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

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1 REVSTAT Saisical Jounal Volume 8, Numbe 2, Novembe 21, IMPROVING ON MINIMUM RISK EQUIVARIANT AND LINEAR MINIMAX ESTIMATORS OF BOUNDED MULTIVARIATE LOCATION PARAMETERS Auhos: Éic Machand Dépaemen de Mahémaiques, Univesié de Shebooke, Shebooke, Canada Ami T. Payandeh Najafabadi Depamen of Mahemaical Sciences, Shahid Beheshi Univesiy, G.C. Evin, , Tehan, Ian Received: Febuay 21 Revised: June 21 Acceped: June 21 Absac: We popose impovemens unde squaed eo loss of he minimum isk equivaian and he linea minimax esimaos fo esimaing he locaion paamee θ of a p-vaiae spheically symmeic disibuion, wih θ esiced o a ball of adius m ceneed a he oigin. Ou consucion of explici impovemens elies on a mulivaiae vesion of Kubokawa s Inegal Expession of Risk Diffeence IERD mehod. Applicaions ae given fo univaiae disibuions, fo he mulivaiae nomal, and fo scale mixue of mulivaiae nomal disibuions. Key-Wods: decision heoy; spheical symmeic disibuion; esiced paamee; minimum isk equivaian esimao; linea minimax esimao; dominaing esimaos; squaed eo loss. AMS Subjec Classificaion: 62F1, 62F3, 62H12.

2 126 Éic Machand and Ami T. Payandeh Najafabadi

3 Impoving on MRE and Linea Minimax Esimaos of Bounded Paamees INTRODUCTION We conside he poblem of esimaing, unde squaed eo loss, he locaion paamee θ of a p-vaiae spheically symmeic disibuion unde he consain θ m, wih m > known. Wih seveal auhos having obained ineesing esuls elaive o his poblem, and moe geneally fo esiced paamee space poblems see Machand and Sawdeman, 24; van Eeden, 26 fo useful eviews, we focus on he deeminaion of benchmak esimaos such as he maximum likelihood esimao MLE, he minimum isk equivaian esimao MRE, and he linea minimax esimao LMX. In his egad, Machand and Peon 21 povide fo he mulivaiae nomal case impovemens on he always inadmissible MLE fo all m, p. These include Bayesian impovemens, bu condiions ae hen equied on m, p. Complemenay findings fo he mulivaiae nomal and paallel findings fo ohe spheically symmeic disibuions, including in paicula mulivaiae suden disibuions, wee obained especively by Foudinie and Machand 21 and Machand and Peon 25; bu again condiions fo he sudied pios π ypically bounday unifom, unifom on sphees, and fully unifom of he fom m c π p fo he Bayes esimao δ π o dominae he MLE ae necessiaed. Hence, he poblem of finding a Bayesian o an admissible impovemen fo any m, p, fo any given spheically symmeic disibuion emains unsolved even fo p 1 o he mulivaiae nomal disibuion. Alenaively, fo he objecive of passing he minimum es of impoving upon he minimum isk equivaian esimao, posiive findings fo he univaiae case p 1 wee obained by Machand and Sawdeman 25, as well as by Kubokawa 25. The fome esablish a geneal dominance esul fo he fully unifom pio Bayes esimao, which acually applies moe geneally fo a wide no necessaily symmeic class of locaion model densiies and locaion invaian losses. The lae povides on he ohe hand a lage class of pios which lead o Bayesian impovemens fo he univaiae vesion of ou poblem of symmeic densiies and squaed eo loss. A key feaue of hese dominance esuls is he use of Kubokawa s 1994 Inegal Expession of Risk Diffeence IERD echnique. Fo mulivaiae seings, a lovely esul by Haigan 24 ells us ha fo mulivaiae nomal disibuions, he fully unifom Bayes pocedue impoves upon he minimum isk equivaian esimao. The esul is acually moe geneal and applies fo convex esiced paamee spaces wih non-empy ineios. Howeve, Haigan s esul does equie nomaliy and hence a spheically symmeic analog emains an open quesion. Moeove, Haigan s esul does no apply o he benchmak linea minimax esimao, which epesens iself a simple impovemen on he minimum isk equivaian esimao.

4 128 Éic Machand and Ami T. Payandeh Najafabadi Wih he above backgound, ou moivaion hee esides in exending he univaiae dominance esuls o he mulivaiae case, exending Haigan s esul fo balls o spheically symmeic disibuions, and consideing impovemens upon he linea minimax pocedue as well. We povide peliminay esuls in his diecion in ems of sufficien condiions fo dominaing eihe he minimum isk equivaian esimao, he linea minimax esimao, o boh. Ou eamen possesses he ineesing feaue of being unified wih espec o he dimension p and he given spheically symmeic disibuion. Moeove, we aive a ou dominance esuls hough a novel mulivaiae vaian of Kubokawa s IERD echnique. The main dominance esuls ae pesened in Secion 2, and vaious examples o illusaions ae pusued in Secion 3. These include univaiae disibuions, he mulivaiae nomal disibuion, and scale mixue of mulivaiae nomal disibuions. 2. MAIN RESULTS Le X be a p-vaiae andom veco wih spheically symmeic densiy 2.1 f x θ 2, whee he locaion paamee θ is consained o a ball ceneed a he oigin and of adius m, say Θ m. We seek impovemens on he minimum isk equivaian MRE esimao δ X X, and he linea minimax esimao δ LMX X m 2 X unde squaed eo loss Lθ, d d θ 2, whee E m 2 +pσ 2 θ X θ 2 p σ 2 <. Heeafe, we denoe he noms of X, x, and θ by R,, and λ especively. Ou esuls bing ino play he ohogonally invaian in θ and nonnegaive quaniies H, λ E θθ T X X E θx T X X and H, λ λe θ X X,, λ. E θ X T X X We will make use of he inequaliy H, λ H, λ fo all, λ, which follows as a simple applicaion of he Cauchy Schwaz inequaliy. Now, we pesen he main dominance esuls of his pape. Theoem 2.1. Fo a model as in 2.1, δ g X g X X dominaes gx, wheneve: i g is absoluely coninuous, nonconsan, and noninceasing; ii and g sup λ [,m] H, λ fo all. Moeove, if condiions i and ii ae saisfied, and [ iii g m 2 pσ 2, 1, m 2 +pσ 2 hen δ g X g X X also dominaes δ X X.

5 Impoving on MRE and Linea Minimax Esimaos of Bounded Paamees 129 Remak 2.1. By viue of he inequaliy H, λ H, λ fo all, λ, condiion ii of Theoem 2.1 can be eplaced by he weake, bu neveheless useful, condiion ii and g sup λ [,m] H, λ fo all. Poof of he Theoem: I is saighfowad o veify ha gx dominaes X unde condiion iii, so ha condiions fo which δ g X dominaes gx, such as i and ii, ae necessaily condiions fo which δ g X g X X also dominaes δ X X. Now, using Kubokawa s IERD echnique, he isk diffeence beween he esimaos δ g X and gx can be wien as 1 2 θ E θ [R θ, g X X R θ, gx ] [E θ g X X θ 2 gx θ 2] X x R p gx θ 2 d g [ gx θ ] T xf x θ 2 d dx g [ gx T x θ T x ] f x θ 2 dx d. {x R p : x } Now, obseve ha condiions i and ii imply ha θ fo all θ Θ m, esablishing he esul. Hee ae some fuhe emaks and obsevaions in elaionship o Theoem 2.1. The noninceasing popey of condiion i is no necessaily esicive. Indeed, fo he mulivaiae nomal case, Machand and Peon 21, heoem 5 esablish ha he noninceasing popey holds fo all Bayesian esimaos associaed wih symmeic, logconcave pio densiies on [ m, m]. The condiions of Theoem 2.1 sugges he bounds ii and ii hemselves sup λ [,m] H, λ and sup λ [,m] H, λ as candidae g funcions. These funcions ae of he fom H, λ and H, λ, whee λ is some funcion aking values on [, m]. All such funcions lead o ange peseving esimaos δ g ; i.e., δ g x m fo all x R p ; since fo all and θ λ: H, λ H, λ λe 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 he above, obseve ha he pojecion of δ X ono Θ m, given by δ gp wih g p m 1, saisfies he condiions of Theoem2.1. We now focus on elaed implicaions fo he esimaos δ H XH X,mX and δ H X H X, mx, which will un ou in seveal cases o be he smalles possible g s saisfying especively condiions ii and ii of Theoem 2.1.

6 13 Éic Machand and Ami T. Payandeh Najafabadi E θθ T X Coollay 2.1. a If H, λ inceases in λ [, m] fo all, and deceases in [, ] fo all λ [, m], hen δ H X H X, mx dominaes boh he linea minimax esimao δ LMX X and he MRE esimao δ X; b If H, λ inceases in λ [, m] fo all, hen δ H X H X, mx dominaes he MRE esimao δ X. Poof: Pa a follows as a diec applicaion of Theoem 2.1 as H, m E θx T X m2 [ m 2 pσ 2, 1, fo θ m. Pa b follows fo wo easons. m 2 +pσ 2 m 2 +pσ 2 Fis, fo any posiive andom vaiable Y wih densiy g Y, and is biased vesion W wih densiy popoional o wg Y w, he aio EY 2 Y > EW W > is EY Y > inceasing in, which implies ha H, m is a deceasing funcion on [,. Secondly, fo θ m, H, m m E θ X E θ X 2 E θ X/m E θ X/m 2 < E θ X/m 2 E θ X/m 2 < 1, since E θ X > E θ X m. 3. EXAMPLES The following subsecions ae devoed o applicaions of Coollay 2.1, wih he key difficuly aising in checking he monooniciy condiions elaive o H and H. We focus on geneal univaiae cases subsecion 3.1, he mulivaiae nomal disibuion subsecion 3.2., and scale mixues of mulivaiae nomal disibuions subsecion Univaiae spheically symmeic disibuions We expess he symmeic univaiae densiies in 2.1 as 3.1 f θ x e qx θ, and esic ouselves o cases whee { q Q q: q is inceasing and convex on,, } and q is concave on,. Examples of such disibuions include nomal, Laplace, exponenial powe densiies wih qy α y β + c, α >, 1 β 2; Hypebolic Secan, Logisic, Genealized logisic densiies wih qy y + 2 α log1 + eαy + c, α > ; and Champenowne densiies wih qy logcoshy + β, β [, 2], also see Machand

7 Impoving on MRE and Linea Minimax Esimaos of Bounded Paamees 131 and Peon, 29; Machand e al., 28. The nex heoem esablishes fo such densiies he applicabiliy of pa a of Coollay 2.1 and dominance of δ H X ove boh he linea minimax esimao, δ LMX X, and he MRE esimao δ X. Theoem 3.1. Fo model densiies as in 3.1 wih q Q, he esimao δ H X H X, mx dominaes boh he linea minimax esimao δ LMX X and he MRE esimao δ X. Poof: By viue of Coollay 2.1, i suffices o show ha H, λ deceases in [, fo all λ [, m], and inceases in λ [, m] fo all. Fis, H, λ can be wien as x f x λ f x+λ dx H, λ λ x 2 f x λ + f x+λ dx qy λ 2 E λ anh +λ qy λ /2, λy whee Y is a andom vaiable wih densiy popoional o y 2 f y λ+f y+λ 1 [, y. Such a family of densiies wih paamee has inceasing monoone likelihood aio in Y. Fuhemoe, since q Q, a esul of Machand e al. 28 Lemma 1, pa e ells us ha he inne funcion of he above expecaion in Y is noninceasing. Hence, we conclude ha, fo all λ [, m], Hλ, deceases on [,. Tuning o he monooniciy of H,, begin by wiing H, λ λ λ λ x f x λ f x+λ dx x 2 f x λ + f x+λ dx y +λf y dy λ y λf y dy +λ y λ +λ2 f y dy + y +λ λ2 f y dy A, λ B, λ, whee A, λ and B, λ ae he numeao and denominao of he above facion, especively. Manipulaions yield: B 2 H, λ, λ A, λb, λ + λa, λb, λ λa, λb, λ 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 + λ l, λ λ G, λf λ + 2A 1, λb 1, λ + λ 2 f +λl, λ + 2 λf +λl, λ + λf +λb 1, λ + λ 2 f +λl, λ,

8 132 Éic Machand and Ami T. Payandeh Najafabadi whee G,λ 2λ +λ λ +λ l, λ yf ydy, λ A 1, λ λ B 1, λ λ yf ydy f ydy + λ y 2 f ydy + +λ λ +λ +λ yf ydy + f ydy, y 2 f ydy. λ y 2 f ydy + +λ y 2 f ydy, Now, obseve ha fo all, λ [, m], he quaniies B 1, λ, A 1, λ, and, λ ae nonnegaive. Hence, o show he posiiviy of H,λ λ, i will suffice o show he posiiviy of G, λ. Bu, we have G, λ +λ λ +λ λ yf y2λ + y dy λ y f ydy 1 [λ, + λ λ 2 y 2 f ydy 1 [,λ, y f y2λ + ydy 1 [,λ which complees he poof Mulivaiae nomal disibuions We conside hee mulivaiae nomal models in 2.1 X N p θ, σ 2 wih θ m. We ake σ 2 1 wihou loss of genealiy since X σ N pθ θ σ, I p wih θ m m σ. We equie he following key popeies elaive o ρλ, E θ T X θ X, X whee λ θ. These popeies involve modified Bessel funcions I v of ode v, and moe specifically aios of he fom ρ v I v+1 /I v, >. Lemma 3.1 Wason, 1983; Machand and Peon, 21. i We have ρλ, λρ p/2 1 λ ; ii ρ p/2 1 is inceasing and concave on [,, wih ρ p/2 1 and lim ρ p/2 1 1; iii ρ p/2 1 / is deceasing in wih lim + ρ p/2 1/ 1/p ; iv ρ p/2 ρ 1 p/2 1 p/.

9 Impoving on MRE and Linea Minimax Esimaos of Bounded Paamees 133 Denoing f p, λ and F p, λ as he pobabiliy densiy and suvival funcions of R X χ 2 pλ 2, we will also equie he following useful popeies. Lemma 3.2. i We have f p, λ p/2 1 λ Ip/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, λρ p/2 1 λ λf p+2, λ; iv he aio F p+2,λ F p,λ deceases in λ [,, fo all p 1 and >. Poof: Pas ii and iii follow diecly fom i, while i consiss of a well known Bessel funcion epesenaion of he noncenal chi-squae disibuion. Pa iv follows fom he ideniy 2 F λ p, λ F p+2, λ F p, λ, and he logconcaviy of F p, on [, see Das Gupa and Saka, 1984; Finne and Roes, We now seek o apply pa a of Coollay 2.1. Theoem 3.2. Fo mulivaiae nomal densiies, he esimao δ H X H X, mx dominaes boh he linea minimax esimao δ LMX X and he MRE esimao δ X. Poof: By viue of Coollay 2.1, i suffices o show ha H, λ deceases in [, fo all λ [, m], and inceases in λ [, m] fo all. Making use of Lemmas 3.1 and 3.2, we obain E θ X E θ T X θ X X H, λ E θ X 2 X 3.2 y E θ θ T X X X y f p y, λ dy y 2 f p y, λ dy y λρ p/2 1λyf p y, λ dy y 2 f p y, λ dy λ2 f p+2 y, λ dy y 2 f p y, λ dy p f p+4 y, λ dy λ 2 + f p+2 y, λ dy { p λ 2 + F } 1 p+4, λ. F p+2, λ 1

10 134 Éic Machand and Ami T. Payandeh Najafabadi The monooniciy popey of H, on [, m] fo all now follows fom he above expession and pa iv of Lemma 3.2. Now, o show ha H, λ deceases in, make use of 3.2 o wie H, λ λ E E Y θ T X X X Y Y λ E ρp/2 1 λy Y, whee Y has densiy popoional o y f p y, λ1 [, y. Since his family of densiies wih paamee has inceasing monoone likelihood aio in Y, we conclude indeed ha H, λ deceases fo fo all λ [, m] by making use of pa iii of Lemma Scale mixues of mulivaiae nomal disibuions We conside hee in his subsecion scale mixues of mulivaiae nomal disibuions whee X admis he epesenaion: X Z z N p θ, zi p, Z having Lebesgue densiy g on R +. The coesponding densiy in 2.1 is of he fom 3.3 } 2πz p/2 x θ 2 exp { gz dz ; 2 z and we fuhe assume ha g is logconcave on eihe R + o some open ineval a, b of R +. Unifom densiies on a, b ae included. Wih such a epesenaion, since X/ Z Z z N p θ/ z, Ip, we infe fom pa i of Lemma 3.2 ha he densiy funcion of R X is given by 3.4 y z y p/2 1 I λ p/2 1 λy exp z { y2 + λ 2 2 z } gz dz. We now seek o apply pa a of Coollay 2.1. Theoem 3.3. Fo scale mixues of mulivaiae nomal densiies as in 3.3 wih g logconcave, he esimao δ H X H X, mx dominaes he MRE esimao δ X. Poof: By viue of Coollay 2.1, i suffices o show ha H, is nondeceasing on [, m] fo all unde he given logconcave assumpion on g.

11 Impoving on MRE and Linea Minimax Esimaos of Bounded Paamees 135 Saing fom he definiion of H and making use of 3.4, we obain H, λ λ E θ R R E θ R 2 R λ /λ y p 2 +1 gz z y p 2 +2 gz z x p 2 +1 gλ 2 x p 2 +2 gλ 2 I p yλ 2 1 z e y2 +λ 2 2z I p yλ 2 1 z dz dy e y2 +λ 2 2 z dz dy I p x 2 1 e 1+x2 2 d dx I p 2 1 x e 1+x2 2 d dx, wih he change of vaiables y, z λx, λ 2. Simple diffeeniaion leads o λ H, λ 1 {A B 2 1 A 2 + A 3 A 4 }, whee B is he above denominao of H, A 1 2λ A 2 2λ A 3 λ 2 A 4 λ 2 /λ x Mx, d dx Mx, d dx x Mx, d dx Mx, d dx x p x 2 +1 g λ 2 I p 2 1 x p 2 +2 g λ 2 I p 2 1 x gλ 2 gλ 2 e 1+x2 2 d dx, e 1+x2 2 d dx, p 2 +1 I p e λ 2 1 λ λ λ 2 d, e λ2 + 2 p 2 +2 I p λ 2 1 λ 2λ 2 d, wih Mx, gλ2 x p 2 +1 I p x 2 1 e 1+x2 2. Obviously, A 3 A 4, because x λ on he domain of inegaion. Fuhemoe, by seing hz g z/gz 1 {z:gz>} z, we have A 1 A 2 2λ 2 λ Mx, d dx x Mx, d dx hλ 2 x Mx, d dx hλ 2 Mx, d dx. Now, since h is inceasing wih he logconcaviy of g, he FKG s inequaliy see Lemma A.1 in he Appendix implies ha A 1 A 2 is nonnegaive wheneve Mx 1, 2 Mx 2, 1 Mx 1, 1 Mx 2, 2, fo x 1 x 2 and 1 2. Fom he definiion of M, manipulaions yield fo non-zeo values of Mx 1, 2

12 136 Éic Machand and Ami T. Payandeh Najafabadi Mx 2, 1 Mx 1, 1 Mx 2, 2 : 1 2 e 1/ 1+1/ 2 { } x 1 x 2 p/2+1 gλ 2 1 gλ 2 Mx 1, 2 Mx 2, 1 Mx 1, 1 Mx 2, 2 2 x1 x2 x1 I p 2 1 I p 2 1 I p I p 2 1 x1 2 I p 2 1 x1 2, 1 I p 2 1 x2 I p 2 1 x2 1 [ I x2 p I x2 p x2 { } I p exp x 2 1 x 2 21/ 1 1/ 2 2 I x1 p 2 1 { } ] 1 2 I x1 exp x 2 p x 2 21/ 1 1/ 2 2 p/2 1 [ { } ] 1 exp x 2 2 x x 1 1/ 1 1/ 2 whee he fome inequaliy follows fom he Ross inequaliy applicaions I see Lemma A.2 in Appendix: p/2 1 x 2 / 1 I p/2 1 x 2 / 2 2/ 1 p/2 1 and I p/2 1x 1 / 1 I p/2 1 x 1 / 2 2 / 1 p/2 1 exp { } x 1 / 1 x 1 / 2, and whee he lae inequaliy follows fom he fac ha x 2 2 x2 1 + x 11/ 1 1/ 2, fo x 1 x 2 and 1 2. APPENDIX The FKG inequaliy due o Fouin, Kaseleyn, and Ginibe 1971 is useful fo Theoem 3.3. Lemma A.1 FKG inequaliy. Suppose a p-vaiae andom vaiable X is disibued wih pobabiliy densiy funcion ξ and wih posiive measue ν. Fo wo poins y y 1,..., y p and z z 1,..., z p, in he 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 mina, b, a b maxa, b. Suppose ha ξ saisfies ξyξz ξy zξy z and ha αy, βy ae nondeceasing in each agumen and α, β and α β ae inegable wih espec o ξ. Then αβξ dν αξ dν βξ dν. The following lemma, efeed o as he Ross inequaliy is due o Joshi and Bissu 1991 and esablishes useful bounds fo a aio of modified Bessel funcions. Lemma A.2. Suppose I v x and I v y ae wo modified Bessel funcions. Moeove, suppose ha y x and v 1 2. Then x v e x y I vx x v y I v y. y

13 Impoving on MRE and Linea Minimax Esimaos of Bounded Paamees 137 ACKNOWLEDGMENTS Éic Machand acknowledges he paial eseach suppo fom NSERC of Canada. We ae gaeful o Dan Kuceovsky and Reinaldo Aellano-Valle fo useful suggesions on he Ross inequaliy and popeies of he H funcion in Theoem 2.1. Thanks o an anonymous eviewe fo consucive commens. REFERENCES [1] Das Gupa, S. and Saka, S.K On TP 2 and log-concaviy, Inequaliies in Saisics and Pobabiliy, IMS Lecue Noes Monog. Se., 5, [2] Fouin, C.M.; Kaseleyn, P.W. and Ginibe, J Coelaion inequaliies on some paially odeed ses, Comm. Mah. Phys., 22, [3] Foudinie, D. and Machand, É. 21. On Bayes esimaos wih unifom pios on sphees and hei compaaive pefomance wih maximum likelihood esimaos fo esimaing bounded mulivaiae nomal means, Jounal of Mulivaiae Analysis, 11, [4] Haigan, J. 24. Unifom pios on convex impove isk, Saisics and Pobabiliy Lees, 67, [5] Finne, H. and Roes, M Log-concaviy and inequaliies fo Chisquae, F and Bea disibuions wih applicaions in muliple compaisons, Saisica Sinica, 7, [6] Joshi, C.M. and Bissu, S.K Some inequaliies of Bessel and modified Bessel funcions, Ausalian Mahemaical Sociey Jounal, Seies A, Pue Mahemaics and Saisics, 5, [7] Kubokawa, T A unified appoach o impoving equivaian esimaos, Annals of Saisics, 22, [8] Kubokawa, T. 25. Esimaion of bounded locaion and scale paamees, Jounal of he Japan Saisical Sociey, 35, [9] Machand, E.; Ouassou, I.; Payandeh, A.T. and Peon, F. 28. On he esimaion of a esiced locaion paamee fo symmeic disibuions, Jounal of he Japan Saisical Sociey, 38, [1] Machand, É. and Peon, F. 21. Impoving on he MLE of a bounded nomal mean, The Annals of Saisics, 29, [11] Machand, É. and Peon, F. 25. Impoving on he MLE of a bounded mean fo spheical disibuions, Jounal of Mulivaiae Analysis, 92, [12] Machand, É. and Sawdeman, W.E. 24. Esimaion in esiced paamee spaces: a eview, A Fesschif fo Heman Rubin, IMS Lecue Noes Monog. Se., 45,

14 138 Éic Machand and Ami T. Payandeh Najafabadi [13] Machand, É. and Sawdeman, W.E. 25. Impoving on he minimum isk equivaian esimao fo a locaion paamee which is consained o an ineval o a half-ineval, Annals of he Insiue of Saisical Mahemaics, 57, [14] van Eeden, C. 26. Resiced paamee space esimaion poblems. Admissibiliy and minimaxiy popeies, Lecue Noes in Saisics, 188, Spinge, New Yok. [15] Wason, G.S Saisics on Sphees, John Wiley and Sons, Inc., New Yok.

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

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