A Ist Stat Math 06 68:469 490 DOI 0.007/s046-04-050-9 Secod-order asymptotic compariso of the MLE ad MCLE of a atural parameter for a trucated expoetial family of distributios Masafumi Akahira Received: 7 November 0 / Revised: 8 December 04 / Published olie: February 05 The Istitute of Statistical Mathematics, Tokyo 05 Abstract For a trucated expoetial family of distributios with a atural parameter θ ad a trucatio parameter γ as a uisace parameter, it is kow that the maximum likelihood estimators MLEs ˆθ γ ML ad ˆθ ML of θ for kow γ ad ukow γ, respectively, ad the maximum coditioal likelihood estimator ˆθ MCL of θ are asymptotically equivalet. I this paper, the stochastic expasios of ˆθ γ ML, ˆθ ML ad ˆθ MCL are derived, ad their secod-order asymptotic variaces are obtaied. The secod-order asymptotic loss of a bias-adjusted MLE ˆθ ML relative to ˆθ γ ML is also give, ad ˆθ ML ad ˆθ MCL are show to be secod-order asymptotically equivalet. Further, some examples are give. Keywords Trucated expoetial family Natural parameter Trucatio parameter Maximum likelihood estimator Maximum coditioal likelihood estimator Stochastic expasio Asymptotic variace Secod-order asymptotic loss Itroductio The first-order asymptotic theory i regular parametric models with uisace parameters was discussed by Bardorff-Nielse ad Cox 994. I higher order asymptotics, uder suitable regularity coditios, the cocept of asymptotic deficiecy discussed by Hodges ad Lehma 970 is useful i comparig asymptotically efficiet estimators i the presece of uisace parameters. Ideed, the asymptotic deficiecies of some asymptotically efficiet estimators relative to the maximum likelihood estimator M. Akahira B Istitute of Mathematics, Uiversity of Tsukuba, Tsukuba, Ibaraki 05-857, Japa e-mail: akahira@math.tsukuba.ac.jp
470 M. Akahira MLE based o the pooled sample were obtaied i the presece of uisace parameters [see, e.g. Akahira ad Takeuchi 98 ad Akahira 986]. O the other had, i statistical estimatio i multiparameter cases, the coditioal likelihood method is well kow as a way of elimiatig uisace parameters [see, e.g. Basu 977]. The cosistecy, asymptotic ormality ad asymptotic efficiecy of the maximum coditioal likelihood estimator MCLE were discussed by Aderse 970, Huque ad Katti 976, Bar-Lev ad Reiser 98, Bar-Lev 984, Liag 984 ad others. Further, i higher order asymptotics, asymptotic properties of the MCLE of a iterest parameter i the presece of uisace parameters were also discussed by Cox ad Reid 987 ad Ferguso 99 i the regular case. However, i the o-regular case whe the regularity coditios do ot ecessarily hold, the asymptotic compariso of asymptotically efficiet estimators has ot bee discussed eough i the presece of uisace parameters i higher order asymptotics yet. For a trucated expoetial family of distributios which is regarded as a typical o-regular case, we cosider a problem of estimatig a atural parameter θ i the presece of a trucatio parameter γ as a uisace parameter. Let ˆθ γ ML ad ˆθ ML be the MLEs of θ based o a sample of size whe γ is kow ad γ is ukow, respectively. Let ˆθ MCL be the MCLE of θ. The it was show by Bar-Lev 984 that the MLEs ˆθ γ ML, ˆθ ML ad the MCLE ˆθ MCL have the same asymptotic ormal distributio, hece they are show to be asymptotically equivalet i the sese of havig the same asymptotic variace. A similar result ca be derived from the stochastic expasios of the MLEs ˆθ γ ML ad ˆθ ML i Akahira ad Ohyauchi 0. But, ˆθ γ ML for kow γ may be asymptotically better tha ˆθ ML for ukow γ i the higher order, because ˆθ γ ML has the full iformatio o γ. Otherwise, the existece of a trucatio parameter γ as a uisace parameter is meaigless. So, it is a quite iterestig problem to compare asymptotically them up to the higher order. I this paper, we compare them up to the secod order, i.e. the order,ithe asymptotic variace. We show that a bias-adjusted MLE ˆθ ML ad ˆθ MCL are secodorder asymptotically equivalet, but they are asymptotically worse tha ˆθ γ ML i the secod order. We thus calculate the secod-order asymptotic losses o the asymptotic variace amog them. Formulatio ad assumptios I a similar way to Bar-Lev 984, we have the formulatio as follows. Suppose that X, X,...,X,... is a sequece of idepedet ad idetically distributed i.i.d. radom variables accordig to P θ,γ, havig a desity f x; θ,γ = axe θux bθ,γ for c <γ x < d, 0 otherwise with respect to the Lebesgue measure, where c < d, a is o-egative ad cotiuous almost surely, ad u is absolutely cotiuous with dux/dx 0 over the iterval γ, d. Let
Secod-order asymptotic compariso 47 γ := d θ 0 < bθ, γ := axe θux dx < γ for γ c, d. The, it is show that for ay γ,γ c, d with γ <γ, γ γ. Assume that for ay γ c, d, γ is a o-empty ope iterval. A family P := P θ,γ θ,γ c, d of distributios P θ,γ [see ] with a atural parameter θ ad a trucatio parameter γ is called a trucated expoetial family of distributios. I Bar-Lev 984, the asymptotic behavior of the MLE ˆθ ML ad MCLE ˆθ MCL of a parameter θ i the presece of γ as a uisace parameter was compared ad also doe with that of the MLE ˆθ γ ML of θ whe γ is kow. As the result, it was show there that, for a sample of size, the ˆθ ML ad ˆθ MCL of θ exist with probability ad are give as the uique roots of the appropriate maximum likelihood equatios. These two estimators were also show to be strogly cosistet for θ with the limitig distributio coicidig with that of the MLE ˆθ γ ML of θ whe γ is kow. I the subsequet sectios, we obtai the stochastic expasios of ˆθ γ ML, ˆθ ML ad ˆθ MCL up to the secod order, i.e. the order o p. We get their secod-order asymptotic variaces, ad derive the secod-order asymptotic losses o the asymptotic variace amog them. The proofs of theorems are located i appedixes. The MLE ˆθ γ ML of θ whe γ is kow Deote a radom vector X,...,X by X, ad let X X be the correspodig order statistics of a radom vector X. Here, we cosider the case whe γ is kow. The, the desity is cosidered to belog to a regular expoetial family of distributios with a atural parameter θ, hece log bθ, γ is strictly covex ad ifiitely differetiable i θ ad λ j θ, γ := j log bθ, γ θ j is the jth cumulat correspodig to for j =,,... For give x = x,...,x satisfyig γ < x := mi i x i ad x := max i x i < d, the likelihood fuctio of θ is give by L γ θ; x := The, the likelihood equatio is b ax i exp θ θ, γ i= ux i. i= ux i λ θ, γ = 0. i=
47 M. Akahira Sice there exists a uique solutio o θ of, we deote it by ˆθ γ ML which is the MLE of θ [see, e.g. Bardorff-Nielse 978 ad Bar-Lev 984]. Let λ i = λ i θ, γ i =,, 4 ad put Z := λ ux i λ, i= U γ := λ ˆθ γ ML θ. The, we have the followig. Theorem For the trucated expoetial family P of distributios with a desity with a atural parameter θ ad a trucatio parameter γ,let ˆθ γ ML be the MLE of θ whe γ is kow. The, the stochastic expasio of U γ is give by U γ = Z λ λ / Z + λ λ λ 4 λ Z + O p, ad the secod-order asymptotic mea ad variace are give by λ E θ Uγ = λ / + O, 5λ V θ Uγ = + λ λ 4 λ + O, respectively. Sice U γ = Z + o p, it is see that U γ is asymptotically ormal mea with mea 0 ad variace, which coicides with the result of Bar-Lev 984. 4 The MLE ˆθ ML of θ whe γ is ukow For give x = x,...,x satisfyig γ<x ad x < d, the likelihood fuctio of θ ad γ is give by Lθ, γ ; x = b ax i exp θ θ, γ i= ux i. 4 Let ˆθ ML ad ˆγ ML be the MLEs of θ ad γ, respectively. From 4 it is see that ˆγ ML = X ad L ˆθ ML, X ; X = sup θ L θ, X ; X, hece ˆθ ML satisfies the likelihood equatio i= 0 = ux i λ ˆθ ML, X, 5 i=
Secod-order asymptotic compariso 47 where X = X,...,X.Letλ = λ θ, γ ad put Û := λ ˆθ ML θ ad T := X γ. The, we have the followig. Theorem For the trucated expoetial family P of distributios with a desity with a atural parameter θ ad a trucatio parameter γ,let ˆθ ML be the MLE of θ whe γ is ukow, ad ˆθ ML be a bias-adjusted MLE such that ˆθ ML has the same asymptotic bias as that of ˆθ γ ML, i.e. ˆθ ML = ˆθ ML + k ˆθ ML, X λ ˆθ ML, X λ ˆθ ML, X, 6 where kθ, γ := aγ e θuγ /bθ, γ. The, the stochastic expasio of Û := λ ˆθ ML θ is give by Û = Û + k λ where k = kθ, γ, δ = λ λ Û = Z λ λ, λ λ / + O p λ δ + k kλ k θ Z λ, λ T + λ Z + O p, δ λ Z T + λ λ λ 4 λ Z ad the secod-order asymptotic mea ad variace are give by E θ,γ Û = λ λ / + O, V θ,γ Û = + 5λ λ λ 4 λ + λ λ uγ + O, respectively. Sice Û = Û = Z + o p, it is see that Û ad Û are asymptotically ormal with mea 0 ad variace, which coicides with the result of Bar-Lev 984. But, it is oted from Theorems ad that there is a differece betwee V θ U γ ad V θ,γ Û i the secod order, i.e. the order, which is discussed i Sect. 6. 5TheMCLEˆθ MCL of θ whe γ is ukow First, it is see from that there exists a radom permutatio, say Y,...,Y of the! permutatios of X,...,X such that coditioally o X = x,the Y,...,Y are i.i.d. radom variables with a desity
474 M. Akahira g y; θ,x = aye θuy bθ, x for x < y < d [see Queseberry 975 ad Bar-Lev 984]. For give X = x, the coditioal likelihood fuctio of θ for y = y,...,y satisfyig x < y i < d i =,..., is Lθ; y x = b ay i exp θ uy i. θ, x i= i= The, the likelihood equatio is uy i λ θ, x = 0. 7 i= Sice there exists a uique solutio o θ of 7, we deote it by ˆθ MCL, i.e. the value of θ for which L θ; y x attais supremum. Let λ i := λ i θ, x i =,,, 4 ad put Z := uy i λ, Ũ 0 := λ ˆθ MCL θ. λ i= The, we have the followig. Theorem For the trucated expoetial family P of distributios with a desity with a atural parameter θ ad a trucatio parameter γ,let ˆθ MCL be the MCLE of θ whe γ is ukow. The, the stochastic expasio of Ũ 0 is give by λ Ũ 0 = Z λ / Z + + λ λ λ 4 Z λ + O p λ λ, T Z ad the secod-order asymptotic mea ad variace are give by E θ,γ Ũ0 = λ λ / + O, V θ,γ Ũ0 = + 5λ λ λ 4 λ + λ λ uγ + O. Remark From Theorems ad, it is see that the secod-order asymptotic mea ad variace of Ũ 0 are the same as those of Û = λ ˆθ ML θ. It is oted that ˆθ MCL has a advatage over ˆθ ML i the sese of o eed of the bias adjustmet.
Secod-order asymptotic compariso 475 Remark As is see from Theorems, ad, the first term of order / i V θ U γ, V θ,γ Û ad V θ,γ Ũ 0 results from the regular part of the desity, which coicides with the fact that the distributio with is cosidered to belog to a regular expoetial family of distributios whe γ is kow. The secod term of order / i V θ,γ Û ad V θ,γ Ũ 0 follows from the o-regular i.e. trucatio part of whe γ is ukow, which meas a ratio of the variace λ = V θ,γ ux = E θ,γ [ ux λ ] to the distace λ uγ from the mea λ of ux to ux at x = γ. 6 The secod-order asymptotic compariso amog ˆθ γ ML, ˆθ ML ad ˆθ MCL From the results i the previous sectios, we ca asymptotically compare the estimators ˆθ γ ML, ˆθ ML ad ˆθ MCL usig their secod-order asymptotic variaces as follows. Theorem 4 For the trucated expoetial family P of distributios with the desity with a atural parameter θ ad a trucatio parameter γ,let ˆθ γ ML, ˆθ ML ad ˆθ MCL be the MLE of θ whe γ is kow, the bias-adjusted MLE of θ whe γ is ukow ad the MCLE of θ whe γ is ukow, respectively. The, the bias-adjusted MLE ˆθ ML ad the MCLE ˆθ MCL are secod-order asymptotically equivalet i the sese that d ˆθ ML, ˆθ MCL := V θ,γ Û V θ,γ Ũ 0 = o 8 as, ad they are secod-order asymptotically worse tha ˆθ γ ML with the secodorder asymptotic losses of ˆθ ML ad ˆθ MCL relative to ˆθ γ ML d ˆθ ML, ˆθ γ ML V := θ,γ Û V θ U γ = λ uγ + o, λ 9 d ˆθ MCL, ˆθ γ ML V := θ,γ Ũ 0 V θ U γ = λ uγ + o λ 0 as, respectively. The proof is straightforward from Theorems, ad. 7 Examples Some examples o the secod-order asymptotic loss of the estimators are give for a trucated expoetial distributio, a trucated ormal distributio ad the Pareto distributio. Example Trucated expoetial distributio Let c =, d =, ax ad ux x for <γ x < i the desity. Sice bθ, γ = e θγ /θ, it follows from that = 0,,
476 M. Akahira From, 5, 6 ad 7 wehave λ = θ log bθ, γ = γ θ, λ = log bθ, γ =, kθ, γ = θ. θ θ ˆθ γ ML = / X γ, ˆθ ML = / X X, ˆθ ML = ˆθ ML / ˆθ ML, ˆθ MCL = X i X. Note that ˆθ ML = ˆθ MCL. I this case, the first part i Theorem 4 is trivial, sice d ˆθ ML, ˆθ MCL = 0. From Theorem 4, we obtai the secod-order asymptotic loss i= as. d ˆθ ML, ˆθ γ ML = d ˆθ MCL, ˆθ γ ML = + o Example Trucated ormal distributio Let c =, d =, ax = e x / ad ux = x for <γ x < i the desity. Sice bθ, γ = πe θ / θ γ, it follows from ad Theorem that =,, λ λ θ, γ = θ + ρθ γ, θ, γ = θ γρθ γ+ ρ θ γ, λ θ, γ = θ γρθ γ ρ θ γ, kθ, γ = ρθ γ, where ρt := φt/ t with x = x φtdt, φt = π e t / for < t <. The, it follows from, 5 ad 7 that the solutios of θ of the followig equatios θ + ρθ γ= X, θ + ρθ X = X, θ + ρθ X = X i i=
Secod-order asymptotic compariso 477 become ˆθ γ ML, ˆθ ML ad ˆθ MCL, respectively, where X = / i= X i.from6, the bias-adjusted MLE is give by ˆθ ML = ˆθ ML + ˆθ ML X ρ ˆθ ML X + ρ ˆθ ML X ˆθ ML X ρ ˆθ ML X. From Theorem 4, we obtai the secod-order asymptotic losses d ˆθ ML, ˆθ MCL = o, d ˆθ ML, ˆθ γ ML = d ˆθ MCL, ˆθ γ ML = θ γ + ρθ γ as. θ γρθ γ ρ θ γ + o Example Pareto distributio Let c = 0, d =, ax = /x ad ux = log x for 0 <γ x < i the desity. The, bθ, γ = /θγ θ for θ = 0,. Lettig t = log x ad γ 0 = log γ, we see that becomes f t; θ,γ 0 = θe θγ 0e θt for t >γ 0, 0 for t γ 0. Hece, the Pareto case is reduced to the trucated expoetial oe i Example. 8 Cocludig remarks I a trucated expoetial family of distributios with a two-dimesioal parameter θ, γ, we cosidered the estimatio problem of a atural parameter θ i the presece of a trucatio parameter γ as a uisace parameter. I the paper of Bar-Lev 984, it was show that the MLE ˆθ γ ML of θ for kow γ,themle ˆθ ML ad the MCLE ˆθ MCL of θ for ukow γ are asymptotically equivalet i the sese that they have the same asymptotic ormal distributio. I this paper, we derived the stochastic expasios of ˆθ γ ML, ˆθ ML ad ˆθ MCL. We also obtaied the secod-order asymptotic loss of the biasadjusted MLE ˆθ ML relative to ˆθ γ ML from their secod-order asymptotic variaces ad showed that ˆθ ML ad ˆθ MCL are secod-order asymptotically equivalet i the sese that their asymptotic variaces are same up to the order o/ as i 8. It seems to be atural that ˆθ γ ML is secod-order asymptotically better tha ˆθ ML after adjustig the bias of ˆθ ML such that ˆθ ML has the same as that of ˆθ γ ML. The values of the secod-order asymptotic losses of ˆθ ML ad ˆθ MCL give by 9 ad 0 are quite simple, which result from the trucated expoetial family P of distributios. The results of Theorems,, ad 4 ca be exteded to the case of a two-sided trucated expoetial family of distributios with a atural parameter θ ad two trucatio parameters γ ad ν as uisace parameters, icludig a upper-trucated Pareto distributio which is importat i applicatios [see Akahira et al. 04]. Further, they
478 M. Akahira may be similarly exteded to the case of a more geeral trucated family of distributios from the trucated expoetial family P. I relatio to Theorem, if two differet bias adjustmets are itroduced, i.e. ˆθ ML + /c i ˆθ ML i =,, the the problem whether or ot the admissibility result holds may be iterestig. Appedix A The proof of Theorem Let λ i = λ i θ, γ i =,,, 4. Sice Z = λ ux i λ, i= U γ := λ ˆθ γ ML θ, by the Taylor expasio we obtai from 0 = λ Z λ U γ λ λ U γ λ 4 6λ / U γ + O p which implies that the stochastic expasio of U γ is give by Sice U γ = Z λ λ / Z + λ λ E θ Z = 0, V θ Z = E θ Z =,, λ 4 λ Z + O p. E θ Z = λ, E θ Z 4 = + λ 4 λ, λ / it follows that E θ U γ = λ λ / + O, hece, by ad 4 E θ U γ = + V θ U γ = + λ 4λ 5λ λ λ 4 λ + O, 4 λ 4 λ + O. 5
Secod-order asymptotic compariso 479 From, ad 5 we have the coclusio of Theorem. Before provig Theorem, we prepare three lemmas the proofs are give i Appedix B. Lemma The secod-order asymptotic desity of T is give by f T t = kθ, γ e kθ,γ t + t kθ, γ t e kθ,γ t + O kθ, γ c θ γ bθ, γ + a θuγ γ e aγ bθ, γ 6 for t > 0, where kθ, γ := aγ e θuγ /bθ, γ, ad E θ,γ T = kθ, γ + Aθ, γ + O, E θ,γ T = k θ, γ + O, 7 where Aθ, γ := cθ γ k θ, γ aγ + kθ, γ with c θ γ = a γ + θaγ u γ. Lemma It holds that E θ,γ Z T = k λ uγ λ + k λ + O, 8 where k = kθ, γ ad λ i = λ i θ, γ i =,. Lemma It holds that E θ,γ Z T = k + O, 9 where k = kθ, γ.
480 M. Akahira The proof of Theorem Sice, for θ, γ c, X λ ˆθ ML, X = λ θ, γ + θ λ θ, γ ˆθ ML θ+ λ θ, γ X γ + θ λ θ, γ ˆθ ML θ + θ λ θ, γ ˆθ ML θ X γ + λ θ, γ X γ + 6 θ λ θ, γ ˆθ ML θ + θ λ θ, γ λ θ, γ ˆθ ML θ X γ +, 0 otig Û = λ ˆθ ML θ ad T = X γ,wehavefrom5 ad 0 0 = λ Z λ Û λ 4 6λ / Û + O p, λ T λ λ Û λ λ ÛT where λ j = λ j θ, γ j =,,, 4 are defied by, hece the stochastic expasio of Û is give by Û = Z λ = Z λ λ T λ 4 6λ Û + O p λ + λ λ λ 4 λ T λ λ / λ λ / Z + O p Û λ Z + δ λ Z T λ ÛT. It follows from ad that E θ,γ Û = λ λ E θ,γ T λ λ / + δ λ E θ,γz T + O.
Secod-order asymptotic compariso 48 Substitutig 7 ad 8 for, we obtai E θ,γ Û = λ k λ + λ + O λ, where k = kθ, γ is defied i Lemma. Wehavefrom E θ,γ Û = E θ,γ Z λ + λ E θ,γ + λ 5λ 4λ λ 4 λ λ T + λ E θ,γ Z T + λ λ λ E θ,γ Z 4 + O λ E θ,γ Z λ + δ E θ,γ Z T. 4 Substitutig, 7, 8 ad 9 for4, we have E θ,γ Û = λ kλ + λ λ kλ uγ λ + k kλ λ + λ 4λ λ λ 4 λ + O. 5 Sice λ λ ˆθ ML, X k ˆθ ML, X λ ˆθ ML, X = λ θ, γ k λ + kλ λ θ, γ λ λ + k k θ λ Û + O p, it follows from 6 that the stochastic expasio of Û is give by Û := λ ˆθ ML θ = λ ˆθ ML θ+ = Û + k λ λ δ + kλ k λ λ ˆθ ML, X k ˆθ ML, X λˆ λ k θ Z + O p, 6
48 M. Akahira where Û is give by, λ i = λ i θ, γ i =,, ad k = kθ, γ. From ad, we have E θ,γ Û = λ k λ = λ + O λ / + λ λ It follows from, 5 ad 6 that E θ,γ Û = kλ λ 4 λ k λ + k λ λ + O. 7 λ uγ λ + k λ k + O θ hece, by 7 V θ,γ Û = E θ,γ Û E θ,γ Û = + 5λ λ λ 4 λ λ kλ λ k + O λ Sice, by it follows that Sice λ θ, γ λ + λ 4λ, uγ λ + k k θ. 8 λ θ, γ = θ log bθ, γ = d axuxe θux dx, bθ, γ γ = aγ eθuγ bθ, γ it is see from 8, 9 ad 0 that V θ,γ Û = + λ θ, γ uγ = kθ, γ λ θ, γ uγ. 9 k θ θ, γ = kθ, γ uγ λ θ, γ, 0 5λ λ λ 4 λ + λ uγ + O λ.
Secod-order asymptotic compariso 48 From 6, 7 ad we have the coclusio of Theorem. The proof of Theorem Sice, from 7 lettig 0 = uy i λ θ, x λ θ, x ˆθ MCL θ i= λ θ, x ˆθ MCL θ 6 λ 4θ, x ˆθ MCL θ + O p, Z = uyi λ θ, x, i= Ũ = ˆθ MCL θ, where λ i := λ i θ, x i =,,, 4, wehave 0 = Z λ Ũ λ Ũ hece the stochastic expasio of Ũ is give by Ũ = = Z Z λ / λ λ / Ũ λ 4 Z + Z + λ 4 6 λ / Ũ + O p Ũ + O p 6 λ λ 4, Z + O p. Sice we obtai λ = λ θ, X = λ θ, γ + Ũ = ˆθ MCL θ = λ ˆθ MCL θ + λ λ T + O p, λ T + O p,
484 M. Akahira where T = X γad λ = λ θ, γ. The, it follows from ad that U 0 = λ ˆθ MCL θ = Z + / Z + λ λ 4 λ λ Z + O p T Z. 4 For give X = x, i.e. T = t := x γ, the coditioal expectatio of Z ad Z is E θ,γ Z t = Eθ,γ [uy i t] λ θ, x = 0, i= [ E θ,γ Z ] t = E θ,γ [uy i λ θ, x t =, i= + [ E θ,γ uyi λ θ, x uy j λ θ, x t ]] i = j i, j hece the coditioal variace of Z is equal to, i.e. V θ,γ Z t =. I a similar way to the above, we have 5 E θ,γ Z t = λ /, E θ,γ Z 4 t = + λ 4. 6 Sice, by 4, 5 ad 6 E θ,γ Ũ 0 T = E θ,γ Z T + E θ,γ Z T + λ = / E θ,γ Z T λ λ 4 λ λ TE θ,γ Z T + O p λ / + O p E θ,γ Ũ 0 T = E θ,γ Z T E θ,γ Z T, 7 λ / E θ,γ Z T
Secod-order asymptotic compariso 485 + E θ,γ Z T + λ λ = + + 4 + O p 5 4 λ 4 TE θ,γ Z T + O p λ 4 λ λ where λ i = λ i θ, X i =,, 4. Sice, for i =,, 4 λ E θ,γ Z 4 T T, 8 λ i = λ i θ, X = λ i θ, γ + λi X γ+ O p = λ i θ, γ + O p = λ i + O p, 9 it follows from 7 that ] E θ,γ Ũ 0 = E θ,γ [E θ,γ Ũ 0 T = = λ λ / + O It is oted from, 7 ad 40 that E θ,γ U γ = E θ,γ Û = E θ,γ Ũ 0 = λ λ / + O E θ,γ. 40 λ / + O I a similar way to the above, we obtai from 7, 8 ad 9 E θ,γ Ũ 0 = + + λ 4λ λ 4 λ kλ Sice, by 9 ad 0 k λ = λ k θ = k = k θ k θ λ uγ + k. λ + O. 4 λ = k λ uγ k θ λ = λ uγ + λ, θ
486 M. Akahira it follows from 4 that hece, by 40 E θ,γ Ũ0 = + λ 4λ λ 4 λ + λ uγ + O λ, V θ,γ Ũ 0 = + 5λ λ λ 4 λ + λ uγ + O λ. 4 From 4, 40 ad 4, we have the coclusio of Theorem. Appedix B The proof of Lemma Sice the secod-order asymptotic cumulative distributio fuctio of T is give by F T t = P θ,γ T t = P θ,γ X γ t γ + t = γ bθ, γ axeθux dx [ ] aγ eθuγ = exp t bθ, γ [ eθuγ t b θ, γ c θ γ bθ, γ + a γ e θuγ + O ] for t > 0, where c θ γ := a γ + θaγ u γ, we obtai 6. From 6, we also get 7 by a straightforward calculatio. The proof of Lemma As is see from the begiig of Sect. 5, they,...,y are i.i.d. radom variables with a desity gy; θ,x = ayeθuy bθ, x for x < y < d. 4 The, the coditioal expectatio of Z give T is obtaied by E θ,γ Z T = λ = λ Eθ,γ [ux i T ] λ i= ux + E θ,γ [uy i T ] λ, 44 i=
Secod-order asymptotic compariso 487 where λ i = λ i θ, γ i =,. Sice, for each i =,...,, by4 d E θ,γ [uy i T ]= uy ayeθuy X bθ, X dy = θ log bθ, X = λ θ, X =: λˆ say, it follows from 44 that E θ,γ Z T = ux + λˆ λ λ, λ hece, from 7 ad 44 E θ,γ Z T = E θ,γ [TE θ,γ Z T ] = λ E θ,γ [ux T ]+ E θ,γ ˆ λ T λ λ k + Aθ, γ + O, 45 where k = kθ, γ. Sice, by the Taylor expasio T + u γ T + O p, ˆλ = λθ, X = λ θ, γ + λ θ, γ T T + O p, ux = uγ + u γ + λ θ, γ it follows from 7 that [ E θ,γ ux T ] = uγ k E θ,γ λˆ T = λ k + + Auγ + u γ k λ A + λ k + O + O, 46, 47 where k = kθ, γ, A = Aθ, γ ad λ = λ θ, γ. From45, 46 ad 47, we obtai 8.
488 M. Akahira The proof of Lemma First, we have E θ,γ Z T = E θ,γ ux i λ T λ For i, wehave = λ + λ + λ + λ i= ux λ ux λ E θ,γ [uy i λ T ] i= ] E θ,γ [uy i λ T i= [ E θ,γ uyi λ ] uy j λ T. 48 i = j i, j E θ,γ [uy i λ T ] = E θ,γ [uy i T ] λ = λ θ, X λ θ, γ λ T = + O p = O p, 49 ad for i = j ad i, j Sice, for i =,..., E θ,γ [ uyi λ uy j λ T ] [ = E θ,γ [uy i λ T ] E θ,γ uy j λ T ] λ T = + O p = O p. 50 E θ,γ [u Y i T ]= d X u y ayeθuy bθ, X dy = bθ, X θ bθ, X = λ θ, X + λ θ, X = ˆ λ + ˆ λ,
Secod-order asymptotic compariso 489 where ˆλ i = λ i θ, X i =,, wehavefori =,..., ] E θ,γ [uy i λ T = E θ,γ [u Y i T ] λ E θ,γ [uy i T ]+λ = λˆ + λˆ λ λˆ + λ = λ + λ T + O p = λ + O p. 5 From 48, 49, 50 ad 5, we obtai E θ,γ Z T = ux λ λ + ux λ O p λ + λ + O p λ + O p λ = + O p, hece, by 7 E θ,γ Z T = E θ,γ[te θ,γ Z T ] =E θ,γt + O Thus we get 9. = k + O. Ackowledgmets The author thaks the referees for their careful readig ad valuable commets, especially oe of them for poitig out the mistake of omittig certai term i the stochastic expasios of the estimators. He also thaks the associate editor for the commet. Refereces Akahira, M. 986. The structure of asymptotic deficiecy of estimators. Quees Papers i Pure ad Applied Mathematics, 75. Kigsto: Quee s Uiversity Press. Akahira, M., Ohyauchi, N. 0. The asymptotic expasio of the maximum likelihood estimator for a trucated expoetial family of distributios I Japaese. Kôkyûroku: RIMS Research Istitute for Mathematical Scieces, Kyoto Uiversity, 804, 88 9. Akahira, M., Takeuchi, K. 98. O asymptotic deficiecies of estimators i pooled samples i the presece of uisace parameters. Statistics ad Decisios,, pp. 7 8. Also icluded i. 00. Joit Statistical Papers of Akahira ad Takeuchi pp. 99 0. New Jersey: World Scietific. Akahira, M, Hashimoto, S., Koike, K., Ohyauchi, N. 04. Secod order asymptotic compariso of the MLE ad MCLE for a two-sided trucated expoetial family of distributios. Mathematical Research
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