Bayes minimax estimation under power priors of location parameters for a wide class of spherically symmetric distributions

: We complement the results of Fourdrinier, Mezoued and Straw- derman in [5] who considered Bayesian estimation of the location parameter θ of a random vector X having a unimodal spherically symmetric density f ( k x − θ k 2 ) for a spherically symmetric prior density π ( k θ k 2 ). In [5], expressing the Bayes estimator as δ π ( X ) = X + ∇ M ( k X k 2 ) /m ( k X k 2 ), where m is the marginal associated to f ( k x − θ k 2 ) and M is the marginal with respect to F ( k x − θ k 2 ) = 1 / 2 R ∞k x − θ k 2 f ( t ) dt , it was shown that, under quadratic loss, if the sampling density f ( k x − θ k 2 ) belongs to the Berger class (i.e.


Introduction
We recall the framework considered by Fourdrinier, Mezoued and Strawderman in [5]. Let X be a random vector in R p with spherically symmetric density around an unknown location parameter θ that we wish to estimate. Any estimator δ is evaluated under the square error loss through the corresponding quadratic risk R(θ, δ(X)) = E θ [ δ(X) − θ 2 ], where E θ denotes the expectation with respect to the density in (1.1). As soon as E 0 [ X 2 ] < ∞, the standard estimator X is minimax, and has constant risk (actually, equal to E 0 [ X 2 ]), which entails that minimaxity of δ will be obtained by proving that the risk of δ is less than or equal to the risk of X, that is, if for any θ ∈ R p (domination of δ over X being obtained if, furthermore, this inequality is strict for some θ). Brandwein and Strawderman [4] gave general conditions on estimators of the form δ a,g (X) = X + a g (X) (1.3) to dominate the standard estimator X, when p ≥ 3. Here g = (g 1 , g 2 , . . . , g p ) is a function from R p into R p and a is a positive constant. Besides the finiteness risk condition for δ a,g (X), these conditions require that (j) 0 < a ≤ 1/[p E 0 ( X −2 )] and involve the existence of a subharmonic function h such that (jj) g 2 /2 ≤ −h ≤ −divg, where div is the divergence operator defined, for x = (x 1 , . . . , x p ) ∈ R p , by divg(x) = p i=1 ∂g i (x)/∂x i , and, denoting by V θ,R the uniform distribution on the ball B θ,R = {x ∈ R p / x − θ ≤ R} of radius R and centred at θ, such that the function (jjj) R → R 2 E V θ,R [h(X)] is nonincreasing in R. Note that (jjj), as (jj), is just a condition on the function h, while (j) is a distributional condition on f in (1.1).
In that context, it should be noticed that, although Brandwein and Strawderman [4] gave examples for which (jjj) holds, this condition may be difficult to prove, in particular, in the situation where Bayesian estimators are concerned. In that situation, a more flexible condition is needed and it would be easier, for instance, to deal with a monotonicity condition such that, for some q ≥ 1, In this paper, we will show that a modification of the Brandwein and Strawderman approach in [4], taking into account the above new monotonicity condition, can be used to obtain domination over X, and hence minimaxity, of generalized Bayes estimators for density prior of the form θ −2k , (1.5) where k is a positive constant. As stated in [8] and [5], for spherical prior densities, the generalized Bayes estimator associated to (1.5) is the posterior mean and can be written under the form δ k (X) = X + ∇M ( X 2 ) m( X 2 ) (1. 6) where ∇ denotes the gradient operator and where and are respectively the marginal densities with respect to the density in (1.1) and the density proportional to F ( x − θ 2 ) with for t ≥ 0. Note that, in (1.6), (1.7) and (1.8) the marginals depend on the observable X through its squared norm since the prior in (1.5) is spherically symmetric around 0. It is easily seen that the finiteness risk condition of X is It is worth noting that this condition is sufficient to guarantee the finiteness of the risk of Bayesian estimator δ k (X) in (1.6) (see [5]).
In the literature, minimaxity of Bayesian estimators is mainly addressed when the sampling in (1.1) is in the Berger class, that is, when there exists a positive constant c such that F (t)/f (t) ≥ c for any t. Thus, for that class, Fourdrinier, Mezoued and Strawderman [5] provide two wide sets of sampling densities (according to the nondecreasing/nonincreasing monotonicity of the ratio F (t)/f (t)) and a wide class of prior densities of the form π( θ 2 ) for which the corresponding Bayes estimators δ π are minimax (it is worth noting that, in [8], the fundamental harmonic prior θ 2−p is shown to be a member of that class when F (t)/f (t) is nondecreasing in t). When one is not restricted to the Berger class, minimaxity of δ π is much more complicated to obtain (and the techniques used in [8] and [5] fail). That difficulty prompted us to consider a different approach based on a modification of the minimaxity result from Brandwein and Strawderman [4] as mentioned above. Also, dealing only with the case where F (t)/f (t) is nonincreasing in t, we only prove minimaxity when the prior densities are power priors of the form (1.5).
As a last remark, note that, if we consider f (t) to be proportional to a density of a positive random variable, then 2 F (t)/f (t) is the reciprocal of the hazard rate. Its monotonicity may be determined in many cases by studying the logconvexity or the log-concavity of f (t) (see e.g. Barlow and Proschan [2]).
In Section 2, we propose a new version of Theorem 2.1 of Brandwein and Strawderman [4]. In Section 3, we give general results on Bayes estimators δ k in (1.6) with respect to the spherical priors in (1.5). In Section 4, we focus on the case where the function F (t)/f (t) is nonincreasing and we show that our result in Section 2 can be applied to obtain minimaxity of δ k . Section 5 is devoted to examples which illustrate the theory while Section 6 is a concluding section. Finally, we provide an appendix which contains technical results needed in the development of the paper.

A minimaxity theorem
To prove the version of Theorem 2.1 of Brandwein and Strawderman [4] below, we will follow the lines of their proof making use of the radial distribution of R = X − θ 2 with density related to f by and whose expectation will be denoted by E.
Theorem 2.1. Let X be a random vector in R p with spherically symmetric density as in (1.1) such that, for some fixed q ≥ 1, Let δ a,g (X) an estimator as in (1.3). Assume that Assume also that there exists a subharmonic function h such that and such that Then δ a,g (X) has a risk smaller than or equal to that of X.
Proof. In the proof of Theorem 2.1 of [4], it is shown that the risk difference at θ between δ a,g (X) and X satisfies Introducing R 2(q−1) in its right hand side Inequality (2.5) can be written as, As ϕ(R) changes sign once from − to + at R 0 = a p and as, by assumption, ψ(R) is nonincreasing, we have Also, by nonpositivity of h, we have ψ(R 0 ) ≤ 0, and hence, we will have d θ ≤ 0 when E[ϕ(R)] ≥ 0, that is, as soon as or, equivalently, as soon as which is the desired result.
Remark 2.1. Note that, for q > 0, the ratio E[R −2(q−1) ]/E[R −2q ] is nonincreasing in q so that, for q ≥ 1, we have Thus, comparing with Condition (j) in Section 1, we can see that the range of values of a is smaller with Condition 2.2 than with Condition (j), since it is respectively described by 0

Generalized Bayes estimators
As mentioned in Section 1, our aim is to study the minimaxity of the Bayes estimator δ k (X) in (1.6) using ). Hence, insofar as Condition and, as a candidate for a suitable function h, set h(x) = ∆M ( x 2 )/(a m( x 2 )), so that, clearly, it suffices to prove that Conditions (2.3) and (2.4) are satisfied for that function h to obtain improvement of δ k (X) over X. Actually, in Section 4, it will turn out that the choice of the upper bound of the value of a will be appropriate for q = k + 1, that is, . (3.1) The first inequality in Condition (2.3) reduces to and the second one to Note that the superharmonicity condition on −h reduces to the subharmonicity of the ratio ∆M ( ≥ 0, and it will be convenient to write this Laplacian as and and to prove separately that and The above conditions involve the marginals m( x 2 ) and M ( x 2 ) through the ratios As noticed by Fourdrinier, Mezoued and Strawderman in [5], where general spherical priors π( θ 2 ) are considered, conditions on π and f are needed to express these quantities as expectations with respect the a posteriori distribution given x. To this end, they rely on formulas of the type where ψ is, either the function f , or the function F and where, for a multi-index α = (α 1 , . . . , α p ) (a p-uple of nonnegative integers) with lengh |α| = α 1 +· · ·+α p , the operator D α = ∂α/∂x α1 is the corresponding partial derivative operator. In order (3.9) to hold the required conditions in [5] are is the space of functions α-times continuously differentiable and bounded on R p \ B r where B r is the ball of radius r centered at 0.
Clearly this is the highest order of derivation which matters so that, as the bi-Laplacian is involved, we have to consider α = 4. Insofar as the prior in is guaranteed while it can be checked that its membership to the Sobolev space W 4,1 loc (R p ) holds for k < p/2 − 2. Hence, in conjunction with , this leads to the following lemma where E x denotes the conditional expectation of θ given x. Note that there is no confusion with the notation E θ introduced after Formula (1.2) since, in the former notation, x is a superscript while, in the latter, θ is a subscript.
Similarly to what was noticed in [6], Lemma 3.1 still holds when requiring only that the assumptions on the generating function f (t) are satisfied except on a finite set T of values of t. This will be implicit in the following and will be used in an example of Section 5.
In order to obtain minimaxity of δ k in (1.6), expressions in (3.10) and (3.14) will play an important role through their inner product with x. This can be seen in the following lemma whose proof is postponed to Appendix A.1.
We have

Bayes minimax estimators
We can now formulate our main results about the minimaxity of the generalized Bayes estimators δ k (X) in (1.6). This minimaxity will be obtained through improvement on the usual estimator X.
Theorem 4.1. Assume that X has a spherically symmetric unimodal density Then the Bayes estimator δ k in (1.6) dominates X (and hence is minimax), under the quadratic loss (1.2), as soon as Remark 4.1.
The assumption 0 < k ≤ p/2 − 3 imposes that p ≥ 7 and is made to guarantee the superharmonicity of the function θ − 2 (k+2) (which implies the superharmonicity of the pior density in (1.5)). It is worth noting that this upper bound for k is implied by Condition (4.2) (see Appendix A.4). Conditions (4.1) and (4.2) express that k lies in a neighborhood of 0 since, in (4.1), according to Remark 2.1, the ratio µ −2 (k+1) /µ −2 k is nondecreasing in k.
Proof. As it was noticed in Section 3, we are reduced to prove Conditions in (3.1), this one can be expressed as After simplifying, it is clear that, through Jensen's inequality, it is sufficient to prove that In the left hand side of Inequality (4.3), as for the first expectation, using the fact that Therefore it follows from (4.4) that Inequality (4.3) holds, and hence Inequality (3.2) holds as well, as soon as Inequality (4.1) is satisfied. To prove (3.7) note that, similarly, by (3.15), (3.13), (3.11) and (3.10) in Lemma 3.1, the term A(x) in (3.5) equals By applying Jensen's inequality to the last expectation of the right hand side of (4.5) and using the fact that F (t)/f (t) ≤ F (0)/f (0), we have A(x) ≥ 0 as soon as The first expectation in the left hand side of (4.6) can be written as where U r,x is the uniform distribution on the sphere of radius r centered at x and ξ(r) is the radial density in (2.1). Note that, as 0 < k ≤ p/2 − 3, the function θ − 2 (k+2) is superharmonic, so that the function Sr,x θ − 2 (k+2) dU r,x is nonincreasing in r. Note also that, by assumption, F (r 2 )/f (r 2 ) is nonincreasing in r. Hence, by the covariance inequality, it follows from (4.7) that where the second equality follows from (1.9), the third one is obtained from an application of Fubini's theorem and, for the fourh one, a change of variable is used. Using (4.8) and applying Jensen's inequality, (4.6) is satisfied if As (4.9) is equivalent to (4.2), Inequality (3.7) is proved. We now turn our attention to Inequality (3.3). Note that F (t) is nonincreasing and that, by unimodality of f ( x − θ 2 ), the function f (t) is nonincreasing in t as well. As an immediate consequence of (3.10) and (3.12) in Lemma 3.1, the right hand side of (3.3) equals, for any x ∈ R p , (4.10) Now, since f (·) and F (·) are nonincreasing functions and since θ −2(k+1) is a nonnegative function, Lemma A.3 in Appendix A.1 guarantees that each integral in (4.10) equals x multiplied by a nonnegative function of x. Hence the left hand side of (4.10) is nonnegative and (3.3) is satisfied. Similarly, by (3.10) and (3.14) in Lemma 3.1, the inner product term in the right hand side of (3.6) equals, for any x ∈ R p , where ζ(k) = −8k 2 (k + 1) (p − 2(k + 1)). As the function ζ(k) is nonpositive, we can conclude, as in (4.10), that the left hand side of (4.11) is nonpositive. Therefore we obtain that B(x) in (3.6) is nonnegative, that is, Inequality (3.8) which, with Inequality (3.7) proved above, provides the subharmonicity of ∆M ( x 2 )/m( x 2 ) according to (3.4). Finally, gathering Conditions (3.2) and (3.3) obtained above, we have completely proved Condition (2.3).
It remains to address Condition (2.4). As it will be more convenient to deal with nonnegative functions, we will be interested in proving the nondecreasing monotonicity in R of for ω( x 2 ) defined in (3.16) which was above shown to be superharmonic, so that the function t → ω(t) is necessarily nonincreasing. Hence, according to Corollary A in Appendix A.1, this desired result will be obtained if we prove that t → r(t) = t k+1 ω(t) is nondecreasing. Note that the monotonicity of the functions ω and r can be expressed respectively as 1 2 1 x 2 x · ∇ω( x 2 ) ≤ 0 (4.13) and 1 2 1 x 2 x · ∇r( x 2 ) ≥ 0 (4.14) since the quantities in the left hand side of (4.13) and (4.14) are the derivatives of ω(t) and r(t) at t = x 2 respectively. It is easily seen, through the expression of ω( x 2 ), that the inner product in (4.13) equals according to (3.17) and (3.18) in Lemma 3.2. Therefore the fact that Inequality (4.13) is satisfied (as mentioned above) can be expressed as it follows from the left hand sides of (4.13) and (4.14) that 1 and hence Inequality (4.14) will be satisfied if and only if Therefore it follows from (4.18) that so that a sufficient condition for (4.17) to hold is As ω(t) is nonincreasing in t and ω(0) = 2 k (see Lemma A.4 in Appendix A.3), this is clearly satisfied.
Remark 4.2. In [5], the minimaxity conditions are weaker but the sampling densities are restricted to the Berger class, that is, there exists a positive constant c such that F (t)/f (t) > c for any t ≥ 0. Here our approach allows to include the case where lim t→∞ F (t)/f (t) = 0. Example 1. Let

(5.4)
It is quite involved to investigate formally this upper bound of the values of k. However, for different values of p and A, Table 1 provides the values k max (p, A) in (5.4).
We can see that the upper bound 1 k max (p, A) is increasing in A, for any fixed p. with γ > 0 and β > 1. Clearly f (t) is nonincreasing. Furthermore we have

Example 2. Consider
which shows that F (t)/f (t) is nonincreasing as well and, by the Lebesgue dominated convergence theorem, implies that since β > 1. Now, through straightforward calculations, we have and, for i > −p, so that, for k < p/2 − 1, Then Condition (4.1) reduces to Also, according to (5.6), we have where g(t) is a nondecreasing and convex function such that lim t→∞ g(t) = ∞ and α > 0.
It is clear that f ( x − θ 2 ) is unimodal. Also, expressing F (t)/f (t) as follows we can see, first, that the function F (t)/f (t) is nonincreasing. Indeed, the function in (5.12) is nondecreasing, since by nondecreasing montonicity and convexity of the function g.
Secondly, for such class of generated functions f (t), it has been shown in [3] that the condition inf t≥0 ∞ t f (s) ds/f (t) > 0 is violated and therefore lim t→∞ (F (t)/f (t)) = 0 since, in our case, F (t)/f (t) is nonincreasing.
Note that a simple example of function g is g(t) = exp(t) so that f (t) ∝ exp(−α t e t ).
It is worth noting that f (t) ∝ exp(−α e t ) is not included in the previous class but gives rise to another example satisfying the conditions of Theorem 4.1. Indeed f (t) is clearly nonincreasing and f ′ (t)/f (t) = −αe t is nonincreasing so that the function F (t)/f (t) is nonincreasing as well. We also have from an application of the Lebesgue dominated convergence theorem.

Concluding remarks
We have seen that, for a sampling density f ( x − θ 2 ) which is unimodal and such that the ratio F (t)/f (t) is nonincreasing (with F (t) in (1.9)), minimaxity of generalized Bayes estimators δ k (X) can be obtained for spherical prior densities θ −2k under conditions involving constants depending on f (·) and k.
We complement the results of Fourdrinier, Mezoued and Strawderman [5] in so far as these authors reduce their framework to the Berger class. However we only deal with the case where F (t)/f (t) is nonincreasing. Indeed, in the case where F (t)/f (t) is nondecreasing, our techniques are unsuitable for this monotonicity. Also, we adopted here a completely different approach since we relied on a modification of the Brandwein and Strawderman approach [4], which may be of an independent interest. Various examples of sampling densities illustrate our findings while the basic example of prior densities is formed of the class of θ −2k with k > 0. A natural scope of a future work is to extend that class of priors.
is nondecreasing for p ≥ 1.
When integrating with respect to F ( θ − x 2 ), the following result is useful.
Lemma A.2. For any function γ, we have Proof. We have the successive equalities Br,x γ(θ) dV r,x (θ) r p+1 f (r 2 ) dr the second inequality following from the change of variable u = r 2 , the third one from the Fubini theorem; in the fourth one, λ(B) is the volume of the unit ball.
The following lemma can be found in Fourdrinier and Righi [7] where the density in (1.2) is considered with fixed x as a function of θ, say θ → f ( θ−x 2 ), so that E x denotes the expectation with respect to that density. Lemma A. 3. Let x ∈ R p fixed and let Θ a random vector in R p with spherically symmetric density f ( θ − x 2 ). Let g be a function from R + into R.
Then there exists a function Γ from R p into R such that provided this expectation exists. Moreover, if the function f is nonincreasing and if the function g is nonnegative, then the function Γ is nonnegative.
A.3. Expressing the functions in Lemma 3.2 We now give expressions, in terms of sphere mean and ball mean, for the functions ω( x 2 ), γ( x 2 ) and δ( x 2 ) defined in Lemma 3.2.