Bayes minimax estimators of a location vector for densities in the Berger class

: We consider Bayesian estimation of the location parameter θ of a random vector X having a unimodal spherically symmetric density f ( k x − θ k 2 ) when the prior density π ( k θ k 2 ) is spherically symmetric and superharmonic. We study minimaxity of the generalized Bayes estimator δ π ( X ) = X + ∇ M ( X ) /m ( X ) under quadratic loss, 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 under the condition inf t ≥ 0 F ( t ) /f ( t ) = c > 0 (see Berger [1]). We adopt a common approach to the cases where F ( t ) /f ( t ) is nonincreasing or nondecreasing and, although details diﬀer in the two settings, this paper complements the article by Fourdrinier and Strawderman [7] who dealt with only the case where F ( t ) /f ( t ) is nondecreasing. When F ( t ) /f ( t ) is nonincreasing, we show that the Bayes esti- mator is minimax provided a k∇ π ( k θ k 2 ) k 2 /π ( k θ k 2 ) + 2 c 2 ∆ π ( k θ k 2 ) ≤ 0 where a is a constant depending on the sampling density. When F ( t ) /f ( t ) is nondecreasing, the ﬁrst term of that inequality is replaced by bg ( k θ k 2 ) where b also depends on f and where g ( k θ k 2 ) is a superharmonic upper bound of k∇ π ( k θ k 2 ) k 2 /π ( k θ k 2 ). Examples illustrate the theory.


Introduction
Let X be a random vector in R p with spherically symmetric density around an unknown location parameter θ ∈ R p which has a spherically symmetric prior π( θ 2 ). (1.2) Consider estimation of the parameter θ under squared error loss for p ≥ 3. We are mainly interested in the minimaxity of the generalized Bayes estimator δ π (X) associated with the prior in (1.2). Maruyama and Takemura [9] proved, through Baranchik type techniques that the Bayes estimator corresponding to the fundamental harmonic prior is minimax for certain f in (1.1). Also Fourdrinier, Kortbi and Strawderman [3] showed such minimaxity results for sampling and prior densities which are both mixtures of normals with a monotone likelihood ratio property when considered as scale families.
By contrast, for spherically symmetric prior densities given in (1.2), Fourdrinier and Strawderman [7] did not utilize the Baranchik representation but used the superharmonicity of the prior π more in the spirit of Stein [10]. Setting (1.4) they treated the case where the sampling densities are such that F (t)/f (t) is nondecreasing in t and bounded from below by some positive constant c (that is, the density in (1.1) belongs to the Berger class [1]). They proved that the risk difference between δ π (X) and X is bounded above by where E θ denotes the expectation with respect to (1.1) and · denotes the inner product in R p . Then, thanks to the superharmonicity of π( θ 2 ), they showed it suffices to prove that in order for the risk difference in (1.5) to be non positive. In contrast, in this paper, we demonstrate that and thus, minimaxity of δ π (X) follows provided that for all x ∈ R p . A benefit of this technique is that it allows us to also deal with the case where F (t)/f (t) is nonincreasing, which constitutes a main contribution of this paper. Finally, we will see below that, as we need to deal with tractable expressions for the derivatives of the marginals m and M , the membership of the prior density π θ 2 in certain Sobolev spaces and the membership of the generating function f in a particular superset of the Schwartz space will be required.
In Section 2, we develop the model and give preliminary calculations involving the derivatives of the marginals related to f and F . Section 3 is devoted to the first main result when F (t)/f (t) is nonincreasing while, in Section 4, we deal with the case where F (t)/f (t) is nondecreasing. Section 5 contains examples illustrating the theory developed in Sections 3 and Section 6 illustrates Section 4. Then we give, in Section 7, some concluding remarks and some perspectives. Finally, the last section is an Appendix containing needed technical material.

Model and generalized Bayes estimators
Let X be a p×1 random vector having a spherically symmetric distribution as in (1.1) about an unknown vector θ ∈ R p which has a spherically symmetric prior density as in (1.2). Consider estimation of the parameter θ under the quadratic 786 D. Foudrinier et al. loss in (1.3). As in [7], the generalized Bayes estimator is the posterior mean and can be written as δ π (X) = X + ∇M (X) m(X) (2.1) where, for any x ∈ R p , is the marginal density and with F (t) given in (1.4). Here ∇ denotes the gradient.
As mentioned above, our aim is to study the minimaxity of the Bayes estimator δ π (X) in (2.1), which follows by proving that δ π (X) dominates X under the loss (1.3). Of course, this only makes sense if the risk of X is finite, that is, if It is worth noting that this last condition, in fact, implies the finiteness of the risk of the Bayes estimator δ π (X) (see Lemma A.1). As stated in Section 1, this domination will be obtained as soon as Inequalities (1.7) and (1.6) will be satisfied.
Note that (1.7) and (1.6) involve the derivatives of the marginals m and M in (2.2) and (2.3). To express these derivatives conveniently, we will rely on formulas of the type where ψ will be, either the function f , or the function F and where, for a multiindex α = (α 1 , . . . , α p ) (a p-uple of nonnegative integers) with lengh |α| = α 1 + · · · + α p , D α is the corresponding partial derivative operator.
As proved in Blouza, Fourdrinier and Lepelletier [2], and used in Fourdrinier and Lepelletier [4], general assumptions on ψ and π in order for Condition (2.4) to be satisfied are that these functions belong to appropriate Sobolev spaces. Thus it will be needed that the prior π( θ 2 ) is in the Sobolev space and that the function ψ x − θ 2 is in the Sobolev space The aim of the membership in the above Sobolev spaces is mainly to control the functions around the origin. However we also need some regularity in a neighborhood of infinity and, to this end, we also assume that there exits r > 0 such that π( θ 2 ) belongs to C α b (R p \ B r ) (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) and there exists ǫ > 0 such that ψ( x − θ 2 ) is in S α,p+ǫ (R p \ B r ) (the space of functions α-times continuously differentiable on R p \ B r such that sup x∈R p \Br ;|β|≤α;γ≤p+ǫ x γ D β ψ x − θ 2 < ∞). Finally the sufficient conditions we use in order that formula (2.4) is satisfied are for a certain ǫ > 0.
It is more convenient to express the above condition on F x − θ 2 in terms of the generating function f . This is done in the following lemma whose the proof is similar to the proofs of Lemma 5 and Lemma 6 in [4].
The derivatives of the marginals m(x) and M (x) are involved through ∇m(x), ∇M (x), ∆M (x). Clearly this is the highest order of derivation which matters and it can be seen, for α = 2, that Lemma 2.1 applies to express these last quantities as expectations with respect to the posterior distribution given x. This is stated in the following lemma whose the proof follows directly from (2.4), as in [6], for various values of α.
where E x (·) denotes the conditional expectation of θ given x.
Similarly to what was noticed in [4], Lemma 2.1 and Lemma 2.2 still hold when requiring only that the assumptions on the generating function f (t) are satisfied except on a finite set T of values of t.
Note also that particular attention should be paid to the membership of the priors to the Sobolev space in Lemma 2.2. As an example, the fundamental harmonic prior π( θ 2 ) = θ 2−p , although it is a smooth function in R p \ {0}, is not in the Sobolev space W 2,1 loc (R p ) so that Equality (2.7) is not valid for that prior. See Subsection 5.2 for more details.

A minimaxity result for nonincreasing F(t)/f(t)
We can now formulate our first result about the minimaxity of the generalized Bayes estimator δ π (X) in (2.1). This minimaxity will be obtained through improvement on the usual estimator X using a condition which relates the prior π( θ 2 ) and the generating function f (t).
Theorem 3.1. Assume that X has a spherically symmetric unimodal density < ∞ and such that, for some ǫ > 0, the generating function f is in S 1,p/2+1+ǫ (R + \ {0}). Assume also that the function Suppose that, for some r > 0, the spherical prior density π( θ 2 ) in (1.2) belongs to the space . Then, under the quadratic loss (1.3), a sufficient condition for the Bayes estimator δ π to dominate X (and hence to be minimax) is that Proof. First, note that Inequality (3.1) implies that ∆π( θ 2 ) ≤ 0 and hence that the prior density π( θ 2 ) is superharmonic. The proof consists in proving (1.7) and (1.6). Consider first (1.7). According to Lemma 2.1, this inequality can be expressed as Through Jensen's inequality, it suffices to prove that Now it is clear that, the conditions on f implying the superharmonicity of π( θ 2 ) entails that Inequality (3.2) holds as soon as and hence, as soon as which is Condition (3.1). Therefore (1.7) is proved.
We now turn our attention to Inequality (1.6). 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 (2.5) and (2.6) in Lemma 2.1, the right hand side of (1.6) equals, for any x ∈ R p , expressing the gradient of π( θ 2 ). Now, by superharmonicity of the prior density π( θ 2 ), we have π ′ ( θ 2 ) ≤ 0 so that, by Lemma A.2 in the Appendix, each integral in (3.3) equals x multiplied by a non positive function. Hence the right hand side of (3.3) is nonnegative and (1.6) is satisfied. This ends the proof of Theorem 3.1.

The case where F(t)/f(t) is nondecreasing
In the case where the ratio F (t)/f (t) is nondecreasing, we need an additionnal assumption on the prior π( θ 2 ), that is, the function ∇π( θ 2 ) 2 /π( θ 2 ) is bounded above by a superharmonic function. Note that the constant F (0)/f (0) in (3.1) will be replaced by the constant d defined by provided this integral is finite. This is stated in the following theorem.
Theorem 4.1. Assume that X has a spherically symmetric unimodal density , for some ǫ > 0. Assume also that the function F (t)/f (t) is nondecreasing and bounded from below by a constant c > 0 and that the constant d in (4.1) is finite.
Suppose that, for some r > 0, the spherical prior density π( θ 2 ) in (1.2) belongs to the space Then, under the quadratic loss (1.2), a sufficient condition for the Bayes estimator δ π to dominate X (and hence to be minimax) is that for all θ ∈ R p , with d in (4.1).
Proof. Note that the conditions which have led, in Theorem 3.1, to Inequality (1.6) remain unchanged; hence we just prove (1.7). Returning to Inequality (3.2), the first term can be written as by assumption on ∇π( θ 2 ) 2 /π( θ 2 ) and where U r,x is the uniform measure distribution on the sphere of radius r centered at x. Now, since g( θ 2 ) is superharmonic, the function is nonincreasing in r and, since F (r 2 )/f (r 2 ) is nondecreasing in r, by the covariance inequality, it follows from the last inequality that with d given by (4.1). Finally, using (4.4), by superharmonicity of π( θ 2 ) and the fact that F (t)/f (t) ≥ c > 0, Inequality (3.2) holds as soon as which is satisfied as soon as (4.2) holds. This ends the proof of the theorem.
Remark 4.1. Bounding ∇π( θ 2 ) 2 /π( θ 2 ) by the superharmonic function g( θ 2 ) in the statement of Theorem 4.1 may appear somewhat ad hoc, but it is quite useful in that it dramatically increases the class of priors to which the theorem applies. To see this, note that, for a prior density of the form which is never superharmonic. However the upper bound Thus the replacement of the assumption that ∇π( θ 2 ) 2 /π( θ 2 ) is superharmonic by superharmonicity of the upper bound g( θ 2 ) allows inclusion of priors of the form ( Of additional interest is the fact that, if ∇π( θ 2 ) 2 /π( θ 2 ) has g( θ 2 ) as an upper bound, then Inequality (4.2) implies so that a natural candidate for g( θ 2 ) is , provided that the Laplacian ∆π( θ 2 ) is subharmonic. This is formulated in the following corollary.
for some ǫ > 0. Assume also that the function F (t)/f (t) is nondecreasing and bounded from below by a constant c > 0 and that the constant d in (4.1) is finite.
Suppose that, for some r > 0, the spherical prior density π( Then, under the quadratic loss (1.2), a sufficient condition for the Bayes estimator δ π to dominate X (and hence to be minimax) is that for all θ ∈ R p , with d in (4.1).
Proof. The corollary immediately follows, as in the proof of Theorem 4.1, from the fact that the upper bound of thanks to Inequality (4.5) and to the superharmonicity of −∆π( θ 2 ).
Corollary 4.1 is applicable to the above example of prior density since it can be shown (see Subsection 6.2) that the bi-Laplacian of ( ). Note that this is the same condition encountered for Inequality (4.2) to hold with g( θ 2 ) = ( θ 2 + b) −k−1 . In each case, since d 2 /c 2 ≥ 1, the latter inequality predominates so that either bound gives the same result.

Illustrating Theorem 3.1
We first consider various examples of sampling densities. Then we provide general conditions for prior densities, which are mixtures of normals, to belong to the space W 2,1 loc (R p ) and to satisfy Condition (3.1).

Sampling densities
In the following, we give examples of generating functions f (t) such that f ′ (t) ≤ 0 and such that F (t)/f (t) is nonincreasing and bounded from below by a constant c > 0, with F (t) in (1.4). Note that, for these examples, it is guaranteed that f ∈ S 1,p/2+1+ǫ (R + \ {0}) for some ǫ > 0 since the densities are elementary functions of the exponential function. For each example, we provide the values of F (0)/f (0) and of c = lim t→∞ F (t)/f (t).
where β > 0 and γ > 0. As f (t) can be written as which is nonincreasing in t. It follows then that where β > 0 and γ > 0. Clearly, this function is nonincreasing since the hyperbolic cosine is nondecreasing, and hence, the sampling density f ( x − θ 2 ) is unimodal. Also we have The nonincreasing monotonicity of F (t)/f (t) can be comfortably found through the one of f ′ (t)/f (t). Indeed, is nonincreasing which implies that F (t)/f (t) is nonincreasing as well (see Fourdrinier and Strawderman [7]). As for the bounds of that function, first, it follows from (5.3) that Secondly, expressing the hyperbolic cosine in (5.3) as where 0 ≤ a ≤ c < 1 and β > 0. We have for the range of a and c. We also have which is nonincreasing in t. Finally, we have Example 4. Let with A > 2. We have for the range of A. We also have

Prior densities
Example 1 -Generalized t densities A first basic example of spherical density prior is with b > 0 and k > 0. Considering the membership of that density in the Sobolev space W 2,1 loc (R p ), it is clear that for b > 0, the function θ → θ 2 + b −k is indefinitely differentiable and is bounded, so that its membership in the space is automatically satisfied. As for the case b = 0, the membership in C 2 b (R p \ B r ) obviously remains and it can be directly checked from the expression θ −2k that the density belongs to the Sobolev space W 2,1 loc (R p ) as soon as k < p/2 − 1, that is, p > 2 (k + 1) (see below an alternative proof through Corollary 5.1).
Through straightforward calculations, it can be seen that Ineqality (3.1) is expressed as which, after simplification, reduces to Example 2 -General mixtures of normals As the density prior in (5.10) is a mixture of normals densities, we consider prior densities of the form and we investigate conditions on the function h : R + → R + such that the density in (5.12) belongs to W 2,1 , for some r > 0, and such that Inequality (3.1) is satisfied. Thus we assume that the mixing function h is such that the function t → t p/2 h(t) has, for any s ≥ 0, a Laplace transform L t p/2 h(t) (s) at s so that the prior in (5.12) can be seen as π( θ 2 ) ∝ L t p/2 h(t) ( θ 2 ) . and, for any u ≥ 0, Also the prior in (5.12) is guaranteed to belong to the space C 2 b (R p \ B r ). However, as it is not a Laplace transform of a multivariate function, it remains to deal with the membership of this prior in the Sobolev space W 2,1 loc (R p ).
Note that formal derivatives of π( θ 2 ) can be taken under the integral sign, so that the membership of the prior in (5.12) in the Sobolev space W 2,1 loc (R p ) reduces to verifying the local integrability of the formal derivatives. Using straightforward calculations, we have successively, for i = 1, . . . , p, and, for j = 1, . . . , p and j = i, where ∂ i (.) and ∂ ji (.) denote the partial derivative operators of the first and second order respectively.
Clearly, to obtain the desired local integrability of π( θ 2 ) and its partial derivatives, it suffices to establish the local integrability of functions of the form for a, b = 0, 1, 2. This is related to the behavior of h at ∞ and can be obtained through an Abelian theorem which relates the behavior of a function at ∞ to the behavior of its Laplace transform at 0. As can be seen, for instance, in Corollary 1.a page 182 of Widder [11]), if α is a function of bounded variation and normalized in any interval [0, b] of R (that is, such that, for any x, α(x) = (α(x − ) + α(x + ))/2 and such that α(0) = 0) and if, for some γ ≥ 0, there exists a constant A such that through an integration by parts. In the case where A = 0, Condition (5.17) and Condition (5.18) should be understood as α(t) = o(t γ ) and L(α)(s) = o s −γ−1 respectively. With this result, we will prove the following lemma.
Proof. According to (5.16) the function ψ a,b ( θ 2 ) can be expressed through the Laplace transform L(t p/2+a h(u))( θ 2 ) and what we have to show is that, for any ball B R centered at 0 and of radius R,  We are now in a position to give condtions on the prior density in (5.12) to belong to W 2,1 loc (R p ) ∩ C 2 b (R p \ B r ). Corollary 5.1. If there exist a constant A and γ ≥ 0 such that then the prior density in (5.12) belongs to W 2,1 loc (R p ) ∩ C 2 b (R p \ B r ) as soon as γ < p/2 (Condition (5.22) being understood as t p/2+2−γ h(t) = o(1) when A = 0).
In order that the generalized Bayes estimator δ π (X), associated to the prior in (5.12), dominate X (and hence to minimax), it remains to give conditions on the mixing density h so that Condition (3.1) is satisfied. We will see below that the main assumption on h is that the function t −→ t h ′ (t)/h(t) is non increasing. This condition has been noticed by [8] and [3] as being equivalent to the property of nondecreasing monotone likelihood ratio when h is considered as a scale family. The following proposition gives conditions on the mixing density h for minimaxity to follow via Theorem 3.1.

Proposition 5.2. Assume that the function t −→ t h ′ (t)/h(t) is nonincreasing. Then Condition (3.1) is satisfied as soon as, for all
Proof. Through straightforward calculations, we have which are functions of the ratios π ′ ( θ 2 ) π( θ 2 ) and π ′′ ( θ 2 ) π( θ 2 ) . Since where E θ 2 denotes the expectation with respect to the distribution of T with density On the other hand, by integration by parts, we can also express according to (5.14) and (5.15), so that Similarly, as an integration by parts leads to Thus Inequality (3.1) may be expressed as Since, by assumption, t → t h ′ (t)/h(t) is nonincreasing, by the covariance inequality, we have Hence (5.27) is satisfied as soon as, for all t, which can be seen to be equivalent to (5.23).
It is worth noting that, the upper bound of u h ′ (u) /h(u) given by Condition (5.23) satisfies ϕ 0 < −2 for p ≥ 3. Note that Fourdrinier and Strawderman [7], to obtain the superharmonicity of priors π( θ 2 ) in (5.12), required that t h ′ (t) /t < −2, which is, of course satisfied here. However, they do not impose a monotone likelihood ratio property on the mixing density h, which we require in this paper.
Example of mixing densities 1-A simple example of mixing density h is given by where b ≥ 0 and k > 0. This mixing density gives rise to the prior θ 2 + b −k considered in Example 1. We can retrieve the Sobolev membership of that prior applying Corollary 5.1. Indeed, when b = 0, with γ = k + 1, we see that the prior belongs to the Sobolev space W 2,1 loc (R p ) as soon as k < p/2 − 1, that is, p > 2 (k + 1). The case where b > 0 is obvious. Now it is easy to see that h satisfies the monotonicity condition (5.23) required in Proposition 5.2 since, which nonincreasing in t. Thus, a sufficient condition to satisfy Inequality (5.23) is which is, according to (5.24), equivalent to (5.28) Note that, as expected, the upper bound on k in (5.28) is smaller than the one in (5.11) ((5.23) is a sufficient condition for (3.1)).

2-A constructive approach to obtain functions h satisfying (5.23).
Let ϕ be a nondecreasing and nonnegative function ϕ(t) such that a primitive of the function t → ϕ(t)/t exists. To any such function ϕ, associate a mixing density where t 0 is some nonnegative number. It is clear that so that it suffices to choose ϕ such that ϕ(t) ≥ ϕ 0 in order (5.23) to be satisfied. Now, noticing that, h(t) is bounded from above by a positive multiple of so that, for γ < p/2, up to a positive multiplicative constant, h(t) t p/2+2−γ is bounded from above by t −ϕ0+p/2+2−γ , the needed Sobolev membership can be obtained using Corollary 5.1. Note that the previous example corresponds to the choice of ϕ(t) = b t − k + p/2 + 1 and to ϕ 0 = −k + p/2 + 1. As another application, the choice of ϕ(t) = a t 2 +b t+c with a, b and c being positive constants such that b 2 −4 a c < 0 leads to the mixing function h(t) ∝ exp(−a t 2 /2) exp(−b t) t −c .

Illustrating Theorem 4.1
Here the improvement on the result of Fourdrinier and Strawderman [7] results from weaker assumptions on the sampling density.

Sampling densities
In [7], examples of sampling densities are given which satisfy stronger conditions than the assumptions of Theorem 4.1 (that is, F (t)/f (t) ≥ c > 0 and the nondecreasing monotonicity of F (t)/f (t)). In particular, they require nondecreasing monotonicity of f ′ (t)/f (t) (which implies the nondecreasing monotonicity of F (t)/f (t)) in addition to the inequality Here, we adapt these examples in order that Condition (6.1) is not necessarily satisfied, but nevertheless d as defined in (4.1) is finite.

Example 1.
Consider the generating function corresponding to a variance mixture of normals, where g(v) is the mixing density of a random variable V . In [7], it is proved that, provided that E[V −p/2 ] < ∞, Condition (6.1) is equivalent to Therefore relaxing Condition (6.3) on the mixing function g extends the scope of sampling densities which gives rise to minimaxity of generalized Bayes estimators in (2.1). In [7], it is shown that, when the distribution of V is an inverse gamma IG(α, β), with α > 0 and β > 0 (i.e. g(v) = β α /Γ(α) v −α−1 exp(−β/v)) Condition (6.1) reduces to p/2 + 1 ≤ α (with no role for β), so that the values of α for which p/2 + 1 > α provide additional new examples of sampling densities. In particular, if the distribution of X is a multivariate t with m degrees of freedom, corresponding to α = β = m/2, relaxing Condition (6.3) gives rise to m < p + 2 allowing the inclusion of more Student distributions. More precisely, it is shown in Proposition A.1, that the constant d in (4.1) is finite and equals provided that α > 2, so that, in the Student case, we have for m ≥ 5. Note that, when m goes to infinity, d goes to one, which corresponds to the normal case.
Example 2. Consider the the generating function where α > 0, 0 < β ≤ 1 and K is the normalizing constant. In [7], it is shown that, Condition (6.1) is satisfied for β in a neighborhood of the form (1 − ǫ, 1] with ǫ > 0 and is not satisfied for β = 1/2. For the constant d, we have and, through the change of variable v = αt β , Also, by the change of variable v = s + α r 2β , we obtain from the above expression. It is clear that these integrals are finite and hence d is finite.
Example 3. Consider the generating function and K is the normalizing constant. In [7], it is shown that, Condition (6.1) reduces to It is clear that, for small values of α and β, Inequality (6.4) cannot be satisfied. However, we show below that the constant d is finite. We have Since, ϕ(·) is a nonincreasing positive function, ϕ(u) ≤ ϕ(0), and hence Also, by monotonicity of ϕ(·), we have ϕ(u) ≥ lim u→+∞ ϕ(u) = µ, and so By evaluating the above integral we obtain which is finite, and hence d 2 is finite.

Example 2 -General mixtures of normals
Condition (4.5) will be satisfied modifying Proposition 5.2 in an obvious way.
Proposition 6.1. Assume that the function t −→ t h ′ (t)/h(t) is nonincreasing. Then Condition (4.5) is satisfied as soon as, for any t ≥ 0, Then the constructive approach in Subsection 5.2 can be pursued choosing a nondecreasing function ϕ such that ϕ(t) ≥ ϕ 1 for any t ≥ 0 and such that a primitive of t → ϕ(t)/t exists. Indeed, it follows from the expression of ∆π( θ 2 )/π( θ 2 ) in (5.26) that Therefore the bi-Laplacian can be expressed as so that a sufficient condition for ∆ (2) π( θ 2 ) to be nonnegative is that, for all t ≥ 0, 2 g(t) + t g ′ (t) ≥ 0. Now, noticing that g(t) = 4 t h(t)(2 − ϕ(t)), it can be checked that Hence the required sufficient condition reduces to Inequality (6.7) can be verified directly for the functions ϕ(t) = b t − k + p/2 + 1 and ϕ(t) = a t 2 + b t + c considered at the end of Subsection 5.2, provided −k + p/2 − 2 ≥ 0, a > 0, b > 0, c ≥ 7/2 and b 2 − 4 a c < 0.

Concluding remarks
In this paper, we have studied minimaxity of generalized Bayes estimators corresponding to a subclass of superharmonic priors π θ 2 when the sampling density is spherically symmetric and is in Berger's class, i.e.
This article is closely related to Fourdrinier and Strawderman [7] in that they too studied minimaxity in the same context. However, this paper differs from [7] in several ways. First, while [7] considered only the case F (t)/f (t) nondecreasing, we consider both the nondecreasing and the nonincreasing cases. Second, the method of proof in this paper is quite different from that of [7]. Here, we give conditions for minimaxity that depend jointly on the sampling density and the prior density (and its derivatives) and are expressed through differential inequalities for the prior distributions with coefficients that depend on f (·). In [7], the conditions on the density and the prior were given separately and the key differential inequality was in terms of f (·), not π(·). A third difference is that the class of sampling densities, even when F (t)/f (t) is nondecreasing, is considerably enlarged in the present paper. Section 6 indicates the degree to which this paper extends the class covered by [7]. A main disadvantage of the results of this paper relative to [7] is that, when the prior density is of the form π θ 2 = ( θ 2 + b) −k , [7] gives minimaxity for k in the range [0, (p − 2)/2] and hence includes the fundamental harmonic prior θ 2−p . In this paper, the range of k is always contained in a proper subinterval of [0, (p − 2)/3].

Appendix A: Miscellaneous results
In the following lemma, we prove that the finiteness of the risk of the generalized Bayes estimator δ π (X) in (2.1) follows from that of X.
Lemma A.1. A sufficient condition for the risk of the Bayes estimator δ π (X) in (2.1) to be finite is that E 0 [ X 2 ] < ∞.
Proof. Recall that the risk of δ π (X) is finite if Thus it suffices to prove that where U R,x is the uniform distribution on the sphere S R,x of radius R and centered at x and σ(S) is the area of the unit sphere. Through the change of variable R = √ v, the right hand side of (A.2) can be written as is nonincreasing in v by the superharmonicity of π( θ 2 ). Now we can express the quantity in brackets in (A.1) as where E 1 is the expectation with respect to the density f 1 (v) proportional to and E 2 is the expectation with respect to the density f 2 (v) proportional to f (v)v p/2−1 .
Indeed the ratio f 2 (v)/f 1 (v) is nondecreasing by the monotonicity of S π ( √ v, x).
which is finite as soon as The next lemma was used in the proofs of Theorems 3.1 and 4.1 to show ∇M · ∇m ≥ 0. A proof can be found in Fourdrinier and Righi [5].
Lemma A.2. Let x ∈ R p fixed and let Θ a random vector in R p with unimodal spherically symmetric density f ( θ − x 2 ). Denote by E x the expectation with respect to that density and 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.
In the following proposition, we provide an expression for the constant d in (4.1) for Example 1 in Section 6.