 + ···+ X n ; S (α) is an α-stable r.v.;and “−→ ” denotes convergence in distribution. If the X i ’s are α-stable, then (1) holds with b = n n 1 α, and 1−µ(n 1 α−1), for α =1, a n = 2 σln n, for α =1,π(2) dand b −1(S n −a n )= X 1. The assumption that the X i ’s are in the DA of an α-stablen law is more general than assuming that they are α-stable distributed, because the former requires only conditions on the tails of the distribution. DA condition (1) is equivalent toP(|X 1| >x )= x −αL (x ),x > 0, (3) where L (x ) is a slowly varying function.4A strong argument in favor of the α-stable distribution for a distributional assumption for heavy-tailed empirical data is that only the α-stable distribution can serve as limiting distribution of sums of i.i.d. r.v.s as proved in Zolotarev (1986). For more details on the α-stable distributions and discussions of the role of the α-stable distribution in financial market and macroeconometric modelling, see Zolotarev (1986), Samorodnitsky and Taqqu (1994), McCulloch (1996), Rachev et al. (1999) and Rachev and Mittnik (2000).2.2 Hill estimator and two practical problems for its appli-cation2.2.1 Hill estimatorThe most popular estimation for α is Hill-estimator (Hill, 1975) which is a simple nonparametric estimator based on order statistic. Because of its simplicity and popularity, we use Hill estimator for constructing our test statistic.5 Given a sample of n observations, X 1,X 2,...,X n the Hill estimator is given as−1k ˆαH = k −1 ln X n +1−j :n −ln X n −k :n , (4)j =13See Samorodnitsky and Taqqu, 1994, p. 5 for definition of domain of attraction.L (cx )4L (x ) is a slowly varying function as x →∞,if for every constant c >0,lim x →∞ L (x ) existsand is equal to 1. See Ibragimov and Linnik (1971, p. 394) for more details on slowly varying functions.5Basically, any consistent estimator can be used to formulate our test statistic.with standard errorkˆαHSD(ˆαH)= ,(5)(k−1)(k−2)where k is number of observations which lie on the tails of distributions of interest to be optimally chosen depending on the sample size,n,and tail–thickness,α, as k=k(n,α),and X j:n denotes the j-order statistic of the sample size n.The asymptotic properties of the Hill estimator are studied by many authors and now well developed:Mason(1982)and Hsing(1992)consider weak consistency of the Hill estimator for independent and dependent case,respectively.The strong consistency is proved by Deheuvels et al.(1988).Goldie and Smith(1987)provide asymptotic normality of the Hill estimator,i.e.√k(ˆ−α−1)∼N(0,ˆα−2).(6)α−1HFor estimating the tail–thickness parameter via the Hill estimator,however,two practical problems should be solved.Thefirst one is how to choose the truncation parameter,k/n,and the other is how to choose the centering parameter from the empirical data.2.2.2Choice of optimal kThe theoretical relation between k andαcan be also driven from the second-order property as(see de Haan and Peng,1998)α−1k=n.(7)Γ(2−α)sin(π∗α/2)The problem for using this relation is not practicable because k is again a function of the unknownα.Note also that the choice of k involves a tradeoff,because it must be sufficiently small so that the observation,X n−k:n is in the tail of the distribution; but if too small,the estimator will lack precision.Mittnik et al.(1998)provide the optimal truncation parameters for some selected sample sizes andαwhich are chosen by simulation.However,certainly in practice,the true value ofαis not known,so that the relation between k andα—whether theoretical and/or simulative—is only of theoretical interest.This is a severe drawback for the Hill estimator as like Pickands-and Dekkers,Einmal and Haan estimator from the viewpoint of empirical relevance.As will be shown,for our MC method based estimation procedure,namely the Hodges-Lehmann estimator,αdo not need to be assumed as known,which is one of the most attractive advantages for our MC method.2.2.3Centering of data for the Hill estimationThe other practical problem is how to choose a centering parameter for re-location of empirical data for the Hill estimator.Note that the Hill estimator is scale-invariant,but not location-invariant so that X has to be centered in a proper way in the beginning of the estimation.The theoretical centering for variables in the DA of an α-stable law is as given in(2)which contains the unknownα.Despite of existence of thefirst moment for1<α<2,the mean can not often serve optimally as a centering parameter because of itsfluctuation,especiallyαis small.Therefore, median is an alternative choice as centering parameter.Regarding centering we perform two simulations.Thefirst simulation shows how the Hill estimator works if one do not take the non-zero mean into account,i.e.a centering is ignored.The second one shows efficiency of the Hill estimator among three centerings,namely true mean,sample mean and sample median.The case for true mean is not of empirical relevance,but it will be a bench mark for the other two sample statistics.Thefirst simulation is designed asα=1.0,1.25,1.5,1.75,1.95,2,β=0,µ=0,0.1,0.5,1andσ=1with sample size of n=100,250,500,1000,5000 10,000replications were made.For estimation we use the usual Hill estimator. The only difference of the design for the second simulation is thatµ=0and we have three re-locations,where the sample is re-located by true mean,by estimated sample mean and by the estimated sample median.The pseudoα-stable r.v.s were generated with the algorithm of Chambers et al.(1976)improved by Weron(1996).6 The results of the simulations are summarized in Table1and2in Appendix.Table 1shows as is expected that if the non-zero mean is ignored for the Hill estimator the lost of efficiency of the estimate is severe for all adaptedαand n.The lost of efficiency is the more larger,the bigger the difference between the sample mean and zero grows.Table2shows that the case of using the median as a centering parameter is almost as efficient as the case of using the true mean for allαand n adopted in the simulation.Furthermore,it is clearly shown that the case of using the median as a centering parameter is more efficient than those using the mean in the sense of mean square error for allαand n adopted in the simulation.The difference of the two root mean squares for the case median and mean is the larger, the smallerαis,which is expected because of largefluctuation for smallαand the difference remains even for large sample size(n=5000).3 Monte-Calro method based estimation and testprocedureIn this Section,we introduce the Monte-Calro method based estimation and test procedure.The technique of MC method,originally proposed by Dwass(1957)for implementing permutation tests and later extended by Barnard(1963)and recently6The same random generator will be used for all the followings simulations.reconsidered by Dufour(2004),provides an attractive method of building exact testsfrom statistics whosefinite sample distribution is intractable but can be simulated.The most compromising advantage of the MC method is that in contrast to boot-strap techniques and other conventional test methods which have only asymptoticjustification,an exactfinite-sample inference can be obtained.Consequently,the validity of this MC method based exact randomized test does not depend on the number of replications made.For more details on the MC method,see Birnbaum (1974),and Dufour and Kiviet(1998).The idea of the MC based estimator goes back to the work of Hodges and Lehmann(1963).For our random sample a assumption needed is given in the following Assump-tion.Assumption1X is an i.i.d. α-stable distributed r.v., i.e. X1,X2,...,X n∼S(α,0,µ,σ).Note that X is not strictlyα-stable r.v.Assumption1is somewhat a generalizationfrom the usual assumption of standard symmetricα-stable r.v.(i.e.,c=1,µ=0), because the centering parameter is also an important issue from the viewpoint of empirical relevance.Now,we test our random sample,{X1,X2,...,X n},forH0(α0):α=α0.(8)To perform this test,we need a test statistic which is free from nuisance parameters under the null hypothesis.A possible statistic can be given asST=ˆα−α0,(9) whereˆαmay be an any αmay be an any consistent estimator forα.The fact thatˆconsistent estimator forαmeans that our MC method can provide any consistent estimator with exact confidence interval forfinite samples,which is usually basedon asymptotic distributions.This is a strong advantage of our MC method.Specifically,we apply the Hill estimator for our test statistic as:ST H=ˆαHM−α0,(10)where the Hill estimator,αHM,is based on re-located data by median and can be estimated as⎡⎛⎞⎤−1k(α0)ˆαHM(α0)=⎣⎝k−1 ln˜X n+1−j:n⎠−ln˜X n−k:n⎦,(11)j=1˜˜with X i:=X i−X med,where X j:n stands for the j-th order statistic of the samplesize n.Note that the truncation parameter of the Hill estimator in(11)do notdepend any more on unknown truncation parameter for tail k/n,butα0 under the null hypothesis.This enables us to use the theoretical relation between k and n as in(7)and/or the optimal k/n ratio reported in Mittnik et al.(1998).This is a further strong advantage of our MC method besides exact confidence intervals for finite samples.To estimate the stability parameter using our Monte Carlo based method,the test statistic in(10)should be nuisance free.Because the estimator in11is location and scale invariant the test statistic10is pivotal as proved in the following lemma.Proposition1[Invariance] Let X1,X2,...,X n be i.i.d. random variables which follow a S(α,β,µ,σ)distribution, and letˆα=a(X1,X2,...,X n)(12)be an estimator of α.If the estimator ˆαis scale invariant, i.e.ˆα=a(cX1,...,cX n)=a(X1,X2,...,X n),for all c>0,(13)then the estimator ˆαhas a distribution which only depends on α,βand µ/σ.If ˆfurthermore, αis location-scale-invariant, i.e.ˆα=a(cX1+d,...,cX n+d)=a(X1,X2,...,X n),for all c>0and d∈R,(14)ˆthen the estimator αhas distribution which only depends on β.Proof1To get the first result, we observe thatX i/σ∼S(α,β,µ/σ,1),i=1,...,n.(15)Then, using the scale-invariance property (13) with c=1/σ,we can write:ˆα=a(X1/σ,...,X n/σ),(16)from which we see that the distribution of ˆαonly depends on α,βand µ/σ.Similarly, under the location-scale invariance condition (14), the result follows on observing that(X i−µ)/σ∼S(α,β,0,1),i=1,...,n,(17) hence, taking c=1/σand d=−d/σ,∗∗ˆα=a(X1,...,X n),(18)∗=(X i−µ)/σ,i=1,...,n.where XiThe MC based estimation and test procedure is as follows:Given a random sample,{X1,X2,...,X n},which corresponds Assumption1.1Determine the set of possibleαunder null hypothesis.From the viewpoint of empirical relevance it will be reasonablyα0 ∈(12].2Calculate test statistics(ˆαHM−α0)for everyα0,where the step length of two neighborhood ofα0 may be0.01,for example.3Generate typically99or999samples for every element of the set by a stable random variable generator and calculate test statistics.4Compute p-values under the every possible null hypothesis.5Take theˆαHM as the estimate forαat which the p-value has its minimum and will be usually zero.This is Hodges-Lehmann estimate.αl andˆ6Take theˆαr as the left and right limit ofη%confidence interval at which the p-value is(1−η)/100,whereˆαr.This is now the exact confidenceαl<ˆinterval for the Hodges-Lehmann estimate in the step6.In comparison with Hill estimator(and also other usually used Pickands-and Dekkers,Einmahl and de Haan estimator)three advantages of our MC method based estimation and test are summarized once more:first,our MC test provides exact confidence intervals forfinite samples,whereas confidence intervals for the Hill estimates are based on the asymptotic distributions.Second,for our estimation the truncation parameter,k/n must not be assumed to be known.Note that the Hill estimator in(4)contains an unknown truncation parameter for tail,k/n which depends again on the unknownα.This is because in our estimation procedure every possibleαunder H0 is checked and the optimal k/n can be chosen by the theoretical relation in(7)and/or the optimal k/n ratio in Mittnik et al.(1998).Third,as will be shown the distribution of our MC estimates has a smaller bias and a standard deviation as well than that of the Hill estimates.4Simulation studyAnalytical derivatives of distributional properties of the MC estimator and/or con-fidence intervals for bothfinite and asymptotic samples are not tractable.In this section,therefore,via simulation we demonstrate three advantages of our estima-tion and test discussed in the previous section.By doing this,the Hill estimation is performed in two cases,namely by known k/n(henceforth,Hill KNOWN)and un-known k/n(henceforth,Hill UNKNOWN).The known case means that the stability parameterαis known and,hence,the truncation parameter k/n is optimal for the Hill estimation,which is not necessarily realistic.For the empirically more relevantunknown case we choose an initialαrandomly between1and2and determine cor-responding k.This is also not very much realistic,because it means that we have any information aboutαof the data of interest.But,the two case(Hill KNOWN,and Hill UNKNOWN)describe two extremes in the real world and,hence,can be seen as best and worst case that can happen in empirical works.In our simulation study,if αis known we take an optimal truncation parameter k/n from the Table in Mittnik et al.(1998).4.1Power function of the Monte Carlo method based testThe theoretical size and power of the MC method based test is considered in Dufour (2004).Although the discrepancy of the correct size and the superior power of the MC method based test over the conventional test goes to zero as sample size goes infinity,the behavior of power function forfinite sample is usually of interest.To check the power of our MC method based test we perform a simulation study by drawing from symmetricα-stable pseudo-r.v.s re-located by the median. As pseudo-empirical data we take the sameα-stable random sample as generated before,and test H0 =α=α0,whereα0 is assumed to be values from1.0to2.0 in steps of0.1.For sample size n=100,250,5001000,2000,5000and10000are selected and the number of replication is10000.For calculating of the test statistic in(9)containing the Hill estimator,we use as optimal k the numbers tabulated in Mittnik et al.(1998).To demonstrate the power function,we select an usual significance level95%.Figures1shows the power functions for the selectedα,n and percentage points as described before.Figure1somewhere here.As is expected,the power converges to the corresponding ideal value for each given significance level,as the sample size grows.A sample size of2000gives a rather satisfactory power.A large lost of power can be observed in extremely small sample sizes.4.2Comparison of confidence intervalsThe most powerful advantage of the MC test is that it is able to provide exact confidence intervals forfinite samples.The usual standard deviation in5is valid asymptotically which is very likely different from those forfinite samples.To com-pare confidence intervals from the MC test and the Hill estimator we perform a simulation designed as before.Table2summarizes the results.Table1somewhere here.Before a comparison of the three estimates some remarks regarding the exactconfidence intervals of our MC test are in order:the exact confidence intervalsforfinite samples are the more asymmetric,the smaller the sample size is and thehigher the quantile is.Note that the asymptotic distributions of the Hill estimateis symmetric at all sample sizes and all quantiles.In comparison of the confidence intervals of our MC test with those of the asymptotic Hill estimates the former are smaller than the latter.For example,the asymptotic confidence interval at95%for sample size of250is[1.19651.8035]for Hill KNOWN and[1.15961.8404]Hill UNKNOWN,whereas the MC test gives an exactinterval[1.31071.7700].Even a large sample size,for example5000,the asymptotic95%confidence interval([1.42221.5778])still wider than that of the MC([1.46011.5506]).The fact that the exact confidence intervals derived by the MC method aresmaller than those of the asymptotic results forfinite samples means the size ofa test based on the asymptotic results suffers from size distortion.To check thesize distortion,a simulation was made.We use the8Hill estimators used in Ta-ble2in Appendix.Forαwe selectα=1.25,1.5,1.75,1.95and for sample sizen=100,250,500,1000,2000,5000.For each combination forαand n100000repli-cations were made.For evaluating the asymptotic distribution of the Hill estimator,√k(ˆ−α−1)∼N(0,ˆα−2),we use the optimal k from Mittnik et al.(1998).Tableα−1H3a-c show the result of the simulation for size distortion of the Hill estimators.Table2a-c somewhere here.As is expected the size distortion is generally severe for all consideredαandsample sizes when the asymptotic distribution is used for tests forfinite sample.This is valid up to a small difference at all for all considered Hill estimators.Fora given sample size the more severe the size distortion the largerαis.And for all consideredαeven large samples,say5000,the size distortion is not removed orrather likely to become more severe.4.3Efficiency of Hodge-Lehmann estimatorTo compare the performances of the MC estimator with the the Hill estimator,weperform simulations by drawing from symmetricα-stable pseudo-random variables.Specifically,we consideredα-values of1.5and sample size of n=100,250,500,1000,2000,5000.For each of the resulting18(α,n)-combinations we generated1000replications of the corresponding random samples.Table1shows the simulation results.Table3somewhere here.As is expected,the MC method based estimator shows smaller MSEs than those of the Hill UNKNOWN and even the Hill KNOWN estimator for all consideredα’s and sample sizes.The efficiency improvement measured by the ratio between the MSE of the Hill KNOWN estimator and the MSE of the MC method based estimator is approximately70–85%depending on sample size.For the case of unknown k/n—more empirically relevant case—the efficiency improvement is more clearly that the ratio of the two RMSE’s amount to20–60%.Bith the mean absolute deviation and standard deviation of the estimates show the same results as those of the RMSE.5An empirical applicationTo illustrate the use of the Monte Carlo test,we employ two empirical data.They are daily return of the exchange rates of Euro against US$(January04.1999-March31.2005,1571observations)and Dow Jones Index for the same period(1599 observations).Figure2shows empirical densities of the two time series(thick solid line)compared with the normal density(thin solid line).Figure2somewhere here.Each of the empirical densities show excessively peaked around the mean and at the same time thicker tails than those of the normal density.In order to check whether the empirical datafits Assumption1,i.e.symmetric, we use a test in ABC?(Journal of Financial Econometrics)andfind that the two series are symmetric.We estimate the two times series by means of our MC method based test pro-cedure.Table4shows the results of the estimation and the Hodge-Lehmann-type estimates and the corresponding confidence interval are graphically demonstrated in Figure3,where the solid line gives p-values at givenαof the empirical data and the three dotted lines give simulated quantiles of90,95and99%.Table4somewhere here.Figure3somewhere here.Additionally,the estimates by means of the Hill estimation are1.82and1.75for Euro/US$and Dow Jones index,respectively.6Concluding RemarksIn this paper we considered estimation and test of stability parameter ofα-stable distributions using Monte Carlo test.Specifically,we show an application of our MC method based on the Hill estimator,but any estimator can be straightforward used as statistic.The main contribution of the paper is to improve the Hill estimator.It turns out that confidence intervals forfinite samples based on our MC method are shorter than those of asymptotic distributions for anyαon which confidence intervals of the Hill estimator are based.Secondly,our estimate do not need to assume the tail ratio k/n to be known.Furthermore,our estimate is more efficient than the Hill estimator in the sense of mean square error.Based on the maximized MC method proposed by Dufour(2004)an estimation and test for the asymmetric parameter of α-stable distributions are studying by the authors.。