Admissibility under the LINEX loss function in non-regular case. Hidekazu Tanaka. Received November 5, 2009; revised September 2, 2010

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1 Scientiae Mathematicae Japonicae Online, e-2012, Admissibility nder the LINEX loss fnction in non-reglar case Hidekaz Tanaka Received November 5, 2009; revised September 2, 2010 Abstract. In the statistical estimation theory, sfficient conditions for admissibility of estimators have been discssed mostly nder qadratic loss fnction. Recently, nder the LINEX loss fnction, sfficient conditions were developed in some families of distribtions. However, they do not inclde some typical non-reglar distribtions so that the dimension of the minimal sfficient statistic is two. The prpose of this paper is to give sfficient conditions for admissibility of estimators in sch non-reglar distribtions nder the LINEX loss fnction. Some examples are also given. 1 Introdction We consider an estimation problem of a real-valed fnction of nknown parameter based on samples obtained from a probability distribtion with nknown parameter. Under qadratic loss fnction, sfficient conditions for linear estimators to be admissible has been investigated by many athors (Karlin [3, Ghosh and Meeden [1, Ralesc and Ralesc [10, Hoffmann [2). Frther, the general theory for admissibility was derived by Plskamp and Ralesc [9 for reglar case, and Sinha and Gpta [15, Kim and Meeden [4 for non-reglar case. On the other hand, nder the LINEX loss fnction, admissibility reslts were limited to whether linear estimator is admissible or not (Rojo [11, Sadooghi-Alvandi and Nematollahi [14, Ko and Dey [5, Sadooghi-Alvandi [13, Pandey [6, Parsian and Farsipor [7). Here, the LINEX loss fnction is defined by { } (1) L(, δ) = b e a(δ g()) a(δ g()) 1, for an estimator δ of fnction g() to be estimated, where a 0 and b > 0 (Varian [18, Zellner [19). It shold be noted that the LINEX loss fnction is regarded as an extension of qadratic loss fnction. Recently, sfficient conditions for admissibility has been developed for reglar case by Tanaka [16. Similar reslt was obtained by Tanaka [17 for the nonreglar case when the dimension of the minimal sfficient statistic is one, which incldes a niform distribtion U(0, ). However, it does not inclde some typical distribtions sch as U(, +1), which will be treated in Section 4. The prpose of this paper is to give sfficient conditions for generalized Bayes estimator (GBE) to be admissible nder the LINEX loss fnction when the dimension of the minimal sfficient statistic is two. This paper is organized as follow. In Section 2, we give preliminaries. In Section 3, we give sfficient conditions for GBE to be admissible nder the LINEX loss fnction and its corollaries. In Section 4, we give some examples. Finally, we give proof of the theorem in Section Mathematics Sbject Classification. Primary 62C15, 62F10; Secondary 62F15. Key words and phrases. LINEX loss fnction, admissibility, non-reglar distribtion.

2 428 HIDEKAZU TANAKA 2 Preliminaries Sppose that a random vector X := (X 1, X 2 ) is distribted according to a probability distribtion, and its probability density fnction (p.d.f.) with respect to the Lebesge measre is given by { q()r(x) (x X ), (2) f(x, ) = 0 (otherwise), where x := (x 1, x 2 ), q() > 0, r(x) > 0 and X := {x α() < x 1 < x 2 < β()} for some fnctions α() and β(). Here, we assme that both α() and β() are strictly increasing. This setp is motivated by the case when the dimension of sfficient statistic is two. Then we consider the admissibilities of estimators of fnction g() based on X nder the LINEX loss fnction (1). We can assme b = 1 withot loss of generality. By the general theory abot admissibility (Sacks [12), it is enogh to focs on GBE w.r.t. improper priors. Let π() be an improper prior density of. Here, we assme π() > 0 for all Θ. We frther restrict estimators to the class (3) := {δ (A1) and (A2) are satisfied}, where (A1) E δ(x) < and E [e aδ(x) < for all Θ, (A2) v E δ(x) g() π()d < and v E [e a(δ(x) g()) π()d < for all < v (, v Θ). Under these conditions, the GBE of g() w.r.t. π() is given by δ π (X), where (4) δ π (x) = 1 a log Θ e ag() q()π()i Θx ()d Θ q()π()i Θ x ()d for all x X := {x x 1 < x 2 }, provided that the integrals exist and are finite, where Θ x := { Θ β 1 (x 2 ) < < α 1 (x 1 )}. 3 Sfficient conditions for generalized Bayes estimator to be admissible In this section, we give sfficient conditions for GBE (4) to be admissible nder the LINEX loss fnction. The following is the main reslt. Theorem 1 Sppose that δ π (X) which is defined in (3). For Θ and x X, pt and F (x, ) := γ() := (e aδ π(x) e ag(t) )q(t)π(t)i Θx (t)dt, eag() F 2 (x, )e aδπ(x) r(x)i X (x)dx, q()π() where I A (x) is the indicator fnction of set A. If γ() < for all Θ and there exists d Θ sch that c d d (5) lim c d γ() = lim d c c γ() =, then δ π (X) is -admissible for g() nder the LINEX loss fnction.

3 ADMISSIBILITY UNDER LINEX LOSS FUNCTION 429 The proof is given in Section 5. The sfficient condition (5) in Theorem 1 is somewhat complicated. By adding some conditions on g(), we give two corollaries. Now we note from (4) that F (x, ) is expressed as 1 F (x, ) = q()π()i Θ x ()d (e ag(s) e ag(t) )q(s)π(s)i Θx (s)q(t)π(t)i Θx (t)dtds. Corollary 1 Sppose that g() is bonded and δ π (X). For Θ and x X, pt 1 F (x, ) := q()π()i Θ x ()d q(s)π(s)i Θx (s)ds and 1 γ() := F 2 (x, )r(x)i X (x)dx. q()π() If γ() < for all Θ and there exists d Θ sch that lim c c d d γ() = lim c d c d γ() =, then δ π (X) is -admissible for g() nder the LINEX loss fnction. The proof is omitted since it can be easily obtained. q(t)π(t)i Θx (t)dt, Next, we consider the class of prior density fnctions treated by Sinha and Gpta [15 and Kim and Meeden [4. Sppose that g() is strictly increasing and differentiable. Then, pt (6) π h () = g () q() h(g()) for positive fnction h( ). In this case, the GBE of g() w.r.t. π h () is given by δ πh (X), where δ πh (x) = 1 a log g(θ) e az h(z)i g(θx )(z)dz g(θ) h(z)i. g(θ x )(z)dz The next corollary is immediately obtained from Theorem 1. Corollary 2 Sppose that g() is strictly increasing and differentiable, and δ πh (X). For Θ and x X, pt F h (x, ) := g() g() (e aδ π h (x) e az )h(z)i g(θx )(z)dz, and e ag() γ h () := g F ()h(g()) h(x, 2 )e aδ πh (x) r(x)i X (x)dx. If γ h () < for all Θ and there exists d Θ sch that lim c c d d γ h () = lim c d c d γ h () =, then δ πh (X) is -admissible for g() nder the LINEX loss fnction.

4 430 HIDEKAZU TANAKA Of corse, same discssion for strictly decreasing fnction g() is possible. Bt we omit it since it is obtained by similar argment. Also, it can be shown that these reslts are essentially regarded as extensions of Kim and Meeden [4 nder sitable conditions (see Tanaka [16). In this paper, we do not discss the detail of the relation. 4 Examples In this section, we give some typical examples. In Examples 1,2 and 3, we treat a location parameter family of distribtions. In Example 4, we treat a scale parameter family of distribtions. Example 1 Sppose that Y 1,..., Y n are i.i.d. random variables according to a niform distribtion U(, + 1), where ( R) is nknown. Then, the p.d.f. of the sfficient statistic (X 1, X 2 ) = (min 1 i n Y i, max 1 i n Y i ) is given by (2), where q() = 1, r(x) = n(n 1)(x 2 x 1 ) n 2 and X = {x < x 1 < x 2 < + 1}. The GBE of w.r.t. π η () = e η (η R) is given by δ πη (x) = 1 a log [ e ax 1 e a(x 2 1) a(x 2 x 1 1) [ 1 a log a(x1 x 2 +1) e ax 1 e a(x 2 1) [ 1 a log η e (η a)x 1 e (η a)(x 2 1) η a e ηx 1 e η(x 2 1) (η = 0), (η = a), (η 0, a). Here, we note that these estimators are location invariant, that is, δ πη (x 1 +, x 2 + ) = δ πη (x 1, x 2 ) for all (x 1, x 2 ) and for all R. Frther, we can easily obtain r(x 1 +, x 2 + ) = r(x 1, x 2 ), F (x 1 +, x 2 +, ) = e (η a) F (x 1, x 2, 0). Hence, we see that γ() = e η F 2 (x, 0)e aδ πη (x) r(x)i X0 (x)dx <, and conseqently, δ πη (X) is -admissible for η = 0 from Theorem 1. We remark that for η 0 Theorem 1 can not tell whether δ πη (X) is -admissible or not. However, we see that δ π0 (X) dominates δ πη (X) for η 0, since δ π0 (X) is the best invariant estimator (Parsian, Farsipor and Nematollahi [8, page 103). Example 2 In Example 1, consider the estimation problem of 1 ( < 0), g() = P (Y 1 1) = 1 (0 < < 1), 0 (1 < ). Sppose that the prior distribtion of is given by π η () = e η (η R). Of corse, in this case, g() is bonded. So, we se Corollary 1. It is easy to derive that { γ() = X 0 F 2 (x, 0)r(x)dx (η = 0), η 2 e η (η 0). From Corollary 1, we see that δ π0 (X) is -admissible nder the LINEX loss fnction.

5 ADMISSIBILITY UNDER LINEX LOSS FUNCTION 431 Example 3 Sppose that Y 1,..., Y n are i.i.d. random variables according to the p.d.f. { C0 e p(y, ) = y ( < y < + 1), 0 (otherwise), where ( R) is nknown and C 0 := 1/(e 1) is the normalizing factor. Then the p.d.f. of the sfficient statistic (X 1, X 2 ) = (min 1 i n Y i, max 1 i n Y i ) is given by (2), where q() = C0 n e n, r(x) = n(n 1)e x1+x2 (e x2 e x1 ) n 2 and X = {x < x 1 < x 2 < +1}. The GBE of w.r.t. π η () = e η (η R) is given by δ πη (x) = { } 1 a log n η e (n η+a)x 1 e (n η+a)(x 2 1) n η+a e (n η)x 1 e (n η)(x 2 1) { } 1 a log a(x1 x 2 +1) e ax 1 e a(x 2 1) 1 a log { 1 a e ax 1 e a(x 2 1) x 1 x 2+1 } (η n, n + a), (η = n + a), (η = n). Of corse, these estimators are location invariant. Frther, by sing the facts r(x 1 +, x 2 + ) = e n r(x 1, x 2 ), F (x 1 +, x 2 +, ) = e (n η+a) F (x 1, x 2, 0), we get γ() = eη C0 n F 2 (x, 0)e aδ πη (x) r(x)i X0 (x)dx <. Therefore, Theorem 1 shows that δ πη (X) is -admissible when η = 0. By the same reason as Example 1, δ π0 (X) dominates δ πη (X) for η 0. Example 4 Sppose that Y 1,..., Y n are i.i.d. random variables according to the p.d.f. { 1 p(y, ) = (ξ < y < (ξ + 1)), 0 (otherwise), where ξ (0, ) is known and (0, ) is nknown. In this case, the p.d.f. of the sfficient statistic (X 1, X 2 ) = (min 1 i n Y i, max 1 i n Y i ) is given by (2), where q() = n, r(x) = n(n 1)(x 2 x 1 ) n 2 and X = {x ξ < x 1 < x 2 < (ξ + 1)}. Consider the estimation problem of g() = log. Let the prior density of be π h () in (6) for h(z) = e εz (ε R). Then, the GBE of log is given by δ πh (x) = [ 1 a log {x2/(ξ+1)} a (x 1/ξ) a a log{(ξ+1)x 1 /(ξx 2 )} [ 1 a log a log{(ξ+1)x1 /(ξx 2 )} (x 1 /ξ) a {x 2 /(ξ+1)} a 1 a log [ ε ε a (x 1 /ξ) ε a {x 2 /(ξ+1)} ε a (x 1 /ξ) ε {x 2 /(ξ+1)} ε (ε = 0), (ε = a), (ε 0, a). By a direct calclation, we get δ πh (x 1, x 2 ) = δ πh (x 1, x 2 ) + log. Frther, we can easily obtain r(x 1, x 2 ) = n 2 r(x 1, x 2 ), F h (x 1, x 2, ) = ε a F h (x 1, x 2, 1). These imply that γ h () = n+ε+1 n(n 1) F 2 h(x, 1)e aδ πh (x) r(x)i X1 (x)dx <. Therefore, we see that δ πh (X) is -admissible for ε = n from Corollary 2. By considering the logarithmic transformations of y 1,..., y n and, the problem is essentially same as Example 3. So, we see that δ πh (X) is not -admissible for ε n.

6 432 HIDEKAZU TANAKA 5 Appendix In this section, we give the proof of Theorem 1. The proof is resemble to Tanaka [16. Proof of Theorem 1. Sppose that there exists an estimator δ(x) sch that E [L(, δ(x)) E [L(, δ π (X)) for all Θ. From the condition (A1), we see that this is eqivalent to [ (7) e ag() E (e aδ(x)/2 e aδπ(x)/2 ) 2 E [a (δ(x) δ π (X)) 2e ag() e aδπ(x)/2 (e aδ(x)/2 e aδπ(x)/2 ). Mltiplying both sides of (7) by π(), and integrating w.r.t. [, v Θ, we obtain over the finite interval v E [(e aδ(x)/2 e aδπ(x)/2 ) 2 e ag() π()d v E [a (δ(x) δ π (X)) 2e ag() e aδπ(x)/2 (e aδ(x)/2 e aδπ(x)/2 ) π()d. An application of the Fbini theorem gives v (8) (e aδ(x)/2 e aδπ(x)/2 ) 2 r(x)i X (x)dxe ag() π()q()d v { r(x)i X (x) a(δ(x) δπ (x)) 2e ag() e aδπ(x)/2 (e aδ(x)/2 e aδπ(x)/2 ) } q()π()i Θx ()ddx, which is garanteed by (A2). Using the ineqality x y e y (e x e y ) for all x, y R, the right hand side in (8) is dominated by (9) 2 r(x)i X (x)(e aδ(x)/2 e aδπ(x)/2 )e aδ π(x)/2 v (e aδπ(x) e ag() )q()π()i Θx ()ddx. Clearly, we see that F (x, v) = 0 for β 1 (x 2 ) > v. Also, it follows from (4) that F (x, v) = (e aδ π(x) e ag(t) )q(t)π(t)i Θx (t)dt = 0 for α 1 (x 1 ) < v. Ths, we get F (x, v) = F (x, v)i Xv (x), conseqently, (9) is rewritten by (10) 2 r(x)(e aδ(x)/2 e aδπ(x)/2 )e aδπ(x)/2 F (x, v)i Xv (x)dx 2 r(x)(e aδ(x)/2 e aδπ(x)/2 )e aδπ(x)/2 F (x, )I X (x)dx.

7 ADMISSIBILITY UNDER LINEX LOSS FUNCTION 433 Pt T () := (e aδ(x)/2 e aδπ(x)/2 ) 2 r(x)i X (x)dx. Then, by applying the Schwarz ineqality to (10) and combining (8), we have v e ag() π()q()t ()d { 1/2 { 1/2 2 T (v)e q(v)π(v)} ag(v) γ 1/2 (v) + 2 T ()e q()π()} ag() γ 1/2 (). By the same argment in Tanaka [16, the proof is completed. ACKNOWLEDGMENTS The athor wold express his appreciation to the referee. This research was partially spported by the Ministry of Edcation, Science, Sports and Cltre, Grant-in-Aid for Yong Scientists (B), , References [1 Ghosh, M. and Meeden, G. Admissibility of linear estimators in the one parameter exponential family. Ann. Statist., 5 (1977), [2 Hoffmann, K. Admissibility and inadmissibility of estimators in the one-parameter exponential family. Statistics, 16 (1985), [3 Karlin, S. Admissibility for estimation with qadratic loss. Ann. Math. Statist., 29 (1958), [4 Kim, B. H. and Meeden, G. Admissible estimation in an one parameter nonreglar family of absoltely continos distribtions. Comm. Statist. Theory Methods, 23 (1994), [5 Ko, L. and Dey, D. K. On the admissibility of the linear estimators of the Poisson mean sing linex loss fnctions. Statist. Decisions, 8 (1990), [6 Pandey, B. N. Testimator of the scale parameter of the exponential distribtion sing linex loss fnction. Comm. Statist. Theory Methods, 26 (1997), [7 Parsian, A. and Farsipor, N. S. Estimation of parameters of exponential distribtion in the trncated space sing asymmetric loss fnction. Statist. Papers, 38 (1997), [8 Parsian, A., Farsipor, N. S. and Nematollahi, N. On the minimaxity of Pitman type estimator nder a LINEX loss fnction. Comm. Statist. Theory Methods, 22 (1993), [9 Plskamp, R. J. and Ralesc, D. A. A general class of nonlinear admissible estimators in the one-parameter exponential case. J. Statist. Plann. Inference, 28 (1991), [10 Ralesc, D. and Ralesc, S. A class of nonlinear admissible estimators in the one-parameter exponential family. Ann. Statist., 9 (1981), [11 Rojo, J. On the admissibility of cx + d with respect to the LINEX loss fnction. Comm. Statist. Theory Methods, 16 (1987), [12 Sacks, J. Generalized Bayes soltions in estimation problems. Ann. Math. Statist., 34 (1963), [13 Sadooghi-Alvandi, S. M. Estimation of the parameter of a Poisson distribtion sing a LINEX loss fnction. Astral. J. Statist., 32 (1990), [14 Sadooghi-Alvandi, S. M. and Nematollahi, N. A note on the admissibility of cx + d relative to the linex loss fnction. Comm. Statist. Theory Methods, 18 (1989),

8 434 HIDEKAZU TANAKA [15 Sinha, B. K. and Gpta, A. D. Admissibility of generalized Bayes and Pitman estimates in the nonreglar family. Comm. Statist. A Theory Methods, 13 (1984), [16 Tanaka, H. Sfficient conditions for the admissibility nder LINEX loss fnction in reglar case. Comm. Statist. Theory Methods, 39 (2010), [17 Tanaka, H. Sfficient conditions for the admissibility nder the LINEX loss fnction in nonreglar case. Accepted for pblication in Statistics. [18 Varian, H. R. A Bayesian approach to real estate assessment. Stdies in Bayesian econometrics and statistics, In honor of Leonard J. Savage. Eds. Fienberg, S. E. and Zellner, A. North-Holland Amsterdam (1975), [19 Zellner, A. Bayesian estimation and prediction sing asymmetric loss fnctions. J. Amer. Statist. Assoc., 81 (1986), Commnicated by Asis SenGpta

Gamma-admissibility of generalized Bayes estimators under LINEX loss function in a non-regular family of distributions

Hacettepe Journal of Mathematics Statistics Volume 44 (5) (2015), 1283 1291 Gamma-admissibility of generalized Bayes estimators under LINEX loss function in a non-regular family of distributions SH. Moradi