ISSN X Quadratic loss estimation of a location parameter when a subset of its components is unknown

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1 Afria Statistia Vol. 8, 013, pages DOI: Afria Statistia ISSN X Quadratic loss estimation of a location parameter when a subset of its components is unnown Idir Ouassou,1 and Mustapha Rachdi Université Cadi Ayyad, Ecole Nationale des Sciences Appliquées, Av. Abdelrim Khattabi, BP. 575, Marraech, Maroc Univ. Grenoble-Alpes, Laboratoire AGIM FRE 3405 CNRS, Equipe GEM, Université de P. Mendès-France de Grenoble, UFR SHS, BP. 47, Grenoble Cedex 09, France Received: October 01, 013; Accepted: November 15, 013 Copyright c 013, Journal Afria Statistia. All rights reserved Abstract. We consider the problem of estimating the quadratic loss δ θ of an estimator δ of the location parameter θ = θ 1,..., θ p ) when a subset of the components of θ are restricted to be nonnegative. First, we assume that the random observation X is a Gaussian vector and, secondly, we suppose that the random observation has the form X, U) and has a spherically symmetric distribution around a vector of the form θ, 0) with dim X = dim θ = p and dim U = dim 0 =. For these two settings, we consider two location estimators, the least square estimator and a shrinage estimators, and we compare theirs unbiased loss estimators with improved loss estimator. Résumé. On considère le problème de l estimation du coût quadratique δ θ d un estimateur δ de la moyenne θ = θ 1,..., θ p ) d une loi à symétrie sphérique lorsque l on sait que certaines composantes θ i de celle-ci sont positives ou nulles. En premier lieu, lorsque X est un vecteur gaussien, on s intéresse à l amélioration de l estimateur sans biais λ 0 de δ θ par des estimateurs de la forme λ 0 X) + hx) en fournissant des conditions sur la fonction h. On étend ensuite cette problématique à un modèle distributionnel où l on dispose d un vecteur résiduel U : la loi de X, U) est supposée à symétrie sphérique autour du couple θ, 0) et l on considère des estimateurs de coût de la forme λ 0 X) + U 4 hx). Key words: Spherical symmetry; Quadratic loss; Least square estimator; Unbiased loss estimator; James-Stein estimation; Minimaxity. AMS 010 Mathematics Subject Classification : Primary 6C0; 6C15; 6C10. Idir Ouassou : idir.ouassou@ensa.ac.ma Mustapha Rachdi: Mustapha.Rachdi@upmf-grenoble.fr, mustapha.rachdi@agim.eu 1 This wor was supported financially by the Académie des Sciences Hassan II from Morocco

2 estimation of a location parameter when a subset of its components is unnown Introduction In this paper we are interested by the estimation of the loss incurred when using the least square estimator and an improved estimator of the location parameter of a spherically symmetric distribution, when a subset of the components of this parameter are restricted to some constraints. The assumption on our model is that the random observation has the form X, U) and has a spherically symmetric distribution around a vector of the form θ, 0) with dim X = dim θ = p and dim U = dim 0 =. In that model, the p-dimensional part of the location, say θ, is unnown and the -dimensional part of the observation is the residual vector. First we construct, for unbiased estimator of the loss, a dominating shrinage-type estimator, in terms of squared-error loss. These results complement these of James-Steintype estimation of a location parameter. An important feature of our results is that the proposed loss estimators dominate the unbiased estimator for the entire class of spherically symmetric distributions under constraints. That is, the domination results are robust with respect to the spherical symmetry distribution under constraints. Notice that the problem of estimating the loss without any constraint on the scale parameter) was first considered by Lehmann and Scheffé 1993), who estimated the power of a statistical test. Recently, Johnstone 1988), Lele 1993) and Fourdrinierand Wells 1995) have discussed this problem in a variety of situations. In this paper, we remain in the contexte of spherically symmetric distribution where a residual vector U is available. The type of constraints we consider are that all or only a subset of the θ i are restricted to be nonnegative. Recall tat Fourdrinierand Wells 1995), Ouassou and Rachdi 011) have studied this problem but without any constraint on θ and they give an estimators class of the form λ 0 c U 4 / X which dominate the unbiased quadratic loss estimator λ 0 X) = p U / of the minimax estimator X. Suppose we wish to estimate θ, by a decision rule δx, U) using the sum of squared-error loss δx, U) θ. This loss is unobservable since it depends on θ; hence one may wish to estimate it by λx, U) from the data. To study how well λ estimates the loss, a further distance measure is needed; for mathematical convenience, we use squared error to evaluate λx). Thus the ris incurred by this latter λ is: Rλ, δ, θ) = λx, U) δx, U) θ ). We say that a loss estimator λ 1 dominates λ if Rλ 1, θ, δ) < Rλ, θ, δ). In section, we study the quadratic loss estimator of an estimator of the scale parameter θ = θ 1,..., θ p ) when some of its components θ i are nonnegative. First, when the random vector X is normally distributed with mean θ and of covariance matrix the identity I p. In this setting, the random vectors X and U are independent then the situation becomes the same as when just the vector X is available, so here, we do not consider the residual vector U. So in the first step, we study the quadratic loss estimation δ 0 X) θ of the maximum lielihood estimator δ 0 X) = δ 01 X),..., δ 0p X)) where, for all i 1 i p), { max0, Xi ) for i = 1,..., s δ 0i X) = for i = s + 1,..., p X i

3 estimation of a location parameter when a subset of its components is unnown. 563 We get the unbiased estimator λ 0 X) of its quadratic loss, which we then we improve by giving some domination conditions for the estimators of the form λ 0 X) + hx). Afterward, we consider a more general class of estimators of θ of the form δ g X) = δ 0 X) + gx). In this case, we give an unbiased estimator λ 0 gx) of the quadratic loss δ g X) θ of the estimator δ g X) and we establish the conditions for which λ 0 gx) is dominated by an estimator of the form λ h g X) = λ 0 gx) + hx). Finally, still for the spherically symmetric distribution context where a residual vector U is available, we estimate the quadratic loss δ g X, U) θ of any estimator of the form δ g X, U) = δ 0 X) + UU gx) of the scale parameter θ, by giving an unbiased estimator λ 0 gx) of this quadratic loss and conditions for which λ 0 gx) is dominated by a more general estimator of the form λ h g X) = λ 0 gx) + U 4 hx).. The Gaussian model framewor Assume that a p-dimensional random vector X = X 1,..., X p ) is observed which is normally distributed with mean vector θ = θ 1,..., θ p ) such that it has s nonnegative components and covariance identity matrix I p. Without loss of generality, we assume that θ 1 0,..., θ s 0 for s p..1. Quadratic loss estimation of the maximum lielihood estimator In this case, the maximum lielihood estimator is δ 0 X) = δ 01 X),..., δ 0p X)) where { 0 if Xi < 0 δ 0i X) = X i if X i 0 X i for i = 1,..., s for i = s + 1,..., p Il is convenientl to write it on the form: δ 0 X) = X+γX) where γx) = γ 1 X),..., γ p X)) with { Xi if X i < 0 for i = 1,..., s γ i X) = 0 if X i 0 0 for i = s + 1,..., p. An unbiased estimator of the quadratic loss δ 0 X) θ of the usual estimator δ 0 X) is given by: λ 0 X) = p + Xi )1 Xi<0. 1) where 1 A denotes the indicator function on the set A. In fact, the ris of δ 0 X) in θ may be written as follows: δ0 X) θ = X θ + γx) = X θ + γx) + X θ) t γx)

4 estimation of a location parameter when a subset of its components is unnown. 564 This decomposition is valid since E θ X θ = p < + which is the ris of X) and that: γx) = E Xi 1 Xi<0 X < + Then, the Schwartz inequality provides that X θ) t γx) exist. As the function γ is wealy differentiable, then the Lemma 5 allows to voice this expectation: X θ) t γx) = div γx) = E by some simple calculation of the divergence of γx). Finally the ris of δ 0 X) is finite and may be written such that: p + X i )1 Xi<0. 1 Xi<0 An alternative class of estimators which improve λ 0 X) are of the form: λ h X) = λ 0 X) hx). where h is some function defined from R p into R. The difference in ris between λ h X) and λ 0 X) is given by Lemma 1 below. In Lemma 1, as in all what follows, the wealy differentiability of the function h is assumed because it is a natural hypothesis as in Stein Lemma cf. Stein; 1981 and Johnstone; 1988). This assumption permits also to give some examples for which the differentiability is not valid. Lemma 1. For every wealy differentiable function h : R R p and for every θ R p, the difference in riss of λ h X) and λ 0 X) is equal to: R λ h, δ 0, θ ) R λ 0, δ 0, θ ) ) = h X) + 4shX) + hx) 4 hx) 1Xi>0 + Xi ) X i θ i )1 Xi<0 provided these expectation exist, where resp. div) denotes the Laplacian operator resp. divergency). Proof.For a fixed θ R p, we have that: R λ h, δ 0, θ ) R λ 0, δ 0, θ ) = λ 0 X) hx) δ 0 X) θ ) λ 0 X) δ 0 X) θ ) = h X) λ 0 X) δ 0 X) θ )hx). Since, λ 0 X) δ 0 X) θ s s s p = p s + I Xi 0 + Xi 1)1 Xi<0 δ 0i X) θ i ) δ 0i X) θ i ) = p s) + 1Xi>0 + Xi ) Xθ i )I Xi<0 X θ. i=s+1

5 estimation of a location parameter when a subset of its components is unnown. 565 then, λ 0 X) δ 0 X) θ )hx) 3) s = p s) hx) + 1Xi>0 + Xi ) X i θ i )1 Xi<0 hx) X θ hx). and, by using the Lemma 6 result on the function h, we get X θ hx) = p hx) + hx). 4) Combining the results above, 3) and 4), the difference in ris equals for every θ h X) + 4shX) + hx) 4 hx) 1Xi>0 + Xi ) X i θ i )1 Xi<0. As a first application of the above we have the following result. Theorem 1. Let h be a two-times wealy differentiable and positive function on R p. If the function h satisfy h + h + 4sh < 0 5) then, the estimator λ h X) dominates λ 0 X), for θ R s + R p s. Proof. The Condition 5) is a consequence of the that the second expectation of the equation ) is negative. Indeed, we have by hypothesis that, for all i = 1,..., s, θ i > 0 and therefore X i θ i 1 Xi<0 0 for i = 1,..., s. Then we have 1 Xi>0 + Xi X i θ i )1 Xi<0) 0 and so hx) 1 Xi>0 + Xi X i θ i )1 Xi<0) 0 since the function h is assumed positive, which is the desired result. Remar 1. The case which there is no positivity constraint on the components of the parameter θr p may be result in the fact that s = 0. Note that in this interpretation, the inequality 5) coincides with the condition given by Johnstone 1988).

6 estimation of a location parameter when a subset of its components is unnown Quadratic loss estimation of a general estimator We consider the estimaton of the loss of a class of shrinage estimators δ, estimators incurred when using the maximum lielihood estimator δ 0 X) as δx) = δ g X) = δ 0 X) + gx) where g is wealy differentiable function from R p to R p. As before we first develop an unbiased estimator of the loss function δ g X) θ of the estimator δ g X). Indeed, by application of Lemma 5 to the function γ + g, we can write δg X) θ = X + γx) + gx) θ = X θ + γx) + gx) = X θ + X θ)γx) + gx)) + γx) + gx) = p + γx) + gx) + div γ + g)x), therefore an unbiased estimator of δ g X) θ has the following form λ 0 gx) = p + γ + g + div γ + g). In the signal of this section we focus to prove the domination of the unbiased estimator λ 0 of the quadratic loss by computing estimators λ h g of the form λ h g X) = λ 0 gx) hx) for some real function unnown h. Then in Lemma we develop the difference in ris between λ h g X) and λ 0 gx). Lemma. For every twice-wealy differentiable function h : R R p and a wealy differentiable function g : R p R p. For every θ R p the difference in ris between λ h g X) and λ 0 gx) is given by: R λ h g, δ g, θ ) R λ 0 g, δ g, θ ) = h X) + hx) + 4 hx), γx) + gx). Proof. Let θ R p. The difference in ris between λ h g X) and λ 0 gx) is equal to R λ h g, δ g, θ ) R λ 0 g, δ g, θ ) = λg X) δ g X) θ ) λ 0 gx) δ g X) θ ) = λ 0 g X) hx) δ g X) θ ) λ 0 gx) δ g X) θ ) = h X) λ 0 g X) δ g X) θ )hx). It is clear, by using Lemma 6 to the function h, that the second expectation in this last expression can be expressed as follows: λ 0 g X) δ g X) θ )hx) = p + γx) + gx) + div γ + g)x) X + γx) + gx) θ ) hx) = p + div γ + g)x) X θ) t γ + g)x) X θ ) hx) = hx)divgx) divg h)x) 1 hx) + divγ X θ) t γx))hx).

7 estimation of a location parameter when a subset of its components is unnown. 567 Otherwise as divγx) X θ) t γx) = I Xi<0 + = s + X i θ i )X i 1 Xi<0 1 Xi 0 + Xi X i θ i )1 Xi<0. Then we are now able to give an expression for the difference in ris h X) + hx) + 4divg h)x) 4hX) divgx) + 4shX) 4hX) 1 Xi>0 + Xi X i θ i )1 Xi<0 which achieves the proof of this Lemma. = h X) + hx) + 4 hx), γx) + gx), The following theorem gives a condition of domination of the estimator λ 0 gx) by λ h g X). Theorem. Let h be a positive function twice-wealy differentiable on R p and p 4. A sufficient condition under which the estimator λ h g X) dominates the unbiased estimator λ 0 gx), for all θ R s + R p s, is that h satisfies de differential inequality h + h + 4sh + 4 h, g 0. Proof. Demonstration follows the same lines as that of Theorem The spherically unimodal framewor We consider the model introduced in section1 where X, U) is a p + random vector having a symmetrically distribution around de p + vector θ, 0) Here dim X = dim θ = p and dim U = dim 0 =. For the constraints on θ, we taes those in paragraph where some components of the mean θ is assumed positive i.e. θ i 0 for i = 1,..., s with s p). We will study the estimation of quadratic loss δ g X, U) θ for any estimator of the form δ g X, U) = X + γx) + U gx) of the mean θ, where g is a function twice-wealy differentiable on R p in R p. General conditions for such an estimator δ g X, U) dominates the estimator X + γx) are given in Fourdrinier et al. 003), Brandwein and Strawderman 1991), Brandwein et al. 1991) and Cellier et al. 1989). The next lemma give an unbiased estimator of the loss δ g X, U) θ of the estimator δ g X, U). Lemma 3. For every wealy differentiable function g : R p R p and for all θ R p, an unbiased estimator of the loss function δ g X, U) θ of the estimator δ g X, U) is equal to λ 0 gx, U) = p U + U divγx) + + U 4 divgx) + γx) + U gx). 6)

8 estimation of a location parameter when a subset of its components is unnown. 568 Proof. let θ R p fixed. They can write δg X, U) θ = X + γx) + U gx) θ = X θ + γx)) + U gx) = X θ + γx) + X θ + U γx)).gx) + U 4 gx). Frome the Lemma 7, for α = 0 and α = 1 respectively, we have and It comes then X θ) t γx) = U U X θ) t gx) = U 4 + div γx) div gx). δg X, U) θ = p U + U div γx) + γx) + U γx) gx) + U X θ) t gx) + U 4 gx) = p U + U + + U 4 div gx) = p U div γx) + γx) + U γx)gx) + U gx) + U div γx) + + U 4 div gx) + γx) + U gx) then the result. Remar. Where there is no positivity constraint on the components of θ can be interpreted by taing γ 0. We thus find the unbiased estimator given by Cellier and Fourdrinier 1995). In the following of this section, we show that the estimator λ 0 gx, U) can be improved through a more general class of estimators of the form λ h g X, U) = λ 0 gx, U) + U 4 h X ) 7) where h ) is a negative real-valued function and twice wealy differentiable. Order to compare two estimators λ h G X, U) and λ0 gx, U) we needs to calculate the difference in their ris R = R λ h g X, U), δ g X, U), θ ) R λ 0 gx, U), δ g X, U), θ ). It is the object of the following lemma.

9 estimation of a location parameter when a subset of its components is unnown. 569 Lemma 4. For every wealy differentiable function g : R p R p, for every twice wealy differentiable function h : R + R and for all θ R p, an unbiased estimator of the difference in ris between λ h g X, U) and λ 0 gx, U) is equal to + 4) + 6) h X ) U h 8 X ) h X )div gx) div h X )gx) ) ) U 6 Proof. See Appendix. p h X ) h X ), γx) + 4 h X )div γx) ). 8) As a first application of the previous result, we have the following theorem where we assume that the distribution of X, U) is unimodal is to say that it have a density of the form x, u) G x θ + u ) where G ) is a nonincreasing function. Theorem 3. We assume that the distribution of X, U) is unimodal. If the function h satisfies the following conditions 1. is a negative function, twice wealy differentiable and concave. h. X i X ) h X ) X X i for all i = 1,..., p then the estimator λ h g X, U) dominates the unbiased estimator λ 0 gx, U), for θ R s R p s, since provided that h X ) s + 4) ) + 4) + 6) h X ) 4 4 div g h)x) h X )div gx) provided these expectation exist. h X ) X 0 9) Proof. Using Lemma 4, we get that the difference in ris U h 8 X ) + 4) + 6) h X ) 4 ) 4 div g h)x) h X )div gx) + 4 p + 4 E θ U 6 h X ) + 4 E θ U 6 h X )div γx) U 6 h X ), γx) ). 10) Using the results of the paper Fourdrinier et al. 003) demonstration of theorem 3.1), we show that the last expectation of 10) is bounded by s U 8 h X ) X. 11)

10 estimation of a location parameter when a subset of its components is unnown. 570 Indeed, U 6 h X ), γx) = U 6 h X ) X i I Xi<0 X i U 6 h X ) X Xi I Xi<0 s U 8 h X ) X because the function h satisfies the same conditions as those of function r of theorem 3.1 du Fourdrinier et al. 003). Then U 6 h X ), γx) s + 4) ) U 8 h X ) X. 1) To conclude we need Lemma 9 to show that, the sum of the second and the third expectation of the expression 10) is negative. Using Lemma 9, we get that, U 6 h X )div γx) s E θ h X ) U 6. 13) In effect U 6 h X )div γx) = U 6 h X ) = I Xi<0 U 6 h X )I Xi<0 s E θ h X ) U 6 the last inequality being acquired by using the fact that the function h ) is nonnegative and the Lemma 9. So, from equation 13), we have 8p + 4) U 6 h X 16 ) + + 4) U 6 h X ) γx) 14) 8p U 6 h X ) 8s + 4) + 4) U 6 h X ) U 6 = 8p s) + 4) h X ) 0 since the function h ) is negative and p s 0. Finally we acquire as upper bound of the difference in ris U 8 h X ) what gives the desired result. + 4) + 6) h X ) 4 g h)x) ) h X )div gx) s + 4) ) h X ) ) X,

11 estimation of a location parameter when a subset of its components is unnown. 571 Remar 3. Where there is no positivity constraint on the components of θ can be interpreted by taing s = 0 and γ 0. We thus find the condition of domination of Theorem 3.1 in Fourdrinier et al. 003). The following corollary gives an example of a function h and function g which satisfies the conditions of Theorem 3. Corollary 1. Assume that p > 4, p > s+8, > + 4s p 8 s and that the distribution is unimodal. For the functions gx) = d X X and h X ) = c X a sufficient condition under which the estimator λ h g X, U) dominates the unbiased estimators λ 0 gx, U), for all θ R s R p s, is that 0 < c < where the constante d satisfy ) 4p 4) p 4) p ) s d ) ). 15) 0 d 1 ) + )p s+8 ) + 4s p s 8 + 4) )p + ) Proof. Let us consider the usual shrinage estimator g of θ with shrinage factor g defined by gx) = d X X, where d is a positive constant, and the shrinage loss estimator used in X the previous corollary with the shrinage function h defined by ht) = c X where c is a positive constante, for every X R p, it easy to chec that and h X ) = div gx) = p p h X ) = c p 4 X i X 4, g i X) = d p X i X. div h g)x) = h X )div gx) + h X ), gx) = c X dp ) X + = cd p 4 X 4. The member of left of the condition of domination 9) becomes c X 4 = c X 4 + 4) + 6) c cp 4) X 4 4p 4) + 4) + 6) p h X i X)g i X) 4 cdp 4) + 6) X cdp ) + X 4 + 4p 4) 4p ) d d + s + 4) ) 16) sc 1 + 4) ) X 4.

12 estimation of a location parameter when a subset of its components is unnown. 57 Hence the sufficient condition of domination is 4p 4) 4p 4) c + 4) + 6) + 6 4p ) + what is identical to p 4) p 4) 0 < c < 4 + 4) + 6) + 4d + 6 ) d + ) p ) + s + 4) ) < 0 s + 4) ). A simple calcultion shows that the right-and side is positive if and only 0 d 1 +)p s+8 4s ) + p s 8) +4) )p+ ) and > + 4s p 8 s Remar 4. The optimal of c is given by c p 4) p 4) = + 4) + 6) + d + 6. This completes the proof. ) p ) + s + 4) ). It is clear that the optimal shrinage factor of the loss estimator depends on the shrinage factor of te point estimate. We found that wih the optimal d, that is, d = p + s, the optimal c is still positive for reasonable values of p and. Comment It is worthwhile to note that the stated condition for domination are too restrictive is certain case s 4). In particular, what is needed is the finiteness of the ris, which is governed by the final term of the expression. Thus, ) if s = 0, it suffices that p 5, 1 and p 4) 0 < c < 4 +4)+6) + 4d p 4) +6 p ) + for same 0 d 1 p 4)+) p+ )+4) which is same bound as in the unrestricted case. If s = 1, or s =, it suffices that p 5 and if s = 4, that p 7. The condition on for s 1 are as in the corollary. In particular, if s = 1 and p = 5, we require 10 Similarly, if s = and p = 6, we also require 10 in general for p = s + 4 we require 10, but, for s = 3 and p = 6, we need 6. For s = 4 and p = 7, we need that 8. For p = s we need to p 9. In general, for fixed s, the value of which is required decreases to 3 as p increases to 5s Appendix The first two lemma are given respectively by Stein 1981) and by Stein 1981) and Johnstone 1988). Lemma 5. For every wealy differentiable function g : R p R p, for every θ R p, we have provided these expectations exist. X θ).gx) = div gx) 17) Lemma 6. For every wealy differentiable function g : R p R,for every θ R p, we have provided these expectations exist. X θ gx) = gx) + pgx)

13 estimation of a location parameter when a subset of its components is unnown. 573 Proof of Lemma 6 We using 17) to the function X θ)gx). The straightforward proof of the two next lemma is given respectively by Fourdrinier and Strawderman 1996) and by Fourdrinierand Wells 1995). Lemma 7. For every wealy differentiable function g : R p R p, for every θ R p and for every integer α, U α X θ).gx) 1 = + α E θ U α+ div gx) Lemma 8. For every twice wealy differentiable function g : R p R +, for every θ R p and for every integer q, we have E R,θ U q X θ gx) = p + q E R,θ U q+ gx) q) + q + ) E R,θ U q+4 gx). Lemma 9. Assume X is a real-valued random variable with symmetric unimodal distribution about θ R +. If f is a nonnegative function on R +, then fx 1 )I X<0 E θ fx ). 18) Proof of Lemma 9 Note that, by the symmetrical unimodal assumption, X has density of the forme x, u) g x θ) ) with g nonincreasing function. The lemma reduces showing that fx ) I X<0 1 ) 0. 19) The left-hand side of 19) can be written as 1 E θ fx ) ) I X<0 I X 0. 0) Conditioning on X, and noting by E the expectation of the distribution of X, the expression 0) is 1 E fx ) g X θ) ) g X θ)) ). So, by hypothesis we have θ > 0 then X θ) X θ). By unimodatily of X the function g ) is decreasing), it is clear that g X θ) ) g X θ)) ). and hence that an upper bound of 19) is given by 1 E fx ) g X θ) ) g X θ)) ) 0 what gives the desired result.

14 estimation of a location parameter when a subset of its components is unnown. 574 Proof of Lemma 4 For θ R p fixed. We can compute using 6) and 7) Rθ) = λ h g X, U) δ g X, U) θ ) λ 0 g X, U) δ g X, U) θ ) λ 0 = g X, U) + U 4 h X ) δ g X, U) θ ) λ 0 g X, U) δ g X, U) θ ) = U 8 h X ) + p U 6 h X ) U 4 X θ h X ) U 6 +4 divγx) + ) + U divgx) h X ) 4 U 4 X θ) γx) + U gx) ) h X ). Using Lemma 8, for q = 4 and gx) = h X ), we obtain U 4 X θ h X ) = p + 4 E θ U 6 h X ) ) + 6) E θ U 8 h X ) and using Lemma 7, for the function γ + U g)h, we get U 4 X θ)γx)h X ) = E θ U 6 div γx)hx)) and U 6 X θ)gx)h X ) = E θ U 8 div gx)hx)) Then the difference in ris be comme U 8 h X ) + p U 6 h X ) + 4 U 6 h X ) γx) + ) + U gx) + 4) + 6) U 8 h X ) U 6 div γx)hx)) p + 4 U 6 h X ) U 8 div gx)hx)) = U {h 8 X ) + 4) + 6) h X ) + 4 } 4 hx)div gx)) div hx)gx)) { p + U 6 h X ) h X ), γx) + 4 } h X )div γx). So the unbiased estimator desired is U h 8 X ) + 4) + 6) h X ) + 4 ) 4 hx) gx)) hx)gx)) +4 U 6 p h X ) divhx)γx)) + 1 ) hx)divγx)

15 estimation of a location parameter when a subset of its components is unnown. 575 then the desired result. References Brandwein, A. C. and Strawderman, W.E., Generalizations of James-Stein estimators under spherical symmetry. The Annals of Statistics. 193): Brandwein, A. C., Ralescu, S. and Strawderman, W.E., Shrinage estimators of the location parameter for certain spherically symmetric distributions. Ann. Inst. Statist. Math. 453), Cellier, D. and Fourdrinier, D., Shrinage estimators under spherical symmetry for the general linear model. Journal of Multivariate Analysis. 5): Cellier, D., Fourdrinier, D. and Robert, C., Robust shrinage estimators of the location parameter for elliptically symmetric distributions. Journal of Multivariate Analysis. 9: Fourdrinier, D. and Wells, M.T., Estimation of a loss function for spherically symmetric distribution in the general linear model. The Annals of Statistics. 3): Fourdrinier, D. and Strawderman, W.E., A paradox concerning shrinage estimators : should a nown scale parameter be replaced by an estimated value in the shrinage factor? Journal of Multivariate Analysis. 59), Fourdrinier, D., Ouassou, I. and Strawderman, W., 003. Estimation of a components vector when some parameters are restricted. Journal of Multivariate Analysis. 86, 14-4, Johnstone, I., On inadmissibility of some unbiased estimates of loss. In Statistical Decision Theory and Related Topics IV S. S. Gupta and J.O.Berger, Eds.), Academic Press, New Yor. 1, Lehmann, E. L. and Scheffé, H., Completeness, similar regions and unbiased estimates. Sanhyā. 10: , Lele, C., Admissibility results in loss estimation. The Annals of Statistics. 1, Ouassou, I. and Rachdi, M., 009. Stein type estimation of the regression operator for functional data. Advances and Applications in Statistical Sciences. 1, no, Stein, C., Estimation of the mean of a multivariate normal distribution. The Annals of Statistics. 9, Ouassou, I. and Rachdi, M Estimation du coût quadratique quand certaines composantes du paramètre de position sont positives C. R. Acad. Sci. Paris, Sér. 1, Volume 349, Issue 17-18, Pages