Improving on minimum risk equivariant and linear minimax estimators of bounded multivariate. location parameters.

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1 Impoving on minimum isk equivaiant and linea minimax estimatos of bounded multivaiate location paametes. Éic Machand a & Ami T. Payandeh Najafabadi b adṕatement de mathématiques, Univesité de Shebooke, Shebooke, CANADA. eic.machand@ushebooke.ca b Depatment of Mathematical Sciences, Shahid Beheshti Univesity, G.C. Evin, , Tehan, IRAN. amitpayandeh@sbu.ac.i Abstact We popose impovements unde squaed eo loss of the minimum isk equivaiant and the linea minimax estimatos fo estimating the location paamete θ of a p-vaiate spheically symmetic distibution, with θ esticted to a ball of adius m centeed at the oigin. Ou constuction of explicit impovements elies on a multivaiate vesion of Kubokawa s Integal Expession of Risk Diffeence (IERD) method. Applications ae given fo univaiate distibutions, fo the multivaiate nomal, and fo scale mixtue of multivaiate nomal distibutions. Keywods: Decision theoy, spheical symmetic distibution, esticted paamete, minimum isk equivaiant estimato, linea minimax estimato, dominating estimatos, squaed eo loss. 21 Mathematics Subject Classification: 62F1, 62F3, 62H12 1. INTRODUCTION We conside the poblem of estimating, unde squaed eo loss, the location paamete θ of a p- vaiate spheically symmetic distibution unde the constaint θ m, withm> known. With seveal authos having obtained inteesting esults elative to this poblem, and moe geneally fo esticted paamete space poblems (see Machand & Stawdeman, 24; van Eeden, 26 fo useful eviews), we focus on the detemination of benchmak estimatos such as the maximum

2 likelihood estimato (mle), the minimum isk equivaiant estimato (me), and the linea minimax estimato (lmx). In this egad, Machand & Peon (21) povide fo the multivaiate nomal case impovements on the (always) inadmissible mle fo all (m, p). These include Bayesian impovements, but conditions ae then equied on (m, p). Complementay findings fo the multivaiate nomal and paallel findings fo othe spheically symmetic distibutions, including in paticula multivaiate student distibutions, wee obtained espectively by Foudinie & Machand (21) and Machand & Peon (25); but again conditions fo the studied pios π (typically bounday unifom, unifom on sphees, and fully unifom) of the fom m c π (p) fo the Bayes estimato δ π to dominate the mle ae necessitated. Hence, the poblem of finding a Bayesian o an admissible impovement fo any (m, p), fo any given spheically symmetic distibution emains unsolved (even fo p =1o the multivaiate nomal distibution). Altenatively, fo the objective of passing the minimum test of impoving upon the minimum isk equivaiant estimato, positive findings fo the univaiate case (p =1) wee obtained by Machand & Stawdeman (25), as well as by Kubokawa (25). The fome establish a geneal dominance esult fo the fully unifom pio Bayes estimato, which actually applies moe geneally fo a wide not necessaily symmetic class of location model densities and location invaiant losses. The latte povides on the othe hand a lage class of pios which lead to Bayesian impovements fo the univaiate vesion of ou poblem of symmetic densities and squaed eo loss. A key featue of these dominance esults is the use of Kubokawa s (1994) Integal Expession of Risk Diffeence (IERD) technique. Fo multivaiate settings, a lovely esult by Hatigan (24) tells us that fo multivaiate nomal distibutions, the fully unifom Bayes pocedue impoves upon the minimum isk equivaiant estimato. The esult is actually moe geneal and applies fo convex esticted paamete spaces with non-empty inteios. Howeve, Hatigan s esult does equie nomality and hence a spheically symmetic analog emains an open question. Moeove, Hatigan s esult does not apply to the benchmak linea minimax estimato, which epesents itself a simple impovement on the 2

3 minimum isk equivaiant estimato. With the above backgound, ou motivation hee esides in extending the univaiate dominance esults to the multivaiate case, extending Hatigan s esult fo balls to spheically symmetic distibutions, and consideing impovements upon the linea minimax pocedue as well. We povide peliminay esults in this diection in tems of sufficient conditions fo dominating eithe the minimum isk equivaiant estimato, the linea minimax estimato, o both. Ou teatment possesses the inteesting featue of being unified with espect to the dimension p and the given spheically symmetic distibution. Moeove, we aive at ou dominance esults though a novel multivaiate vaiant of Kubokawa s IERD technique. The main dominance esults ae pesented in Section 2, and vaious examples o illustations ae pusued in Section 3. These include univaiate distibutions, the multivaiate nomal distibution, and scale mixtue of multivaiate nomal distibutions. 2. Main esults Let X be a p-vaiate andom vecto with spheically symmetic density f( x θ 2 ), (1) whee the location paamete θ is constained to a ball centeed at the oigin and of adius m, say Θ m. We seek impovements on the minimum isk equivaiant (MRE) estimato δ (X) =X, and the linea minimax estimato δ lmx (X) = m2 X unde squaed eo loss L(θ, d) = d θ 2, m 2 +pσ 2 whee E θ ( X θ 2 )=pσ 2 <. Heeafte, we denote the noms of X, x, andθ by R,, andλ espectively. Ou esults bing into play the othogonally invaiant in θ and nonnegative quantities H(t, λ) = E θ(θ T X X t) E θ (X T X X t) and H (t, λ) = λe θ( X X t) E θ ; t,λ. We will make use of the ( X T X X t) inequality H(t, λ) H (t, λ) fo all t,λ, which follows as a simple application of the Cauchy-Schwatz inequality. Now, we pesent the main dominance esults of this pape. Theoem 1. Fo a model as in (1), δ g (X) =g( X )X dominates g() X, wheneve: 3

4 (i) g is absolutely continuous, nonconstant, and noninceasing; (ii) and g() sup λ [,m] H(, λ) fo all. Moeove, if conditions (i) and (ii) ae satisfied, and (iii) g() [ m2 pσ 2 m 2 +pσ 2, 1), then δ g (X) =g( X )X also dominates δ (X) =X. Remak 1. By vitue of the inequality H(t, λ) H (t, λ) fo all t,λ, condition (ii) of Theoem 1 can be eplaced by the weake, but nevetheless useful, condition (ii ) and g() sup λ [,m] H (, λ) fo all. Poof of the Theoem. It is staightfowad to veify that g()x dominates X unde condition (iii), so that conditions fo which δ g (X) dominates g() X, suchas(i) and (ii), ae necessaily conditions fo which δ g (X) =g( X )X also dominates δ (X) =X. Now, using Kubokawa s IERD technique, the isk diffeence between the estimatos δ g (X) and g() X can be witten as 1 2 (θ) = 1 [R(θ, g( X )X) R(θ, g()x)] 2 [ Eθ g( X )X θ 2 g()x θ 2] = 1 2 = 1 2 E θ = = ( X x R p ) t g(t)x θ 2 dt g (t)[g(t) x θ] T xf( x θ 2 ) dt dx g (t) [ g(t)x T x θ T x]f( x θ 2 ) dx dt. {x R p : x t} Now, obseve that conditions (i) and(ii) implythat (θ) fo all θ Θ m, establishing the esult. Hee ae some futhe emaks and obsevations in elationship to Theoem 1. 4

5 The noninceasing popety of condition (i) is not necessaily estictive. Indeed, fo the multivaiate nomal case, Machand and Peon (21, theoem 5) establish that the noninceasing popety holds fo all Bayesian estimatos associated with symmetic, logconcave pio densities on [ m, m]. The conditions of Theoem 1 suggest the bounds (ii) and (ii ) themselves sup λ [,m] H(, λ) and sup λ [,m] H (, λ) as candidate g functions. These functions ae of the fom H(, λ()) and H (, λ()), wheeλ( ) is some function taking values on [,m]. All such functions lead to ange peseving estimatos δ g ; i.e., δ g (x) m fo all x R p ; since fo all and θ = λ() : H(, λ()) H (, λ()) = λ() E θ( X X ) E θ ( X T X X ) λ() m, and since δ g (x) m fo all x R p wheneve g() m fo all >. Finally, as a consequence of the above, obseve that the pojection of δ (X) onto Θ m,givenbyδ gp with g p () = m 1, satisfies the conditions of Theoem 1. We now focus on elated implications fo the estimatos δ H (X) =H( X,m)X and δ H (X) = H ( X,m)X, which will tun out in seveal cases to be the smallest possible g s satisfying espectively conditions (ii) and (ii ) of Theoem 1. Coollay 1. (a) If H(, λ) inceases in λ [,m] fo all, and deceases in [, ] fo all λ [,m], then δ H (X) =H( X, m) X dominates both the linea minimax estimato δ lmx (X) and the MRE estimato δ (X); (b) If H (, λ) inceases in λ [,m] fo all, then δ H (X) =H ( X, m) X dominates the MRE estimato δ (X). Poof. Pat (a) follows as a diect application of Theoem 1 as H(,m)= E θ(θ T X) E θ (X T X) = m2 m 2 +pσ 2 [ m2 pσ 2, 1), fo θ = m. Pat (b) follows fo two easons. Fist, fo any positive andom vaiable Y m 2 +pσ 2 with density g Y, and its biased vesion W with density popotional to wg Y (w), the atio E(Y 2 Y>t) = E(Y Y >t) E(W W > t) is inceasing in t, which implies that H (,m) is a deceasing function on [, ). Secondly, fo θ = m, H (,m)=m E θ( X ) E θ ( X 2 ) E θ (X) = m. = E θ( X/m ) E θ ( X/m 2 ) < E θ( X/m ) 2 E θ ( X/m 2 ) < 1, sincee θ( X ) > 5

6 3. Examples The following subsections ae devoted to applications of Coollay 1, with the key difficulty aising in checking the monotonicity conditions elative to H and H. We focus on geneal univaiate cases (subsection 3.1), the multivaiate nomal distibution (subsection 3.2.), and scale mixtues of multivaiate nomal distibutions (subsection 3.3) Univaiate spheically symmetic distibutions We expess the symmetic univaiate densities in (1) as f θ (x) =e q(x θ), (2) and estict ouselves to cases whee q Q = {q : q( ) is inceasing and convex on (, ), and q ( ) is concave on (, )}. Examples of such distibutions include nomal, Laplace, exponential powe densities with q(y) = αy β + c, α >, 1 β 2; Hypebolic Secant, Logistic, Genealized logistic densities with q(y) = y + 2 log(1 + α eαy )+c, α>; and Champenowne densities with q(y) =log(cosh(y)+β), β [, 2], (also see Machand & Peon, 29; Machand et al., 28). The next theoem establishes fo such densities the applicability of pat (a) of Coollay 1 and dominance of δ H (X) ove both the linea minimax estimato, δ lmx (X), and the MRE estimato δ (X). Theoem 2. Fo model densities as in (2) with q Q, the estimato δ H (X) =H( X,m)X dominates both the linea minimax estimato δ lmx (X) and the MRE estimato δ (X). Poof. By vitue of Coollay 1, it suffices to show that H(, λ) deceases in [, ) fo all λ [,m], and inceases in λ [,m] fo all. Fist,H(, λ) can be witten as H(, λ) = λ x(f (x λ) f (x + λ))dx x 2 (f (x λ)+f (x + λ))dx = λ 2 E λ ( tanh((q(y + λ) q(y λ))/2) λy 6 ),

7 whee Y is a andom vaiable with density popotional to y 2 (f (y λ)+f (y + λ))1 [, ) (y). Such a family of densities with paamete has inceasing monotone likelihood atio in Y. Futhemoe, since q Q, a esult of Machand et al. (28) (Lemma 1, pat e) tells us that the inne function of the above expectation in Y is noninceasing. Hence, we conclude that, fo all λ [,m], H(λ, ) deceases on [, ). Tuning to the monotonicity of H(,), begin by witing x(f (x λ) f (x + λ))dx H(, λ) = λ x 2 (f (x λ)+f (x + λ))dx λ = λ (y + λ)f (y)dy (y λ)f +λ (y)dy (y + λ λ)2 f (y)dy + (y +λ λ)2 f (y)dy A(, λ) = λ B(, λ), whee A(, λ) and B(, λ) ae the numeato and denominato of the above faction, espectively. Manipulations yield: B 2 H(, λ) (, λ) λ = A(, λ)b(, λ)+λa (, λ)b(, λ) λa(, λ)b (, λ) = [l(, λ)+a 1 (, λ)][b 1 (, λ)+λ(λf ( λ)+λf ( + λ) f ( λ)+f ( + λ))] + [λ(f ( λ)+f ( + λ)) + A 1 (, λ)][b 1 (, λ)+λl(, λ)] = λg(, λ)f ( λ)+2a 1 (, λ)b 1 (, λ)+λ 2 f ( + λ)l(, λ) + 2 λf ( + λ)l(, λ)+λf ( + λ)b 1 (, λ)+λ 2 f ( + λ)l(, λ), whee G(, λ) = 2λ +λ yf λ (y)dy +λ yf λ (y)dy + λ y2 f (y)dy + +λ y2 f (y)dy, l(, λ) = +λ yf λ (y)dy, A 1 (, λ) =λ( f λ (y)dy+ f +λ (y)dy), and B 1 (, λ) = λ y2 f (y)dy+ +λ y2 f (y)dy. Now, obseve that fo all, λ [,m], the quantities B 1 (, λ), A 1 (, λ), and(, λ) ae nonnegative. Hence, to show the positivity of H(,λ), it will suffice to show the positivity of G(, λ). But, λ we have G(, λ) +λ λ yf (y)(2λ + y)dy +λ λyf (y) dy 1 [λ, ) ()+ λ λ yf (y)(2λ + y) dy 1 [,λ) () λ 2y 2 f (y) dy 1 [,λ) (), which completes the poof. 7

8 3.2. Multivaiate nomal distibutions We conside hee multivaiate nomal models in (1) X N p (θ, σ 2 ) with θ m. Wetakeσ 2 =1 without loss of geneality (since X σ N p(θ = θ,i σ p) with θ m = m ). We equie the following σ key popeties elative to ρ(λ, ) =E θ ( θt X X = ), whee λ = θ. These popeties involve X modified Bessel functions I v of ode v, and moe specifically atios of the fom ρ v (t) =I v+1 (t)/i v (t), t>. Lemma 1. (Watson, 1983; Machand & Peon, 21) (i) We have ρ(λ, ) =λρ p/2 1 (λ); (ii) ρ p/2 1 ( ) is inceasing and concave on [, ), withρ p/2 1 () = and lim t ρ p/2 1 (t) =1; (iii) ρ p/2 1 (t)/t is deceasing in t with lim t + ρ p/2 1(t)/t =1/p ; (iv) ρ p/2 (t) =ρ 1 p/2 1 (t) p/t. Denoting f p (,λ) and F p (,λ) as the pobability density and suvival functions of R = X χ 2 p(λ 2 ), we will also equie the following useful popeties. Lemma 2. (i) We have f p (, λ) =( λ )p/2 1 I p/2 1 (λ)exp{ 2 +λ 2 2 }; (ii) 2 f p (, λ) =λ 2 f p+4 (, λ)+pf p+2 (, λ); (iii) f p (, λ) ρ p/2 1 (λ) =λf p+2 (, λ); (iv) the atio F p+2 (, λ) F p(, λ) deceases in λ [, ), fo all p 1 and >. Poof. Pats (ii) and (iii) follow diectly fom (i), while (i) consists of a well known Bessel function epesentation of the noncental chi-squae distibution. Pat (iv) follows fom the identity 2 F λ p (, λ) = F p+2 (, λ) F p (, λ), and the logconcavity of F p (, ) on [, ) (see Das Gupta and Saka, 1984; Finne and Rotes, 1997). We now seek to apply pat (a) of Coollay 1. 8

9 Theoem 3. Fo multivaiate nomal densities, the estimato δ H (X) =H( X,m)X dominates both the linea minimax estimato δ lmx (X) and the MRE estimato δ (X). Poof. By vitue of Coollay 1, it suffices to show that H(, λ) deceases in [, ) fo all λ [,m], and inceases in λ [,m] fo all. Making use of Lemmas 1 and 2, we obtain H(, λ) = E θ( X E θ ( θt X X )) X E θ ( X 2 X ) = = ye θ ( θt X X = y)f X p(y, λ)dy y 2 f p (y, λ)dy yλρ p/2 1 (λy)f p (y, λ)dy y 2 f p (y, λ)dy λ 2 f p+2 (y, λ)dy = y 2 f p (y, λ)dy = { p λ + f p+4 (y, λ)dy 2 f p+2 (y, λ)dy } 1 = { p λ 2 + F p+4 (, λ) F p+2 (, λ) } 1. (3) The monotonicity popety of H(, ) on [,m] fo all now follows fom the above expession and pat (iv) of Lemma 2. Now, to show that H(, λ) deceases in, make use of (3) to wite H(, λ) = λe ( E Y ( θt X X = Y ) X ) Y = λe ( ρ p/2 1(λY ) ), Y whee Y has density popotional to yf p (y, λ)1 [, ) (y). Since this family of densities with paamete has inceasing monotone likelihood atio in Y, we conclude indeed that H(, λ) deceases fo fo all λ [,m] by making use of pat (iii) of Lemma 1. 9

10 3.3. Scale mixtues of multivaiate nomal distibutions We conside hee in this subsection scale mixtues of multivaiate nomal distibutions whee X admits the epesentation: X Z = z N p (θ, zi p ), Z having Lebesgue density g on R +. The coesponding density in (1) is of the fom (2πz) p/2 exp{ x θ 2 }g(z) dz ; (4) 2z and we futhe assume that g is logconcave on eithe R + o some open inteval (a, b) of R +. Unifom densities on (a, b) ae included. With such a epesentation, since X/ Z Z = z N p (θ/ z,i p ), we infe fom pat (i) of Lemma 2 that the density function of R = X is given by y z ( y λ )p/2 1 I p/2 1 ( λy z ) + λ 2 exp{ y2 }g(z) dz. (5) 2z We now seek to apply pat (a) of Coollay 1. Theoem 4. Fo scale mixtues of multivaiate nomal densities as in (4) with g logconcave, the estimato δ H (X) =H ( X,m)X dominates the MRE estimato δ (X). Poof. By vitue of Coollay 1, it suffices to show that H (, ) is nondeceasing on [,m] fo all unde the given logconcave assumption on g. Stating fom the definition of H and making use of 5, we obtain H (, λ) = λe θ(r R ) E θ (R 2 R ) = λ = y p 2 +1 g(z) z y p 2 +2 g(z) z x p 2 +1 g(λ 2 t) /λ /λ I p 2 1 ( yλ ) y 2 +λ 2 z e 2z dzdy I p 2 1 ( yλ z y2+λ2 ) e 2z 1+x 2 ) e 2t dzdy I t p 2 1 ( x dtdx t, x p 2 +2 g(λ 2 t) I t p 2 1 ( x 1+x2 ) e 2t dtdx t with the change of vaiables (y, z) =(λx, λ 2 t). Simple diffeentiation leads to λ H (, λ) = 1 B 2 {A 1 1

11 A 2 + A 3 A 4 }, whee B is the above denominato of H, A 1 = 2λ A 2 = 2λ /λ A 3 = λ 2 A 4 = λ 2 /λ /λ /λ xm(x, t) dtdx M(x, t) dtdx xm(x, t) dtdx M(x, t) dtdx /λ /λ x p 2 +1 g (λ 2 t)i p 2 1 ( x 1+x2 )e 2t dtdx, t x p 2 +2 g (λ 2 t)i p 2 1 ( x 1+x2 )e 2t dtdx, t g(λ 2 t) ( t λ ) p 2 +1 I p 2 1 ( λ2 + 2 λt )e 2λ 2 t dt, g(λ 2 t) ( t λ ) p 2 +2 I p 2 1 ( λ2 + 2 λt )e 2λ 2 t dt, with M(x, t) = g(λ2 t) t x p 2 +1 I p 2 1 ( x t 1+x 2 ) e 2t. Obviously, A 3 A 4, because x λ integation. Futhemoe, by setting h(z) =( g (z)/g(z)) 1 {z:g(z)>} (z), wehave A 1 A 2 = 2λ /λ 2λ M(x, t)dtdx /λ /λ xm(x, t)dtdx h(λ 2 t) xt M(x, t)dtdx /λ h(λ 2 t) tm(x, t)dtdx. on the domain of Now, since h is inceasing with the logconcavity of g, the FKG s inequality (see Lemma 3 in the Appendix) implies that A 1 A 2 is nonnegative wheneve M(x 1,t 2 )M(x 2,t 1 ) M(x 1,t 1 )M(x 2,t 2 ), fo x 1 x 2 and t 1 t 2. Fom the definition of M, manipulations yield fo non-zeo values of M(x 1,t 2 )M(x 2,t 1 ) M(x 1,t 1 )M(x 2,t 2 ): t 1 t 2 e (1/t 1+1/t 2 ) (x 1 x 2 ) p/2+1 g(λ 2 t 1 )g(λ 2 t 2 ) {M(x 1,t 2 )M(x 2,t 1 ) M(x 1,t 1 )M(x 2,t 2 )} = I p 2 1 ( x 1 )I t p 2 1 ( x 2 ) I 2 t p 2 1 ( x 1 1 = I p 2 1 ( x 1 t 2 )I p 2 1 ( x 2 t 2 ) [ I p 2 1 ( x 2 t 1 ) I p 2 1 ( x 2 )I t p 2 1 ( x 2 ) exp{ (x 2 1 x 2 1 t 2)(1/t 1 1/t 2 )} 2 t 2 ) I ] p 2 1 ( x 1 t 1 ) I p 2 1 ( x 1 t 2 ) exp{ (x2 1 x 2 2)(1/t 1 1/t 2 )} I p 2 1 ( x 1 t 2 )I p 2 1 ( x 2 t 1 )( t 2 t 1 ) p/2 1 [ 1 exp{(x 2 2 x x 1 )(1/t 1 1/t 2 )} ], whee the fome inequality follows fom the Ross inequality applications (see Lemma 4 in Appendix): I p/2 1(x 2 /t 1 ) I p/2 1 (x 2 /t 2 ) (t 2/t 1 ) p/2 1 and I p/2 1(x 1 /t 1 ) I p/2 1 (x 1 /t 2 ) (t 2/t 1 ) p/2 1 exp{x 1 /t 1 x 1 /t 2 }, and whee the 11

12 latte inequality follows fom the fact that (x 2 2 x x 1 )(1/t 1 1/t 2 ), fo x 1 x 2 and t 1 t 2. Acknowledgments Éic Machand acknowledges the patial eseach suppot fom NSERC of Canada. We ae gateful to Dan Kuceovsky and Reinaldo Aellano-Valle fo useful suggestions on the Ross inequality and popeties of the H function in Theoem 1. Thanks to an anonymous eviewe fo constuctive comments. Appendix The FKG inequality due to Fotuin, Kasteleyn, & Ginibe (1971) is useful fo Theoem 4. Lemma 3. (FKG inequality) Suppose a p-vaiate andom vaiable X is distibuted with pobability density function ξ and with positive measue ν. Fo two points y =(y 1,,y p ) and z =(z 1,,z p ), in the sample space of X, we define y z =(y 1 z 1,,y p z p ) and y z =(y 1 z 1,,y p z p ), whee a b = min(a, b), a b = max(a, b). Suppose that ξ satisfies ξ(y)ξ(z) ξ(y z)ξ(y z) and that α(y), β(y) ae nondeceasing in each agument and α, β and αβ ae integable with espect to ξ. Then αβξdν αξdν βξdν. The following lemma, efeed to as the Ross inequality is due to Joshi & Bissu (1991) and establishes useful bounds fo a atio of modified Bessel functions. Lemma 4. Suppose I v (x) and I v (y) ae two modified Bessel functions. Moeove, suppose that y x and v 1 2. Then e x y ( x y )v I v(x) I v (y) (x y )v. 12

13 Refeences [1] Das Gupta, S. & Saka, S.K. (1984). On TP 2 and log-concavity. Inequalities in Statistics and Pobability, IMS Lectue Notes Monog. Se., 5, [2] Fotuin, C. M., Kasteleyn, P. W., & Ginibe, J. (1971). Coelation inequalities on some patially odeed sets. Comm. Math. Phys., 22, [3] Foudinie, D. & Machand, É. (21). On Bayes estimatos with unifom pios on sphees and thei compaative pefomance with maximum likelihood estimatos fo estimating bounded multivaiate nomal means. Jounal of Multivaiate Analysis, 11, [4] Hatigan, J. (24). Unifom pios on convex impove isk, Statistics & Pobability Lettes, 67, [5] Finne, H. & Rotes, M. (1997). Log-concavity and inequalities fo Chi-squae, F and Beta distibutions with applications in multiple compaisons, Statistica Sinica, 7, [6] Joshi, C. M. & Bissu, S. K. (1991). Some inequalities of Bessel and modified Bessel functions. Austalian Mathematical Society. Jounal. Seies A. Pue Mathematics and Statistics, 5, [7] Kubokawa, T. (1994). A unified appoach to impoving equivaiant estimatos. Annals of Statistics, 22, [8] Kubokawa, T. (25). Estimation of bounded location and scale paametes. Jounal of the Japan Statistical Society, 35, [9] Machand, E., Ouassou, I., Payandeh, A.T. & Peon, F. (28). On the estimation of a esticted location paamete fo symmetic distibutions, Jounal of the Japan Statistical Society, 38,

14 [1] Machand, É. & Peon, F. (21). Impoving on the MLE of a bounded nomal mean. The Annals of Statistics, 29, [11] Machand, É. & Peon, F. (25). Impoving on the MLE of a bounded mean fo spheical distibutions. Jounal of Multivaiate Analysis, 92, [12] Machand, É & Stawdeman, W. E. (24). Estimation in esticted paamete spaces: a eview. A Festschift fo Heman Rubin, IMS Lectue Notes Monog. Se., 45, [13] Machand, É & Stawdeman, W. E. (25). Impoving on the minimum isk equivaiant estimato fo a location paamete which is constained to an inteval o a half-inteval. Annals of the Institute of Statistical Mathematics, 57, [14] van Eeden, C. (26). Resticted paamete space estimation poblems. Admissibility and minimaxity popeties. Lectue Notes in Statistics, 188, Spinge, New Yok. [15] Watson, G. S. (1983). Statistics on Sphees. John Wiley & Sons, Inc., New Yok. 14

IMPROVING ON MINIMUM RISK EQUIVARIANT AND LINEAR MINIMAX ESTIMATORS OF BOUNDED MULTIVARIATE LOCATION PARAMETERS

REVSTAT Saisical Jounal Volume 8, Numbe 2, Novembe 21, 125 138 IMPROVING ON MINIMUM RISK EQUIVARIANT AND LINEAR MINIMAX ESTIMATORS OF BOUNDED MULTIVARIATE LOCATION PARAMETERS Auhos: Éic Machand Dépaemen