Linear Algebra and its Applications

Transcription

1 Liear Algebra ad its Applicatios 437 () Cotets lists available at SciVerse ScieceDirect Liear Algebra ad its Applicatios joural homepage: Simple regressio i view of elliptical models M. Arashi a,, S.M.M. Tabatabaey b, H. Soleimai b a Faculty of Mathematics, Shahrood Uiversity of Techology, P.O. Box 36, Shahrood, Ira b Departmet of Statistics, Faculty of Mathematical Scieces, Ferdowsi Uiversity of Mashhad, Mashhad, Ira ARTICLE INFO Article history: Received August Accepted 7 May Available olie 6 Jue SubmittedbyR.A.Brualdi AMS classificatio: Primary: 6H Secodary: 6F Keywords: Elliptically cotoured distributio Prelimiary test estimator Simple liear regressio Shrikage estimator ABSTRACT For the simple liear model Y = θ+βx+ε where the error vector follows the elliptically cotoured distributio, we cosider the urestricted, restricted, prelimiary test ad shrikage estimators for the itercept parameter, θ whe it is suspected that the slope parameter β may be β o. The exact bias ad MSE expressios are derived ad the mea-square relative efficiecy is take to determie the relative domiace properties of the proposed estimators i compariso. I the cotiuatio, the optimal level of sigificace of the prelimiary test estimator is tabulated ad some graphical result are also displayed. Elsevier Ic. All rights reserved.. Itroductio Cosider a simple liear model Y = θ + βx + ε = Aη + ε, A =[, x], η = (θ, β), (.) where Y = (Y,...,Y ) is the respose vector ad x = (x,...,x ) is a fixed vector of kow costats, while ε = (ε,...,ε ) is the -vector of radom errors distributed accordig to the law belogig to the class of elliptically cotoured distributios (ECDs), EC (,σ V,ψ)for σ R + ad u-structured kow matrix V S(), where S() deotes the set of all positive defiite matrices of order ( ) with the followig characteristic fuctio φ ε (t) = ψ ( σ t ) Vt (.) Correspodig author. address: m_arashi_stat@yahoo.com (M. Arashi) /$ - see frot matter Elsevier Ic. All rights reserved.

2 676 M. Arashi et al. / Liear Algebra ad its Applicatios 437 () for some fuctios ψ :[, ) R say characteristic geerator [9]. If ε has a desity, the it is of the form ( ) f (ε) σ V g σ ε V ε, (.3) where g(.) is a o-egative fuctio over R + such that f (.) is a desity fuctio w.r.t (with respect to) a σ -fiite measure μ o R p. I this case, otatio ε EC (,σ V, g) would probably be used. Some of the well-kow members of the class of ECDs are the multivariate ormal, Kotz Type, Pearso Type II ad VII, multivariate Studet s t, multivariate Cauchy, Logistic, Bessel ad geeralized slash distributios. Datig back to Kelker [4], there are may kow results cocerig ECDs, i particular the mathematical properties ad its applicatio to statistical iferece. These results have bee put forward by Muirhead [] ad Fag et al. [9] amog others. It is sometimes difficult to have complete aalysis of the regressio model with ECD errors of the type (.) or(.3). To overcome such difficulties, oe may cosider ay of the three sub-classes of ECDs, amely, (i) scale mixture of ormal distributios, (ii) Laplace class of mixture of ormal distributios, ad (iii) siged measure mixture of ormal distributios. Geeral formula for the above mixture of distributios is give by f ε (x) = W(t)φ N (,t σ V)(x)dt, (.4) where φ N (,t σ V)(.) is the pdf (probability desity fuctio) of N (, t σ V). (a) If W(τ) = (Ɣ(γ /)) ( γσ the we have f (ε) = ) γ/ τ (γ +) e γσ τ, <γ,σ,τ < (.5) ( ) Ɣ +γ V ( (πγ ) / Ɣ (γ/) σ + ε V ) ε (+γ), γσ (.6) where E(ε) = ad E(εε ) = γσ γ V = σ e (b) Chu [7] cosidered for γ>. W(t) = (π) σ V t p L [f (s)], (.7) L [f (s)] deotes the iverse Laplace trasform of f (s) with s =[x (σ V) x/]. For some examples of f (.) ad W(.) see Arashi ad Tabatabaey [5]. The iverse Laplace trasform of f (.) exists provided that the followig coditios are satisfied. (i) f (t) is differetiable whe t is sufficietly large. (ii) f (t) = o(t m ) as t, m >. Although, it is rather difficult to derive the iverse Laplace trasform of some fuctios, we are able to hadle it for may desity geerators of elliptical desities. We refer the readers to Debath ad Batta [8] for more specific details. The mea of ε is the zero-vector ad the covariace-matrix of ε is

3 M. Arashi et al. / Liear Algebra ad its Applicatios 437 () ε = Cov(ε) = Cov(ε t)w(t)dt { = W(t)Cov N p (, t σ V) } dt ( ) = t W(t)dt σ V (.8) provided the above itegral exists. Comparig the models (.3) ad (.4), sice ε = Cov(ε) = ψ ()σ V, it ca be cocluded that ψ () = t W(t)dt. Now suppose that X EC (μ, V, g). The it is importat to poit out that sice x f (x)dx =, usig Fubii s theorem we have = W(t)φ N (μ,t V)(x)dtdx = = x W(t) φ N (μ,t V)(x)dxdt x W(t)dt. Thusfor oegative fuctiow(.), it is a desity. For oegative fuctio W(.), the elliptical models ca be iterpreted as a scale mixture of ormal distributios. (c) Srivastava ad Bilodeau [7] cosidered the siged measure, W(t) such that (i) t W + (dt) <, (ii) t W (dt) <, (.9) where W + W is the Jorda decompositio of W i positive ad egative parts. Note that from (i) (ii) of (.9), t W(dt) < (.) ad thus, Cov(ε) exists uder the sub-class defied above. This subclass cotais the subclass defied by (b). Remark.. Regardig the above classificatios, we should take the followig otes:. I all the above classes we have ( ε = ψ ()σ V = ) t W(t)dt σ V resultig i ψ () = t W(t)dt.. The subclass (a) is either cotaied i subclass (b) or i the subclass (c). However, subclass (b) i cotaied i the subclass(c). Thus, all the implicatios about the subclass (c) ca be used for the subclass (b). 3. For the subclass (c) we ca assure that ψ () = t W(t)dt exists. However it may ot exist for the subclass (b).

4 678 M. Arashi et al. / Liear Algebra ad its Applicatios 437 () Throughout the paper, we assume that σ ε = ψ ()σ. (.) There have bee may studies i the area of the improved estimatio followig the semial work of Bacroft [6] ad later Ha ad Bacroft []. They developed the prelimiary test estimator that uses ucertai o-sample prior iformatio (ot i the form of prior distributios), i additio to the sample iformatio. Stei [9] elegat approach domiates the usual maximum likelihood estimators uder the squared error loss fuctio. I a series of papers Saleh ad Se [5,6] explored the prelimiary test approach to Stei-rule estimatio. May authors have cotributed to this area, otably Judge ad Bock [3], Stei [8], Kha [5 7], Kibria [8], Kibria ad Saleh [9,], Ahmed et al. [,], Saleh ad Kibria [3,4], Hassazadeh Bashtia et al. [,], Arashi et al. [4] ad Arashi [3]. The recet book of Saleh [] presets a comprehesive discussio of this area.. Estimatio ad testig For coveiece we express some otatios due to the rest of the work. Let K = V, K = x V x, K 3 = V x = x V, K = A V A. (.).. Estimator of η Based o the LS/ML priciple, the urestricted estimator of η = (θ, β) is give by η = ( A ) V ( A A ) V Y = K K 3 V Y K 3 K x = V Y θ. (.) β Theorem.. Assume i the simple liear model (.), Y θ,β,σ EC (η, σ V, f ); thewehave η EC (η, σ K, f ). Proof. Uder the assumptio Y θ,β,σ N (η, σ τ V, f ), the exact distributio of η follows N (η, σ τ K ), where K = (A V A) = K K 3 K 3 K = K K 3. K K K3 K 3 K Thus we get ( f Y (y) = W(τ)N η, σ τ K ) dτ, which completes the proof.

5 M. Arashi et al. / Liear Algebra ad its Applicatios 437 () Also the ubiased estimator of σε is S u give by S = u m (Y A η) V (Y A η) ; (m = ). (.3).. Test of itercept parameter At this step, first we propose test statistic of the parameter η, ad the we focus o the problem of estimatio of the itercept parameter i a more precise setup. Theorem.. Let ={(η, σ, V) : η R,σ R +, V > }, ad ω ={(η, σ, V) : η = η o = (θ o,β o ),η o R,σ R +, V > }. Moreover, suppose y f (y) has a fiite positive maximum yf. The the LR criterio for testig the hypothesis H o : η = η o is give by L [ ] = S ( η η u o) K( η η o ) ad it has the followig modified geeralized o-cetral F distributio g,m (L ) = r ( ) (+r) L (r) m K () r ( ) ( )( ), r! B r+, m + m L (+r+m) where = ξ/σ ε for ξ = (η η o) K(η η o ),ad ( K (h) r ( ) = ) r t r h e t W(t)dt. (.4) Proof. For the test of the ull hypothesis H o : η = η o vs H A : η = η o,let σ ε = (Y Aη o) V (Y Aη o ). The usig Theorem. we have = max ( ) ω L(y) max L(y) = Su f (yf ) σ ε f (y f ) [ ] (Y A η) V ( (Y A η) msu = (Y A η o ) = V (Y A η o ) msu + (η η o) K(η η o ) = ( + m L ). Hece, L is the LR test for testig the uderlyig ull hypothesis. For its o-ull distributio, we ote that uder ormality L follows the o-cetral F-distributio with (, m) d.f. ad o-cetrality parameter t = (η η o) K(η η o ). The itegratig over t w.r.t. the siged measure W, the proof is completed. t σ Accordigly, we have )

6 68 M. Arashi et al. / Liear Algebra ad its Applicatios 437 () Corollary... Uder H o, the pdf of L g,m (L ) = ( ) m ( )( B, m + m L is give by ) (m+), which is the cetral F-distributio with (, m) d.f. Corollary... The power fuctio at γ -level of sigificace of L, say, modified geeralized o-cetral F cumulative distributio fuctio of the statistic L is give by G p,m (l γ ; ) = [ r! K() r ( )I x (p + r), m ], r where I x (.,.) is the icomplete Beta fuctio, x = l γ m+lγ ad l γ = F,m (γ ). Straightforward cosequeces of Theorem., gai the test statistics for idividuals H o : θ = θ ad H o : β = β o. I order to test the ull hypothesis H o : β = β o, agaist a alterative H A : β = β o, oe uses the test statistic L,defiedby L = ( β β o ) ( ) K 4 K ; Su K 4 =. K K K3 The the exact distributio of L uder H o has the cetral F-distributio with (, m) d.f. Similarly, for the test of H o : θ = θ o agaist H A : θ = θ o oe uses the test-statistic L = ( θ θ o ) ( ) K 5 K ; Su K 5 =. K K K3 (.5) The exact distributio of L uder H o is cetral F-distributio with (, m) d.f. Note that based o the virtue of (.5), oe ca directly coclude the followig result. Lemma.. The LR criterio L for testig the hypothesis H o : θ = θ o has the followig distributio g,m (L ) = r where = ξ/σ ε for ξ = K 5(θ θ o ). ( ) (+r) L (r ) m K () r ( ) ( )( ), r! B r+, m + m L (+r+m) Now we tur our attetio to estimatio of the itercept parameter θ whe it is suspected that the slope parameter β may be β o. As a special case it covers the two-sample problem of estimatig oe mea whe it is suspected that the two meas may be equal. Also, oe-sample estimatio of mea is obtaied by lettig x = ad prior iformatio θ = θ o.3. Estimators of θ I additio to θ ad Su, we preset a few more estimators of θ ad σ ε. First of all ote that we have θ = K V Y K K 3 β = K Y K β, K = K V, K = K K 3. (.6)

7 M. Arashi et al. / Liear Algebra ad its Applicatios 437 () Replacig V by I i (.6), results θ = Ȳ x β as i Saleh [, p. 56]. If we suspect β to be β o, the the restricted estimator (RE) of θ is give by ˆθ = K Y K β o. (.7) Now followig Saleh [], we defie the estimators give below: Prelimiary test estimator (PTE) of θ is give by ˆθ PT = ˆθ I(L < F,m(α)) + θ I(L F,m(α)) = θ + ( β β o )K I(L < F,m(α)), (.8) where F,m (α) is the α-level upper critical value of a cetral F-distributio with (, m) d.f. ad I(A) is the idicator fuctio of the set A. Shrikage type estimator (SE) of θ is give by ˆθ S = θ + c( β β o )K, c > (.9) L 3. Properties of itercept parameter I this sectio, we derive the exact bias ad MSE expressios for the proposed estimators of the itercept parameter. Lemma 3. (Saleh, []). If Z N(, ), the E E( Z ) = e π [ ] Z = ( ), Z + ( ( ) ) where (.) is the cdf of the stadard ormal distributio. 3.. Bias expressios of the estimators The biases of θ ad ˆθ are obvious ad give by b ( θ ) =, b ( ˆθ ) = K (β β o). (3.) For the PTE, we have b 3 ( ˆθ PT ) = E [ θ + ( β β o )K I(L < F,m(α)) θ ] = K [ E ( β β o )I(L < F,m(α)) ] ( ) = E t E t σε Z ZI χ < m /m F,m(α) t K 4 [ = K K4 E t (β β ) ( )] χ 3 K 4 I χm /m ( ) = K K4 σ ε G () 3,m 3 F,m(α);, (3.)

8 68 M. Arashi et al. / Liear Algebra ad its Applicatios 437 () where t = t = t (β β o) K 4 σ ε G (h) (q, p,m ) = r= x = ad G (h) p,m(.;.) is give by Ɣ( p+m+r ) Ɣ( p+r )Ɣ(m/) K(h) r pq m + pq. [ p + r ( )I x, m ] ; Fially for the bias expressio of SE, takig Z = ( β β o ) K 4,wehave t σε [ ] b 4 ( ˆθ S ) = E θ + c( β β o )K L θ S u ] = K [c( E β β o ) ( β β o ) K 4 [ ]} = ck K S 4 E t {E Z u Z t. (3.3) (β β Sice Z t N( t, ), t = o ) K 4, t σ 3. the expressio i (3.3) simplifies to [ ] b 4 ( ˆθ S ) = ck K S 4 W(t)E Z u Z t dt = ck K 4 W(t)E msu t χ t σ m ad Z t is idepedet of S u t, usig Lemma [ ] msu t σ Z t σ m t E Z t dt = ck K Ɣ( m+ ) [ ] t 4 Ɣ( m ) W(t) σ Z m E Z t dt = ck K Ɣ( m+ σ 4 m ) Ɣ( m ) t W(t)( ( t ))dt. (3.4) 3.. MSE expressios of the estimators Usig Theorem. we get M ( θ ) = σ ε K (K K K 3 ). (3.5) For the restricted estimator, applyig Theorem. we have [ M ( ˆθ ) = E ( θ θ)+ K ( β β o ) ] = M ( θ ) + K E( β β o ) + K [ E ( θ θ)( β β o ) ] [ = σε K (K K K 3 ) + K K σ ] ε + (β β K K K3 o ) K K 3 σε K K K3

9 M. Arashi et al. / Liear Algebra ad its Applicatios 437 () = (K K K 3) + K 4 (K K K 3 ) (K K K 3 ) σ ε = ( K + K 4 ) σ ε. (3.6) For the MSE of PTE, usig equatio (3..9b) of Saleh [] wecaobtai M 3 ( ˆθ PT ) = E [ ( θ θ)+ K ( β β o )I(L < F,m(α)) ] = M ( θ ) + K [ E ( β β o ) I(L < F,m(α)) ] +K E[( θ θ)( β β o )I(L < F,m(α))] { [ ( ) ]} = M ( θ ) + K Z K4 Et E (t σε )Z I χ < m /m F,m(α) t { [ ( ) ]} K Z K4 Et E (t σε )Z I χ < m /m F,m(α) t { [ ( ) ]} +K Z K4 σ ε E t E t σε ZI χ < m /m F,m(α) t [ ( )] = σε K (K K K 3 ) + σε K K4 G () 3,m 3 F,m(α); { ( ) ( )} σε K K4 G () 3,m 3 F,m(α); + G () 5,m 5 F,m(α);. (3.7) Fially, for the shrikage estimator, usig Lemma 3. we have [ M 4 ( ˆθ S ) = E θ + c( β β o )K L θ [ ] [ = M ( θ ) + c K E ( β β o ) L + ck E ( θ θ)( β β o ) L = M ( θ ) + c K ( ) K4 E S u { [ ( )]} ck Z K 4 E t t σ E S u Z Z t Z ] = σ K (K K K 3 ) + c k k4 σ { ck K4 σ E t E[S u t]e t [t π e t + t { ( t ) } ] t { ( t )}} t, (3.8) where m+ Ɣ( E t E [S u t] = ) σ Ɣ( m ) W(t)t dt m E t t π e t + t { ( t ) } t { ( t )} = π t t e W(t)dt. (3.9) ]

10 684 M. Arashi et al. / Liear Algebra ad its Applicatios 437 () Compariso I this sectio we compare the proposed estimators with respect to their MSE fuctios. The measquare relative efficiecy (MRE) of ˆθ compared to θ may be writte as MRE( ˆθ ; θ ) = M ( θ ) M ( ˆθ ) = (K K K3 ) σε (K + K 4 )σ ε K K 4 K = (K 4 + K )(K K K3 ) K =. K 4 + K (4.) The efficiecy is a decreasig fuctio of.uderh o : β = β o it has the maximum MRE( ˆθ ; θ ) = K K 4. (4.) I geeral to compare ˆθ ad θ,usig(4.) MRE( ˆθ ; θ )>wheever <( K 3 K ). The relative efficiecy of ˆθ PT compared to θ is give by MRE( ˆθ PT ; θ ) =[ + g( )], (4.3) where g( ) = K { ( K G () 3,m K ( ( + G () 5,m ) 3 F,m(α); 5 F,m(α); Uder H o, it has the maximum value { MRE( ˆθ PT ; θ ) = I geeral, MRE( ˆθ PT K ( K G () 3,m K ; θ ) accordig as G () ( 3,m 3 F,m(α); ) G () ( ) 3,m 3 F,m(α); () G 5,m The relative efficiecy of ˆθ S compared to θ,isgiveby ) ( )) } G () 3,m 3 F,m(α);. (4.4) )}. 3 F,m(α); (4.5) ( 5 F,m(α); ). (4.6) MRE( ˆθ S ; θ ) =[ + h( )], (4.7) where { h( ) = M ( θ ) c k k4 σ ck m+ Ɣ( K4 σ ) σ Ɣ( m ) πm } t e t W(t)dt. (4.8)

11 M. Arashi et al. / Liear Algebra ad its Applicatios 437 () Fig.. Graph of bias fuctio for PTE. Fig.. Graph of bias fuctio for PTE. It is a decreasig fuctio with respect to.uderh o, it simplifies to { [ MRE( ˆθ S ; θ ) = + M ( θ ) c k k4 σ + 4ψ ()ck K4 wheever by Remark. < c m+ Ɣ( ) ]} (4.9) Ɣ( m ) πm 4Ɣ( m+ ) πmɣ( m )ψ (). (4.)

12 686 M. Arashi et al. / Liear Algebra ad its Applicatios 437 () Fig. 3. Graph of bias fuctio for SE. Fig. 4. Graph of risk fuctio for UE ad RE. 4.. Optimum level of sigificace of ˆθ PT Followig Sectio 3..4 of Saleh [], deote the relative efficiecy of ˆθ PT compared to θ by MRE(α, ). Its maximum value occurs at = asgivei(4.5), i.e. max MRE(α, ) = MRE(α, ). Subsequetly, i order to obtai prelimiary test estimator with a miimum guarateed efficiecy E say, we adopt the followig procedure: If, we always choose θ.however,i geeral, is ukow, so there is o way to choose a estimator that is uiformly best. For this reaso, we select a estimator with miimum guarateed efficiecy, such as E, ad look for a suitable α from the set A = { α MRE(α, ) E }. The estimator chose maximizes MRE(α, ) over all

13 M. Arashi et al. / Liear Algebra ad its Applicatios 437 () Fig. 5. Graph of risk fuctio for PTE. Fig. 6. Graph of risk fuctio for PTE. α A ad. Thus, we solve the followig equatio for the optimum α : mi MRE(α, ) = E(α, (α)) = E. (4.) The solutio α obtaied this way gives the PTE with miimum guarateed efficiecy E. 5. Numerical example I this sectio, we proceed by a umerical example based o the multivariate Studet s t (Mt) distributio, a well-kow member of ECDs. First of all assume that ε i the model (.), follows a Mt distributio with the scale matrix

14 688 M. Arashi et al. / Liear Algebra ad its Applicatios 437 () Fig. 7. Graph of risk fuctio for SE. Fig. 8. Graph of MRE (RE vs UE) V = ad ν degrees of freedom with the pdf as i (.6). The we have W(t) = ν(νt/)ν/ e νt/ Ɣ(ν/).

15 M. Arashi et al. / Liear Algebra ad its Applicatios 437 () Fig. 9. Graph of MRE (PTE vs UE). Fig.. Graph of MRE (PTE vs UE). The respective expressios for G p,m(q, (h) ), E (h) [ χp ( ) ] ad E (h) [ χp 4 ( ) ] ca be foud i Kha [5]. Further assume that x = (6834). Accordig to the result of Sectio 3, the graphs of PTE ad SE biases vs are displayed i Figs. 3. As it ca be realized, whe the both level of sigificace α ad degrees of freedom ν icrease the bias of PTE decreases. The bias of SE performs the same as ν icreases. Similar coclusios ca be made for the MSE graphs i Figs For the MRE graphs i Figs. 8, it ca be cocluded that the efficiecy of ˆθ relative to θ is a decreasig fuctio as discussed i Sectio 4. MRE( ˆθ PT; θ ) is a decreasig fuctio relative to ad also for small level of sigificace α, the UE performs better tha the PTE. This sceario has a little bit

16 69 M. Arashi et al. / Liear Algebra ad its Applicatios 437 () Fig.. Graph of MRE (SE vs UE). Table Maximum ad miimum guarateed efficiecies for = 6. α ξ e max E mi mi E max E mi mi E max E mi mi E max E mi mi E max E mi mi E max E mi mi E max E mi mi chage for the degrees of freedom ν; its behavior ca be verified from Fig.. Fially the shrikage estimator performs better tha the urestricted estimator as ν icreases. To coclude this sectio, Table 5 gives selected values of ξ = K the procedure of choosig the level α of sigificace. K K ad α =.5(.5).35 for

17 M. Arashi et al. / Liear Algebra ad its Applicatios 437 () Refereces [] S.E. Ahmed, K.A. Doksum, S. Hossai, J. You, Shrikage, pretest ad absolute pealty estimators i partially liear models, Aust. N. Z. J. Stat. 49 (7) [] S.E. Ahmed, A.A. Hussei, P.K. Se, Risk compariso of some shrikage M-estimators i liear models, J. Noparametr. Statist. 8 (6) [3] M. Arashi, Prelimiary test ad Stei estimators i simultaeous liear equatios, Liear Algebra Appl. 436 () 95. [4] M. Arashi, A.K.Md.E. Saleh, S.M.M. Tabatabaey, O mathematical characteristics of some improved estimators of the mea ad variace compoets i elliptically cotoured models, J. Ira. Statist. Soc. () [5] M. Arashi, S.M.M. Tabatabaey, A ote o classical Stei-type estimators i elliptically cotoured models, J. Statist. Pla. Iferece 4 () 6 3. [6] T.A. Bacroft, O biases i estimatio due to the use of the prelimiary tests of sigificace, A. Math. Statist. 5 (944) 9 4. [7] K.C. Chu, Estimatio ad decisio for liear systems with elliptically radom process, IEEE Tras. Automat. Cotrol 8 (973) [8] L. Debath, D. Bhatta, Itegral Trasforms ad their Applicatios, Chapma ad Hall, Lodo, New York, 7. [9] K.T. Fag, S. Kotz, K.W. Ng, Symmetric Multivariate ad Related Distributios, Chapma ad Hall, Lodo, New York, 99. [] C.P. Ha, T.A. Bacroft, O poolig meas whe variace is ukow, J. Amer. Statist. Assoc. 563 (968) [] M. Hassazadeh Bashtia, M. Arashi, S.M.M. Tabatabaey, Usig improved estimatio strategies to combat multicolliearity, J. Statist. Comput. Simulatio 8 () () [] M. Hassazadeh Bashtia, M. Arashi, S.M.M. Tabatabaey, Ridge estimatio uder the stochastic restrictio, Comm. Statist. Theory Methods 4 () [3] G.C. Judge, M.E. Bock, Statistical Implicatios of Pre-test ad Stei-rule Estimators i Ecoometrics, North Hollad, Amsterdam, 978. [4] D. Kelker, Distributio theory of spherical distributios ad locatio-scale parameter geeralizatio, Sakhya 3 (97) [5] S. Kha, A ote o a optimal tolerace regio for the class of multivariate elliptically cotoured locatio-scale model, J. Calcutta Statist. Assoc. Bull. 53 (5) 5 3. [6] S. Kha, Estimatio of the parameters of two parallel regressio lies uder ucertai prior iformatio, Biom. J. 45 (3) [7] S. Kha, Estimatio of parameters of the simple multivariate liear model with Studet-t error, J. Statist. Res. 39 () [8] B.M.G. Kibria, Performace of the shrikage prelimiary tests ridge regressio estimators based o the coflictig of W, LR ad LM tests, J. Statist. Comput. Simulatio 74 () (4) [9] B.M.G. Kibria, A.K.Md.E. Saleh, Optimum critical value for pretest estimators, Comm. Statist. Comput. Simulatio 35 () (6) [] B.M.G. Kibria, A.K.Md.E. Saleh, Prelimiary test ridge regressio estimators with Studet s t errors ad coflictig test-statistics, Metrika 59 () (4) 5 4. [] R.J. Muirhead, Aspect of Multivariate Statistical Theory, Joh Wiley, New York, 98. [] A.K.Md.E. Saleh, Theory of Prelimiary Test ad Stei-type Estimatio with Applicatios, Joh Wiley, New York, 6. [3] A.K.Md.E. Saleh, B.M.G. Kibria, O some ridge regressio estimators: a oparametric approach, J. Noparametr. Statist 3 (3) () [4] A.K.Md.E. Saleh, B.M.G. Kibria, Estimatio of the mea vector of a multivariate elliptically cotoured distributio, Calcutta Statist. Assoc. Bull. 6 () [5] A.K.Md.E. Saleh, P.K. Se, Noparametric estimatio of locatio parameters after a prelimiary test o regressio, A. Statist. 6 (978) [6] P.K. Se, A.K.Md.E. Saleh, O some shrikage estimators of multivariate locatio, A. Statist. 3 (985) 7 8. [7] M. Srivastava, M. Bilodeau, Stei estimatio uder elliptical distributio, J. Multivariate Aal. 8 (989) [8] C. Stei, Estimatio of the mea of a multivariate ormal distributio, A. Statist. 9 (98) [9] C. Stei, Iadmissibility of the usual estimator for the mea of a multivariate ormal distributio, i: Proceedigs of the Third Berkeley Symposium o Math. Statist. ad Prob., vol., Uiversity of Califoria Press, Berkeley, 956, pp