ESTIMATION OF MOMENT PARAMETER IN ELLIPTICAL DISTRIBUTIONS

Transcription

1 J. Japa Statist. Soc. Vol. 33 No ESTIMATION OF MOMENT PARAMETER IN ELLIPTICAL DISTRIBUTIONS Yosihito Maruyaa* ad Takashi Seo** As a typical o-oral case, we cosider a faily of elliptically syetric distributios. The, the oet paraeter ad its cosistet estiator are preseted. Also, the asyptotic expectatio ad the asyptotic variace of the cosistet estiator of the geeral oet paraeter are give. Besides, the uerical results obtaied by Mote Carlo siulatio for soe selected paraeters are provided. Key words ad phrases:asyptotic expasio, cosistet estiator, elliptical distributio, kurtosis paraeter, oet paraeter, Mote Carlo siulatio, perturbatio ethod.. Itroductio The geeral oet paraeter icludes the iportat kurtosis paraeter i the study of ultivariate statistical aalysis for elliptical populatios. The kurtosis paraeter, especially with relatio to the estiatio proble, has bee cosidered by ay authors. Mardia 970, 974 defied a easure of ultivariate saple kurtosis ad derived its asyptotic distributio for saples fro a ultivariate oral populatio. Also, the testig orality was cosidered by usig the asyptotic result. The related discussio of the kurtosis paraeter uder the elliptical distributio has bee give by Aderso 993, ad Seo ad Toyaa 996. Heze 994 has discussed the asyptotic variace of the ultivariate saple kurtosis for geeral distributios. Here we deal with the estiatio of the geeral oet paraeters i elliptical distributios. I particular, we ake a geeralizatio of the results of Aderso 993 ad give a extesio of asyptotic properties i Mardia 970, 974, Seo ad Toyaa 996. I geeral, it is ot easy to derive the exact distributio of test statistics or the percetiles for the testig proble uder the elliptical populatios, ad so the asyptotic expasio of the statistics is cosidered. Especially that give up to the higher order icludes ot oly the kurtosis paraeter but the ore geeral higher order oet paraeters as well. The we have to speculate about the estiatio of the oet paraeters as a practical proble. The preset paper is orgaized i the followig way. First, the probability desity fuctio, characteristic fuctio ad subclasses of the elliptical distributio are explaied. Secodly, we prove the results cocerig the oets ad defie the oet Received October 8, 00. Revised April, 003. Accepted October 7, 003. *Departet of Matheatics, Graduate School of Sciece, Tokyo Uiversity of Sciece, -3, Kagurazaka, Shijuku-ku, Tokyo 6-860, Japa. **Departet of Matheatical Iforatio Scieces, Faculty of Sciece, Tokyo Uiversity of Sciece, -3, Kagurazaka, Shijuku-ku, Tokyo 6-860, Japa.

2 6 YOSIHITO MARUYAMA AND TAKASHI SEO paraeter. These are doe i Sectio. Although Berkae ad Betler 986, 987 coputed the oets of a elliptically distributed rado vector i a differet way, we ca cofir that our results coicide with theirs. I Sectio 3, we costruct the cosistet estiator of the geeral oet paraeters by the oet ethod. I additio, the asyptotic properties of those estiators for the elliptical distributio are preseted i the two cases for which the populatio covariace atrix is kow ad ukow. Fro the results, aother estiator that has bias correctio with respect to the saple size is proposed. I the last istace, we ivestigate the bias ad the MSE of the estiators by Mote Carlo siulatio applied to soe selected paraeters, ad oreover evaluate the cofidece itervals for the geeral oet paraeter with the asyptotic properties i Sectio 4.. Moets for elliptical populatio Let the p-variate rado vector X be distributed as the p-variate elliptical distributio with paraeters µ ad Λ, which is deoted by E p µ, Λ, Λ is soe positive defiite syetric atrix. If the probability desity fuctio of X exists, it has the for fx =c p Λ / g x µ Λ x µ for soe o-egative fuctio g, c p is positive costat. The characteristic fuctio is of the for φθ = exp[iθ µ]ψθ Λθ for soe fuctio ψ, i =. Note that the expectatio ad the covariace atrix of X are EX =µ ad CovX =Σ= ψ 0Λ, respectively. Elliptical distributios iclude the several special cases, for istace, the ultivariate oral distributio which is described as N p µ, Λ with the probability desity fuctio [ fx =π p/ Λ / exp ] x µ Λ x µ, the ultivariate t distributio with ν degrees of freedo ad the probability desity fuctio ν + p Γ fx = ν Λ / + ν+p/ Γ νπ p/ ν x µ Λ x µ, the cotaiated oral distributio with the probability desity fuctio fx = ω [ π p/ Λ / exp ] x µ Λ x µ [ ω + πτ p/ Λ / exp ] τ x µ Λ x µ, 0 ω,

3 ESTIMATION OF MOMENT PARAMETER IN ELLIPTICAL DISTRIBUTIONS 7 ad so o, which are referred to e.g., Cabais et al. 98, Muirhead 98. Now we will calculate the oets of elliptically distributed rado variates. Suppose that A is a osigular atrix satisfyig A A = Λ. The we ca express A X µ =RU, the vector U has the uifor distributio o the uit sphere U U = such that U is idepedet of the scalar R which is oegative. The cetral oets of X ca be foud fro the oets of R ad U. I oral case, R has the chi-square distributio with p degrees of freedo. So the -th order oets of R are give as E R = p, p p p p + +. The, for exaple, we have the secod, fourth ad sixth order oets of U give by EUU = p I p, EU i U j U k U l = EU i U j U k U l U s U t = pp + δ ij δ kl, 3 pp + p +4 δ ij δ kl δ st, δ ij is the Kroecker s delta, ad the su occurs over all ways of groupig the subscripts, δ ij δ kl = δ ij δ kl + δ ik δ jl + δ il δ jk, 3 δ ij δ kl δ st = δ ij δ kl δ st + δ ij δ ks δ lt + δ ij δ kt δ ls + δ ik δ jl δ st + δ ik δ js δ lt 5 I geeral, we have the followig result. 5 + δ ik δ jt δ ls + δ il δ jk δ st + δ il δ js δ kt + δ il δ jt δ ks + δ is δ jk δ lt + δ is δ jl δ kt + δ is δ jt δ kl + δ it δ jk δ ls + δ it δ jl δ ks + δ it δ js δ kl. Lea. The odd order oets of U are 0, the -th order oets are of the for EU i U j U k U l U u U v = p δ ij δ kl δ uv, d d =.

4 8 YOSIHITO MARUYAMA AND TAKASHI SEO O the other had, the -th order oets of R i the elliptical distributios are give by ER = 4 p ψ 0, which is due to Hayakawa ad Puri 985. Fro the, we have the followig result. Lea. For elliptical populatio, the odd order oets of X µ are 0, the -th order oets are of the for [ ] E X i µ i X j µ j X u µ u X v µ v = K + σ ij σ kl σ uv, Σ=σ ij, ad K ψ 0 {ψ 0}, d =. The result of Lea coicides with which Berkae ad Betler 987 derived by successive differetiatios of φθ, ad also whe i = j = = u = v, we have the oets which are the sae as that give by Berkae ad Betler 986. Here we defie K as the -th order oet paraeter. Especially, K is siply deoted by κ ad called a kurtosis paraeter. The elliptical distributios are characterized by a kurtosis paraeter which is icluded i the geeral oet paraeter. Fro the relatio betwee the oets ad the cuulats e.g., see Kedall et al. 987, for exaple, the fourth, sixth ad eighth cuulats are give by d κ ijkl = κ 3 σ ij σ kl, κ ijklst =K 3 3κ 5 σ ij σ kl σ st, κ ijklstuv =K 4 4K 3 3κ +6κ σ ij σ kl σ st σ uv. 3. Asyptotic properties I this sectio, we give careful cosideratio to the cosistet estiators of the geeral oet paraeter i the elliptical distributios by the oet ethod. We also give the asyptotic expectatio ad the asyptotic variace of those estiators. Suppose that X,...,X are idepedet ad idetically distributed rado vectors accordig to E p µ, Λ. A ultivariate coefficiet of kurtosis i the 05

5 ESTIMATION OF MOMENT PARAMETER IN ELLIPTICAL DISTRIBUTIONS 9 sese of Mardia 970 is β,p E [ {X µ Σ X µ} ], ad the affie ivariat saple aalogue of β,p is obtaied by ˆβ,p { Xi X S X i X }, i= X is the saple ea vector. S is the saple covariace atrix, S X i XX i X. i= Now β,p ca be calculated as pp + κ + with Lea. For the cosistet estiator of κ, we ca propose ˆκ = pp +ˆβ,p. Further, as geerality of this result, β,p E [ {X µ Σ X µ} ] = p K +. Therefore, we have a cosistet estiator of the -th order oet paraeter give by 3. ˆK = p ˆβ,p, ˆβ,p = { Xi X S X i X }. i= This represetatio is a extesio of the estiator of kurtosis paraeter discussed by Aderso 993. We first cosider the asyptotic properties of the cosistet estiator, which is described by ˆK or ˆβ,p whe Σ is kow. I this case, it is assued without loss of geerality that Σ = I p. The replacig S with Σ, we ca write ˆβ,p = i= T i, Ti =X i X X i X. I order to avoid the depedece of X i ad X, we defie X i = X k. k i

6 0 YOSIHITO MARUYAMA AND TAKASHI SEO The, we ca write T i = Xi X i Xi X i. Note that X i is idepedet of X i. To obtai the expectatio of ˆK by the perturbatio ethod, we put X i = Y, the, T i ca be expaded as T i = X ix i + γ + γ + O p 3/, γ = X ix i Y X i, γ = X ix i Y Y X ix i + X i X i Y X i. Therefore, calculatig the expectatio with respect to X i ad Y, we get the asyptotic ea of T = ˆK K up to the order as ET = { K K + } 3. + O 3/. If the uderlyig distributio is oral the K = 0, the asyptotic expectatio of T ca be reduced as ET = + O 3/. Siilarly, calculatig the expectatio for the expasio of Ti 4, we ay coe by the variace. Therefore, we have the followig result. Theore. Suppose that ˆK is the cosistet estiator of the -th order oet paraeter K with kow Σ, which is give by 3.. The the asyptotic variace of T = ˆK K is p σ T = p K + K +. Here, we correct the bias of the estiator ˆK as follows. The odified estiator ca be obtaied by K = ˆK + ˆK ˆK

7 ESTIMATION OF MOMENT PARAMETER IN ELLIPTICAL DISTRIBUTIONS Note that the bias i K is order with the sae variace up to the order as that of ˆK. Next, we cosider the case whe Σ is ukow, that is the ai purpose of this paper. I order to avoid the depedece of X i, X ad S, we put T i = X i X S X i X, X i = X k, S i = k i Xk X i Xk X i. k i The, we ca write ad also S = { S i + } Xi X i Xi X i, S = S i S i Xi X i Xi X i S i +. Xi X i S i Xi X i Therefore, we ote that Ti = T i +, T i Further, let T i = Xi X i S i Xi X i. X i = S i = I p + Y, M, the, we ca expad T i as T i = X ix i + a + a + O p 3/,

8 YOSIHITO MARUYAMA AND TAKASHI SEO Therefore, T i a = X iy X imx i, a =X ix i +X imy + X im X i + Y Y. is expaded as T i = X ix i + b + b + O p 3/, b = X ix i a, b = X i X i { a +X ix i a 4 X } ix i X 3 i X i. By usig the asyptotic expaded joit probability desity fuctio of Y ad M e.g., see Wakaki 997, we ca calculate the expectatio for the expasio of Ti. The we obtai the asyptotic ea of T = ˆK K upto the order as ET = 3.5 c + O 3/, c = K + p + K p +κ + K + + K +. Siilarly, we ay coe by the variace. I order to avoid the depedece of X i, X j, X ad S, we defie X i,j = X k, S i,j = k i,j Xk X i,j Xk X i,j. k i,j The, T i ca be writte as { } T i = Q i + Qi 3 Q j Q i,j Q i,j + 3 Q j + Q j, Q i = + Xi X i,j S i,j Xi X i,j, Q j = + Xj X i,j S i,j Xj X i,j, Q i,j = + Xi X i,j S i,j Xj X i,j,

9 ESTIMATION OF MOMENT PARAMETER IN ELLIPTICAL DISTRIBUTIONS 3 ad E ˆβ,p = E Ti 4 + E T i Tj. Further, let X i,j = S i,j = I p + Ỹ, M, the, T i ca be expaded as T i = X ix i X ix i f + X ix i e + O p 3/, e = f + X ix i f + X ix i X i X i X ix j X ix i X i X j, ad T f =X iỹ + X i MX i, f =X ix i +X i MỸ + X i M X i + Ỹ Ỹ. Substitutig i i Ti, we ca get the expasio of Ti 4 up to the order. Further calculatig the expectatio for the expasio of Ti 4 with respect to X i, Y ad M, ad the doig for the expasio of Ti Tj with respect to X i, X j, Ỹ ad M, we ay coe by the variace up to the order. Therefore, we have the followig result. Theore. Suppose that ˆK is a cosistet estiator of the -th order oet paraeter K with ukow Σ, which is give by 3.. The the asyptotic variace of T = ˆK K is 3.6 σ T = p + p K + +3 κ + K p p + K + κ +{p + κ + p} K + p + p K + K + +.

10 4 YOSIHITO MARUYAMA AND TAKASHI SEO This result is essetially the extesio of that discussed by Seo ad Toyaa 996. I the case whe Σ is ukow, aother estiator with the bias correctio ca be obtaied as 3.7 K = ˆK ĉ,, ĉ = ˆK + p + ˆK p +ˆκ + ˆK + + ˆK +. Further, if the uderlyig distributio is oral the K = 0, we have the variace of ˆK which is the sae as that give by Mardia Nuerical calculatio This sectio ivestigates the bias ad the MSE for the estiators of the geeral oet paraeter by Mote Carlo siulatio for soe selected values of paraeters. Coputatios are ade for each case the populatio covariace atrix Σ is ukow. For elliptical populatios, we cosider the followig six types of distributios. E : Cotaiated oral ω =0., τ= 3. E : Cotaiated oral ω =0.4, τ= 3. E3 : Cotaiated oral ω =0.7, τ= 3. E4 : Multivariate oral. E5 : Multivariate t ν = 3. E6 : Copoud oral U,, the rado vector X fro E6 is the product of a oral vector Z which has N p 0,I p ad the iverse of a rado variable accordig to the uifor distributio o the iterval [, ]. We ote that the theoretical values of K are coputed easily usig the appropriate forula, i.e., for the cotaiated oral distributio, K = +ωτ {+ωτ }, for the ultivariate t distributio with ν degrees of freedo is give by K = ν ν, ν >, ad for the copoud oral distributio, K = + / +. The oet paraeters of each distributio are obtaied as follows Table, because we eed the up to the sixth order for coputig.

11 ESTIMATION OF MOMENT PARAMETER IN ELLIPTICAL DISTRIBUTIONS 5 Table. Theoretical values of K. κ K 3 K 4 K 6 E E E E E E We first focus our cocetratio o the ea of T. Table shows the approxiate values with 3.5 for soe paraeters p ad whe =3adΣis ukow. O the other had, whe we ay assue that Σ = I p without ay loss of geerality, the saple ea for T based o 0,000 siulatios is also obtaied. It ay be foud fro Table that the siulatio results early coicide with the approxiate values for E4 ad E6. But i the case of E5, the covergece is slow ad the approxiate expressio 3.5 is ot always precise. I the other cases, both values agree for a sufficietly large. Table. Siulatio results for the ea of T whe Σ is ukow. E E E3 E4 E5 E6 \p AE SV AE SV AE SV AE SV AE : Approxiated values for the expectatio of T. SV : Siulated values for the ea of T. Next, Table 3 gives the bias for ˆK 3 ad aother estiator K 3 by 3.7 based o 0,000 siulatios whe Σ is ukow. The MSE is doe by Table 4. If Σ is kow, we ake use of 3.4. It ay be see fro siulatio results i Tables 3 ad 4 that the expectatio of the estiators coverges to the oet paraeter whe the saple size is large. Especially i oral case E4, it is oted that the odified estiator K 3 rapidly approaches K 3. The values obtaied for the estiators were acceptable for a large ad iproved as ω icreased for the cotaiated oral E, E ad E3, ad whe icreased for the copoud oral E6. Also fro Tables 3 ad 4, we ote that ˆK 3 is uderestiated for elliptical populatios. The bias of K3 is actually saller i agitude tha that of ˆK 3. It ay be oted that the size of p does ot have uch effect o the bias of ˆK. These results are true for sall saple sizes, ot ore tha

12 6 YOSIHITO MARUYAMA AND TAKASHI SEO = 50, whe Σ is kow. As for the MSE, whe Σ is ukow with large p, the MSE of K is about the sae as that of ˆK. It ca be see that the bias as well as the MSE for K is reduced whe the saple size is large ad the covariace atrix is ukow. As far as we ca judge these results, K is better tha ˆK. Table 3. Siulatio results for the bias of the estiators whe Σ is ukow. p E E E3 E4 E5 E6 ˆK 3 K3 ˆK3 K3 ˆK3 K3 ˆK3 K3 ˆK3 K3 ˆK3 K Table 4. Siulatio results for the MSE of the estiators whe Σ is ukow. p E E E3 E4 E5 E6 ˆK 3 K3 ˆK3 K3 ˆK3 K3 ˆK3 K3 ˆK3 K3 ˆK3 K Furtherore, the asyptotic orality of T also eables us to easily costruct the cofidece itervals for K. A cofidece iterval for K with cofidece coefficiet α is approxiately [ ] 4. ˆK ˆσ T z α/, ˆK + ˆσ T z α/ ˆσ T is give by substitutig ˆK for K i σt gaied as 3.3 i Theore or 3.6 i Theore, because K is geerally ukow. z α/ is the twotailed 00α% poit of the stadard oral distributio. I the sae way, we have aother iterval for K by eas of K, that is to say [ ] σ K T σ 4. z α/, K + T z α/, σ T is doe by substitutig K for K.,

13 ESTIMATION OF MOMENT PARAMETER IN ELLIPTICAL DISTRIBUTIONS 7 Figure. Cofidece iterval E4. : ˆK, : K By way of illustratio, we thik over the oral case E4 whe Σ is ukow. Figure presets the cofidece itervals for K 3 with oial cofidece coefficiet α =0.95. Those are the cofidece liit with ˆK 3 by 4. ad with K 3 by 4.. It ca be see fro those siulatio results that the large saple size has the sall rage of the cofidece iterval. The cofidece iterval by eas of K 3 has a slightly sall rage by cotrast with that of ˆK 3. Moreover, the actual cofidece coefficiets of the iterval i the circustaces are displayed i Table 5. It ca tur out to be that both of the two values coe closer to the oial cofidece coefficiet α = 0.95 whe the saple size is large. The cofidece coefficiet with the help of ˆK 3 is a little bigger tha that of K 3. Note that the largeess of p does ot affect the itervals. I additio, whe p is large, σt decreases ootoously. I the other cases E to E6, the siulatio results are show i Table 6, ad also the, we are able to see the results siilar to E4. Table 5. Actual cofidece coefficiets i E ˆK K The facts etioed above ay be applied to the case whe Σ is kow, but details are abbreviated i the preset paper. Here, we cite oly oe or two istaces. For the ea of T, it was see that the siulatio results alost agree with the approxiate values 3. i each case. Also we foud that the MSE of ˆK or K is ot as sall as that i the case of a ukow Σ.

14 8 YOSIHITO MARUYAMA AND TAKASHI SEO Table 6. Cofidece iterval for K 3 whe Σ is ukow. p = E E E3 lower upper α lower upper α lower upper α = 00 ˆK K = 00 ˆK K = 500 ˆK K = 000 ˆK K p = E4 E5 E6 lower upper α lower upper α lower upper α = 00 ˆK K = 00 ˆK K = 500 ˆK K = 000 ˆK K α : Actual cofidece coefficiets. As a result, we coclude to suggest K as the better estiator of K. Ackowledgeets The authors wish to express their sicere gratitude to the referees for the coets that were very useful for the iproveet of the paper. Refereces Aderso, T. W Nooral ultivariate distributios:iferece based o elliptically cotoured distributios, Multivariate Aalysis. Future Directios, ed. Rao, C. R., 4, Elsevier Sciece Publishers. Berkae, M. ad Betler, P. M Moets of elliptically distributed rado variates, Statist. Probab. Lett., 4, Berkae, M. ad Betler, P. M Characterizig paraeters of ultivariate elliptical distributios, Co. Statist. Siulatio Coput., 6, Cabais, S., Huag, S. ad Sios, G. 98. O the theory of elliptically cotoured distributios, J. Multivariate Aal,, Hayakawa, T. ad Puri, M. L Asyptotic distributios of likelihood ratio criteria for testig latet roots ad latet vectors of a covariace atrix uder a elliptical populatio, Bioetrika, 7, Heze, N O Mardia s kurtosis test for ultivariate orality, Co. Statist. Theory Methods, 3, Kedall, M. G., Stuart, A. ad Ord, J. K The Advaced Theory of Statistics, Vol., 5th ed., Charles Griffi, Lodo. Mardia, K. V Measures of ultivariate skewess ad kurtosis with applicatios, Bioetrika, 57,

15 ESTIMATION OF MOMENT PARAMETER IN ELLIPTICAL DISTRIBUTIONS 9 Mardia, K. V Applicatios of soe easures of ultivariate skewess ad kurtosis i testig orality ad robustess studies, Sakhyá, B36, 5 8. Muirhead, R. J. 98. Aspects of Multivariate Statistical Theory, Joh Wiley, NewYork. Seo, T. ad Toyaa, T O the estiatio of kurtosis paraeter i elliptical distributios, J. Japa. Statist. Soc., 6, Wakaki, H Asyptotic expasio of the joit distributio of saple ea vector ad saple covariace atrix fro a elliptical populatio, Hiroshia Math. J, 7,