Generalized Bayes minimax estimators of location vectors for spherically symmetric

distributions with residual vector

Dominique FOURDRINIER ^{∗} Othmane KORTBI ^{†} and William E. STRAWDERMAN ^{‡}

May 13, 2010

Abstract

We consider estimation of the mean vector, θ, of a spherically symmetric distribution with known scale parameter under quadratic loss and when a residual vector is available. We show minimaxity of generalized Bayes estimators corresponding to superharmonic priors with a non decreasing Laplacian of the form π(∥θ∥^{2}), under certain conditions on the generating function f(·) of the sampling distributions. The class of sampling distributions includes certain variance mixtures of normals.

AMS 2010 subject classifications. Primary 62C20, 62C15, 62C10.

Keywords and phrases: Spherical symmetry, Bayes estimators, Minimax estimators, Quadratic loss.

∗Université de Rouen, LITIS, EA 4108, France. The support of the ANR grant 08-EMER-002 is gratefully acknowledged.

†Université de Sherbrooke, Département de mathématiques, Canada.

‡Rutgers University, Department of Statistics, USA.

1 Introduction

This paper is devoted to the study of minimaxity of generalized Bayes estimators, under quadratic loss, of the location parameter of a spherically symmetric distribution. More specifically, we consider the model where (X, U) is a p + k random vector having a spherically symmetric distribution around the p + k vector (θ, 0) with θ being the unknown parameter of interest. Here dim X= dim θ = p and dim U = dim 0 = k. Suppose the density function of (X, U) is of the form

f (∥x − θ∥^{2} + ∥u∥^{2}) , (1.1)

where f(·) is known, and consider generalized Bayes estimators of θ for priors of the form

π (∥θ∥^{2}) , (1.2)

under the quadratic loss function

L (θ, δ) = ∥δ − θ∥^{2} . (1.3)

The fundamental tool used in this work is the development of an expression for the risk based on a differential expression involving the marginal which has become a basic tool to prove minimaxity. This differential expression has been extended to non normal models such as (1.1) by several authors (see Fourdrinier et al. [7, 4, 3, 8], Maruyama [9]). The main difference between the setup of this paper and the others is that here f(·) is known and so the scale parameter σ^{2} = E∥X − θ∥^{2}/p is fixed and known. However the residual vector U is also available. Because of this, results herein are intermediate between the previous results; however distributional robustness results for generalized Bayes minimax estimators falls somewhat closer to those for a known scale without a residual vector.

The general line of research pertinent to this work is the development of (generalized) Bayes minimax estimators. In the case of normal distribution with known scale, see for example Fourdrinier et al. [6]. When the scale is unknown, see Strawderman [11]. For variance mixture of normals with known scale and no residual vector, see Strawderman [12], Fourdrinier et al. [2], and Maruyama [9]. For general spherically symmetric distributions, with no residual vector see Fourdrinier et al. [3] and for spherically symmetric distributions with residual vector and unknown scale, see Fourdrinier et al. [5], and Maruyama [10].

In this paper, we establish minimaxity of generalized Bayes estimators for broad classes of spherically symmetric distributions which are not restricted to variance mixtures of normals. Minimaxity results are obtained for unimodal spherically symmetric superharmonic priors of the form (1.2) under the additional condition of monotonicity of the Laplacian of the prior. This class contains the fundamental harmonic prior ∥θ∥^{2−p} and, more generally, priors of the form (a + ||θ||^{2})^{−b} for 0 < b < (p − 2)/2 and a ≥ 0.

Conditions on the sampling density are that ∫_{s}^{∞} f(t) dt/2 f(s) is bounded below by a certain positive function c(s), that f′(t)/f(t) is non decreasing, and that

∫_{0}^{∞} f(v + s) v^{p/2} dv ≤ 4 c(s) ∫_{0}^{∞} −f′(v + s) v^{p/2} dv < ∞

for all s > 0. This class includes variance mixtures of normal distributions satisfying mild conditions on the mixing density.

In Section 2, we develop the model, give preliminary risk calculations and we establish the main result. Section 3 gives examples to illustrate the theory. In Section 4, we give some concluding remarks.

2 Main result

2.1 Risk considerations

Any estimator δ(X, U) of θ is evaluated by its risk function associated to the loss (1.3), that is by

R (θ, δ) = E_{θ} [∥δ(X, U) − θ∥^{2}] (2.1)

where E_{θ} denotes the expectation with respect to the density (1.1). For the rest of this paper, we assume

E_{θ} [∥X − θ∥^{2}] < ∞ (2.2)

which guarantees that the standard estimator X has finite risk and is minimax. As any estimator δ can be written as δ(X, U) = X + g(X, U), the finiteness of its risk is guaranteed by

E_{θ} [∥g (X, U)∥^{2}] < ∞ , (2.3)

and δ will be itself minimax provided the difference in risk

Δ_{θ}(δ) = E_{θ} [∥δ (X, U) − θ∥^{2} − ∥X − θ∥^{2}]
(2.4)  = E_{θ} [2 (X − θ)′ g (X, U) + ∥g (X, U)∥^{2}]

is non positive.

To express the risk difference between δ and X, we introduce first the function F defined, for any t > 0, by

F (t) = \frac{1}{2} ∫_{t}^{∞} f (l) dl . (2.5)

We have the following lemma whose proof is similar to the version found in Fourdrinier et al. [7].

Lemma 2.1. Let g be a weakly differentiable function from ℝ^{p+k} into ℝ^{p}. Then

E_{θ} [(X − θ)′ g (X, U)] = E_{θ} [\frac{F (∥X − θ∥^{2} + ∥U∥^{2})}{f (∥X − θ∥^{2} + ∥U∥^{2})} divg_{X} (X, U)] (2.6)

provided these expectations exist.

The next result follows immediately from the Lemma 2.1.

Lemma 2.2. Assume that E_{θ} [∥g (X, U)∥^{2}] < ∞. The risk difference in (2.4) equals

Δ_{θ} (δ) = E_{θ} [2\frac{F (∥X − θ∥^{2} + ∥U∥^{2})}{f (∥X − θ∥^{2} + ∥U∥^{2})} div_{X}g (X, U) + ∥g (X, U)∥^{2}] . (2.7)

A sufficient condition for minimaxity of estimators δ with finite risk is given by the following lemma when p ≥ 3. For that, let use the notation s = ∥u∥^{2}.

Lemma 2.3. Assume that F/f is bounded below by a positive function c(s) and that E_{θ} [∥g (X, U)∥^{2}] < ∞ for all θ ∈ ℝ^{p}. Then the estimator δ (X, U) is minimax provided

2c(s) div_{x}g (x, u) + ∥g (x, u)∥^{2} ≤ 0 (2.8)

for all x ∈ ℝ^{p} and all u ∈ ℝ^{k}.

The proof follows immediately from Lemma 2.2.

2.2 Form of the Bayes Estimators

The Bayes estimator under loss (1.3), for the priors given by (1.2) is of the form

δ_{π} (X, U) = X + \frac{∫_{ℝ^{p}} (θ − X) f (∥X − θ∥^{2} + ∥U∥^{2}) π (∥θ∥^{2}) dθ}{∫_{ℝ^{p}} f (∥X − θ∥^{2} + ∥U∥^{2}) π (∥θ∥^{2}) dθ} . (2.9)

Now, define for any (x, u) ∈ ℝ^{p} × ℝ^{k},

M (x, u) = ∫_{ℝ^{p}} F (∥x − θ∥^{2} + ∥u∥^{2}) π (∥θ∥^{2}) dθ (2.10)

and

m (x, u) = ∫_{ℝ^{p}} f (∥x − θ∥^{2} + ∥u∥^{2}) π (∥θ∥^{2}) dθ (2.11)

Then the generalized Bayes estimator δ_{π} defined above can be written as

δ_{π} (X, U) = X + \frac{∇_{X}M (X, U)}{m (X, U)} (2.12)

where ∇_{X} denote the gradient with respect of the variable X.

Applying Lemma 2.3 to the Bayes estimator δ_{π} gives the following result.

Lemma 2.4. Assume that F/f is bounded below by a positive function c(s) and that E_{θ} [||∇_{X}M(X, U)/m(X, U)| |^{2}] < ∞ for all θ ∈ ℝ^{p}. Then the estimator δ_{π} (X, U) in (2.12) is minimax provided

2c(s) \frac{Δ_{x}M (x, u)}{m (x, u)} − 2c(s) \frac{∇_{x}M (x, u)′ ∇_{x}m (x, u)}{m^{2} (x, u)} + \frac{∥∇_{x}M (x, u)∥^{2}}{m^{2} (x, u)} ≤ 0 (2.13)

for all x ∈ ℝ^{p} and all u ∈ ℝ^{k}.

PROOF The lemma follows from the expression of div_{x}(∇_{x}M(x, u)/m(x, u)), where we denote by Δ_{x} the Laplacian with respect to the variable x. □

Note that it follows from the spherical symmetry of f and π that, the functions m (x, u) and M (x, u) depend on (x, u) through t = ∥x∥^{2} and s. Then, we write (with a slight abuse of notation)

m (x, u) = m (t, s) and M (x, u) = M (t, s)

so that

∇_{x}m(x, u) = 2 m′(t, s) x and ∇_{x}M(x, u) = 2 M′(t, s) x.

2.3 Minimaxity of generalized Bayes Estimators

First, we will need the following result which essentially gathers results in Lemma 3.1 - 3.3 of Fourdrinier and Strawderman [3].

Lemma 2.5. Let m (x, u) and M (x, u) be as defined in (2.11) and (2.10) and let · denote the inner product in ℝ^{p}. Then we have

1.

x · ∇_{x}m (x, u) = −2 ∫_{0}^{∞} H (v, ∥x∥^{2}) v^{p/2} f′(v + s) dv

and

x · ∇_{x}M (x, u) = ∫_{0}^{∞} H (v, ∥x∥^{2}) v^{p/2}f (v + s) dv

where, for v > 0,

H (v, ∥x∥^{2}) = λ (B) ∫_{B_{\sqrt{v},x}} x · θ π′(||θ||^{2}) dV_{\sqrt{v},x} (θ) (2.14)

and V_{\sqrt{v},x} is the uniform distribution on the ball B_{\sqrt{v},x} of radius \sqrt{v} centered at x and λ (B) is the volume of the unit ball.

2. For any x ∈ ℝ^{p}, the function H (v, ∥x∥^{2}) in (2.14) is non decreasing in v provided that Δπ(||θ||^{2}) is non decreasing in ∥θ∥^{2}.

3. For any v > 0 and any x ∈ ℝ^{p}, the function H (v, ∥x∥^{2}) in (2.14) is non positive provided that π′ (∥θ∥^{2}) ≤ 0.

Now, our main result is given in the following theorem.

Theorm 2.1. Suppose θ ∈ ℝ^{p} has a superharmonic prior π(∥θ∥^{2}), where π(∥θ∥^{2}) is non increasing and Δπ(∥θ∥^{2}) is non decreasing in ∥θ∥^{2}. Then the Bayes estimator δ_{π} is minimax under quadratic loss (1.3) provided that \frac{f′}{f} is non decreasing, \frac{F (v+s)}{f(v+s)} bounded below by a positive function c(s) and for all s > 0

∫_{0}^{∞} f (v + s) v^{p/2} dv ≤ 4c (s) ∫_{0}^{∞} −f′ (v + s) v^{p/2} dv < ∞ . (2.15)

PROOF Due to the superharmonicity of π(∥θ∥^{2}), for any (x, u) ∈ ℝ^{p} × ℝ^{k}, we have Δ_{x}M(x, u) ≤ 0 so that, by Lemma 2.4, it suffices to prove that

−2c(s) ∇_{x}M(x, u)′∇_{x}m(x, u) + ∥∇_{x}M(x, u)∥^{2} ≤ 0 (2.16) for all x and all u. Using notation for m (x, u) and M (x, u) defined above, (2.16) reduces to

−2c(s) M′(t, s) m′(t, s) + (M′(t, s))^{2} ≤ 0 . (2.17)

Now, since we have supposed that π(∥θ∥^{2}) is non increasing in ∥θ∥^{2}, Lemma 2.5 implies that M′(t, s) ≤ 0. So, Inequality (2.17) reduces to

−2 c(s) m′(t, s) + M′(t, s) ≥ 0 , (2.18)

which can be written as

−2 c(s) x · ∇_{x}m(x, u) + x · ∇_{x}M(x, u) ≥ 0 . (2.19)

Using again Lemma 2.5, the left hand side of the last inequality may be expressed as

4 c(s) ∫_{0}^{∞} H(v, ∥x∥^{2}) v^{p/2}f′(v + s) dv + ∫_{0}^{∞} H(v, ∥x∥^{2}) v^{p/2} f(v + s) dv
(2.20)  = 4 c(s) E[H(v, ∥x∥^{2}) \frac{f′(v + s)}{f(v + s)}] + E[H(v, ∥x∥^{2})] ,

where E denotes the expectation with respect to the density proportional to v^{p/2}f (v + s). As the function \frac{f′(v+s)}{f(v+s)} is assumed to be non decreasing in v and as H(v, ∥x∥^{2}) is also non decreasing in v according to Lemma 2.5, since Δπ(∥θ∥^{2}) is assumed to be non decreasing in ∥θ∥^{2}, the first expectation in (2.20) satisfies

E [H(v, ∥x∥^{2})\frac{f′(v + s)}{f(v + s)}] ≥ E [H(v, ∥x∥^{2})] E [\frac{f′(v + s)}{f(v + s)}]

by the covariance inequality. Therefore Inequality (2.19) will be satisfied as soon as

4c(s)E [\frac{f′(v + s)}{f(v + s)}] + 1 ≤ 0 (2.21)

since H (v, ∥x∥^{2}) ≤ 0 by Lemma 2.5. Finally, Inequality (2.21) is equivalent to the condition given in the theorem. □

It is worth noting that, since f′(v+s)/f(v+s) is non decreasing, it follows that F (v + s)/f(v + s) is non decreasing as well and hence the lower bound function c(s) can be replaced by F (s)/f(s).

8 Kortbi, Fourdrinier, Strawderman

3 Examples

3.1 Examples of sampling distributions

Example (S1).

Assume that

f(v + s) = K (\frac{1}{v + s + a})^{b} , (3.1)

where b > 1, a > 0 and K is the normalizing constant. Note that

\frac{f′(v + s)}{f(v + s)} = (\frac{−b}{v + s + a}) (3.2)

is non decreasing in v for all value of b > 0. For the lower bound function, as noted above, we can take

c(s) = \frac{F (s)}{f(s)} = \frac{s + a}{2(b − 1)} . (3.3)

Also, we have

(3.4)  E [\frac{f′(v + s)}{f(v + s)}] = −b \frac{∫_{0}^{∞} v^{p/2} (\frac{1}{v+s+a})^{b+1} dv}{∫_{0}^{∞} v^{p/2} (\frac{1}{v+s+a})^{b} dv}
= \frac{−b}{s + a} \frac{∫_{0}^{∞} z^{p/2} (\frac{1}{z+1})^{b+1} dz}{∫_{0}^{∞} z^{p/2} (\frac{1}{z+1})^{b} dz}
= \frac{−b}{s + a} \frac{B (p/2 + 1, b − p/2)}{B (p/2 + 1, b − p/2 − 1)}

where B (α, β) is the beta function with parameters α > 0 and β > 0. Then (3.4) becomes

(3.5)  E [\frac{f′(v + s)}{f(v + s)}] = \frac{−b}{s + a} \frac{Γ (b − p/2)}{Γ (b + 1)} \frac{Γ (b)}{Γ (b − p/2 − 1)}
= −\frac{b − p/2 − 1}{s + a} .

So, the last condition in the theorem becomes

2\frac{b − p/2 − 1}{b − 1} ≥ 1 (3.6)

which is satisfied for p ≤ b − 1.

Example (S2). Variance mixtures of normals.

Assume that

f(v + s) = K ∫_{0}^{∞} w^{−(p+k)/2} exp (−\frac{v + s}{2w}) h(w) dw , (3.7)

where h is a mixing density and K is the normalizing constant. Then, we have

(3.8)  \frac{f′(v + s)}{f(v + s)} = \frac{−\frac{1}{2} ∫_{0}^{∞} w^{−(p+k)/2−1} exp (−\frac{v+s}{2w}) h(w) dw}{∫_{0}^{∞} w^{−(p+k)/2} exp (−\frac{v+s}{2w}) h(w) dw}
= −\frac{1}{2}E_{v} [W^{−1}] ,

where E_{v} denotes the expectation with respect to the density f_{v}(w)∝w^{−(p+k)/2} exp (−\frac{v+s}{2w}) h(w) which has increasing monotone likelihood ratio in v. Since −W^{−1} is increasing, the monotonicity of \frac{f′(v+s)}{f(v+s)} follows. Again, for the lower bound function c(s) we take

(3.9)  c(s) = \frac{F (s)}{f(s)} = \frac{∫_{0}^{∞} w^{−(p+k)/2+1} exp (−\frac{s}{2w}) h(w) dw}{∫_{0}^{∞} w^{−(p+k)/2} exp (−\frac{s}{2w}) h(w) dw}

= \frac{E_{s} [W^{−p/2+1}]}{E_{s} [W^{−p/2}]} ,

where E_{s} denotes the expectation with respect to the density f_{s}(w) ∝ w^{−k/2} exp (−\frac{s}{2w}) h(w). Now, to verify the last condition of the theorem, we use Fubini’s theorem to obtain

∫_{0}^{∞} v^{p/2}f(v + s) dv = K ∫_{0}^{∞} ∫_{0}^{∞} v^{p/2} exp (−\frac{v}{2w}) dv

× w^{−(p+k)/2} exp (−\frac{s}{2w}) h(w) dw

= K2^{p/2+1}Γ (p/2 + 1) ∫_{0}^{∞} w^{−k/2+1} exp (−\frac{s}{2w}) h(w) dw .

Similarly,

∫_{0}^{∞} v^{p/2}f′(v + s) dv = −\frac{K}{2} ∫_{0}^{∞} ∫_{0}^{∞} v^{p/2} exp (−\frac{v}{2w}) dv

× w^{−(p+k)−1/2} exp (−\frac{s}{2w}) h(w) dw

= −K2^{p/2}Γ (p/2 + 1) ∫_{0}^{∞} w^{−k/2} exp (−\frac{s}{2w}) h(w) dw .

10 Kortbi, Fourdrinier, Strawderman Then, we have

(3.10) E [\frac{f′(v + s)}{f(v + s)}] = −\frac{1}{2} \frac{∫_{0}^{∞} w^{−k/2} exp (−\frac{s}{2w}) h(w) dw}{∫_{0}^{∞} w^{−k/2+1} exp (−\frac{s}{2w}) h(w) dw}
= −\frac{1}{2E_{s} [W]} ,

and so the last condition in the theorem reduces to

g(s) = \frac{E_{s} [W] E_{s} [W^{−p/2}]}{E_{s} [W^{−p/2+1}]} ≤ 2 , (3.11)

for all s > 0. Also, we have the next result. Lemma 3.1. Assume that the mixing density h of the sampling distribution in (3.7) is such that

−t\frac{h′(t)}{h(t)} ≤ α (3.12)

for some α ≥ 2 − (p + k)/2. Assume also that the mixing density h satisfies

lim_{t→∞} t^{β}h(t) = c (3.13)

for some β > 2 − k/2 and some constant c ≠ 0. Then

g(s) ≤ \frac{p + k + 2α − 4}{k + 2β − 4} , (3.14)

provided there exist K > 0 such that t^{β}h(t) ≤ K for all t > 0. PROOF Let, for any β and for any s > 0

I_{β}(s) = ∫_{0}^{∞} w^{β−k/2} exp (−\frac{s}{2w}) h (w) dw , (3.15)

then we can see that g(s) = \frac{I_{1}(s)}{I_{0}(s)} \frac{I_{(−p/2)}(s)}{I_{(−p/2+1)}(s)}. Now, using integration by parts we can see that

I_{−p/2}(s) = \frac{2}{s} (\frac{p + k}{2} − 2) ∫_{0}^{∞} t^{(p+k)/2−3} exp (−\frac{st}{2}) h (\frac{1}{t}) dt

− \frac{2}{s} ∫_{0}^{∞} t^{(p+k)/2−4} exp (−\frac{st}{2}) h′ (\frac{1}{t}) dt .

So, using condition (3.12) we can see that

\frac{I_{−p/2}(s)}{I_{−p/2+1}(s)} ≤ \frac{1}{s} (p + k + 2α − 4) .

Also, we can write

\frac{I_{1}(s)}{I_{0}(s)} = s\frac{∫_{0}^{∞} t^{k/2−3} exp (−\frac{t}{2}) h (\frac{s}{t}) dt}{∫_{0}^{∞} t^{k/2−2} exp (−\frac{t}{2}) h (\frac{s}{t}) dt} ,

which imply in turn that

(3.16) g(s) ≤ \frac{1}{s} (p + k + 2α − 4) × s \frac{∫_{0}^{∞} t^{k/2−3} exp (−\frac{t}{2}) h (\frac{s}{t}) dt}{∫_{0}^{∞} t^{k/2−2} exp (−\frac{t}{2}) h (\frac{s}{t}) dt}

≤ (p + k + 2α − 4) lim_{s→∞} \frac{∫_{0}^{∞} t^{k/2−3} exp (−\frac{t}{2}) h (\frac{s}{t}) dt}{∫_{0}^{∞} t^{k/2−2} exp (−\frac{t}{2}) h (\frac{s}{t}) dt} ,

since I_{1}(s)/I_{0}(s) is increasing in s. Now the condition t^{β}h(t) ≤ K for all t > 0, allows us to apply the Lebesgue dominated convergence theorem. Then, according to (3.13), we have

lim_{s→∞} \frac{∫_{0}^{∞} t^{k/2−3}exp (−\frac{t}{2}) h (\frac{s}{t}) dt}{∫_{0}^{∞} t^{k/2−2} exp (−\frac{t}{2}) h (\frac{s}{t}) dt}=\frac{∫_{0}^{∞}t^{k/2+β−3} exp (−\frac{t}{2}) lim_{s→∞} (\frac{s}{t})^{β} h (\frac{s}{t}) dt}{∫_{0}^{∞}t^{k/2+β−2} exp (−\frac{t}{2}) lim_{s→∞} (\frac{s}{t})^{β} h (\frac{s}{t}) dt}

=\frac{1}{2} \frac{Γ (k/2 + β − 2)}{Γ (k/2 + β − 1)}
(3.17)  =\frac{1}{k + 2β − 4} .

Combining (3.16) and (3.17), we have (3.14). □

Finally, according to Lemma 3.1, Inequality 3.11 reduces to

\frac{p + k + 2α − 4}{k + 2β − 4} ≤ 2 . (3.18)

Therefore, we can give general conditions for Bayes estimators to be minimax in this case of sampling distributions.

Corollary 3.1. Assume that a prior π satisfies conditions of the Theorem 2.1. Assume also that f is given by (3.7) such that E_{0} [W^{−p/2}] < ∞ and that the mixing density h satisfy (3.12) and (3.13). Then the corresponding generalized Bayes estimator is minimax provided that (3.18) is satisfied.

Example (S3). Generalized multivariate Student distribution.

We consider the particular case of the inverse gamma mixing, when 1/W ∼ Gamma(α, β) with α > 0 and β > 0. Thus, the mixing density is given by

h (w) ∝ w^{−α−1} exp (−β/w) ,

which in turn implies that

f (v + s) ∝ (\frac{1}{v + s + 2β} )^{(p+k)/2+α} .

Note that, the case of a = 2β and b = \frac{p+k}{2} + α, corresponds to the Example (S1). Also, note that the case of α = β = n/2, corresponds to the multivariate t with n degrees of freedom.

Now to prove minimaxity, we have to apply Corollary 3.1. For that, we have for k/2 + m > −α and for any s > 0

E_{s} [W^{−m}] = \frac{2^{m}}{(s + 2β)^{m}} \frac{Γ(k/2 + m + α)}{Γ(k/2 + α)} < ∞ .

Also, we can see that −t\frac{h′(t)}{h(t)} ≤ α + 1 and lim_{t→∞} t^{α+1}h(t) = c, where c is the normalizing constante of h. Then, the condition (3.18) is satisfied if p ≤ k +2α−2. For the multivariate t distribution with n degrees of freedom, the condition (3.18) is satisfied in the cases when n ≥ p − k + 2.

Example (S4). Kotz distribution.

In this example, we consider the class of spherical distributions with generating function f given by

f (v + s) = K (v + s)^{m} exp (−\frac{v + s}{2}) , (3.19)

where m > −p/2 and K is the constant of normalisation.

Note that, for the Kotz distribution, when m > 0, it is not a variance mixture of normals, while, when m < 0, it is (the mixing distribution is the β(p/2 + m, −m) distribution). However, in the last case, Condition (3.13) cannot be satisfied since the beta distribution has compact support. So, Corollary 3.1 cannot apply.

But, we can see that

\frac{f′ (v + s)}{f (v + s)} = \frac{m}{v + s} − \frac{1}{2} , (3.20)

which is non increasing for m > 0 and non decreasing for m < 0 in v. Thus, we can apply the Theorem 2.1 only in the case m < 0. In this case, we can see that

\frac{F (s)}{f (s)} = \frac{s}{2} ∫_{0}^{∞} (1 + u)^{m} exp (−us/2) du . (3.21)

In order to determine the conditions for which Inequality (2.21) is satisfied, we use integration by parts to obtain

∫_{0}^{∞} (\frac{m}{v + s} − \frac{1}{2}) v^{p/2} (v + s)^{m} exp (−(v + s)/2) du

= −\frac{p}{2} ∫_{0}^{∞} v^{p/2−1} (v + s)^{m} exp (−(v + s)/2) du

= −\frac{p}{2}s^{p/2+m} ∫_{0}^{∞} u^{p/2−1} (1 + u)^{m} exp (−us/2) exp (−s/2) du .

Also, we have

∫_{0}^{∞} v^{p/2} (v + s)^{m} exp (−(v + s)/2) du

= s^{p/2+m+1} ∫_{0}^{∞} u^{p/2} (1 + u)^{m} exp (−us/2) exp (−s/2) du ,

which in turn implies

E [\frac{f′ (v + s)}{f (v + s)}] = −\frac{p}{2s} \frac{∫_{0}^{∞} u^{p/2−1} (1 + u)^{m} exp (−us/2) du}{∫_{0}^{∞} u^{p/2} (1 + u)^{m} exp (−us/2) du} .

Then, Inequality (2.21) reduces to

\frac{J_{0}(s)J_{p/2−1}(s)}{J_{p/2}(s)} ≥ \frac{1}{p} (3.22)

for all s > 0, where J_{β}(s) = ∫_{0}^{∞} u^{β} (1 + u)^{m} exp (−us/2) du. This last integral can be written as

J_{β}(s) = Γ(β + 1)U(β + 1, m + β + 2, s/2) (3.23)

(see [1], page 504-515) where U is the Kummer’s function of the second kind. Now, we will use the following recurrence relation for the function U,

(b − a)U(a, b, z) + U(a − 1, b, z) = zU(a, b + 1, z)

for the case b − a ≤ 0, which imply that U(a − 1, b, z) ≥ zU(a, b + 1, z). In our setting, b − a = m + 1. So if we assume that m + 1 ≤ 0, then

\frac{J_{p/2−1}(s)}{J_{p/2}(s)} = \frac{Γ(\frac{p}{2}) U(\frac{p}{2}, \frac{p}{2} + m + 1, \frac{s}{2})}{Γ(\frac{p}{2} + 1) U(\frac{p}{2} + 1, \frac{p}{2} + m + 2, \frac{s}{2})} ≥ \frac{s}{p} .

On the other hand, J_{0}(s) is a decreasing function in s and for large s it can be written as \frac{2}{s} (1 + O (\frac{1}{s})), so we have for all s > 0

J_{0}(s)\frac{J_{p/2−1}(s)}{J_{p/2}(s)} ≥ \frac{2}{p} (1 + O (\frac{1}{s})) ≥ \frac{2}{p}

Then, the condition 2.21 is satisfied for all −p/2 < m ≤ −1.

Example (S5).

We consider the class of spherical distributions with generating function f given by

f (v + s) = K exp (−α (v + s) + βφ (v + s)) , (3.24)

where α > 0, β > 0, φ(0) < ∞, φ′(t) ≥ −M, where M > 0 and K is the constant of normalisation. Also, we assume that φ is a positive, non increasing and convex function.

Clearly f′(v + s)/f(v + s) = −α + βφ′(v + s) is non decreasing in v. Note that, ∫_{0}^{∞} f (v + s) v^{p/2} dv < ∞ and also ∫_{0}^{∞} −f′ (v + s) v^{p/2} dv < ∞ since |φ′| is bounded. Finally, condition (2.21) becomes

∫_{0}^{∞} f (v + s) v^{p/2} dv ≤ 4αc(s) ∫_{0}^{∞} f (v + s) v^{p/2} dv

+ 4βc(s) ∫_{0}^{∞} −φ′(v + s)f (v + s) v^{p/2} dv ,

which is satisfied if we have 4αc(s) ≥ 1 for all s > 0. Then, as we show below, the generalized Bayes estimator is minimax for any α > 0 and for any function φ satisfying the above conditions with β ≤ α \frac{log(2)}{M}. Indeed, we have for all s > 0

c(s) = \frac{F (s)}{f (s)} = \frac{1}{2α exp (βφ(s))}E_{α} [exp (βφ(u + s))] ,

where E_{α} denotes the expectation with respect to the density α exp(−αu). As the function exp (βφ(u + s)) is convex in u, we have by Jensen inequality

c(s) ≥ \frac{1}{2α exp (βφ(s))} exp (βφ(\frac{1}{α} + s)) .

Hence 4αc(s) ≥ 1 for all s > 0 as soon as

2 exp (β (φ(\frac{1}{α} + s) − φ(s))) ≥ 1

which is equivalent to

β ≤ \frac{log(2)}{φ(s) − φ(\frac{1}{α} + s)} (3.25)

all s > 0. Using the mean value theorem, we have φ(s) − φ(\frac{1}{α} + s) = −φ′(k_{s})\frac{1}{α} ≤ M \frac{1}{α} by assumption on φ. Then, β ≤ α \frac{log(2)}{M} implies (3.25) for all s > 0.

3.2 Examples of priors

Several examples of priors which satisfy the conditions of Theorem 2.1 are given in Fourdrinier and Strawderman [3]. Thus, as a first example, these conditions are satisfied by the following priors π(∥θ∥^{2}) = (A + ∥θ∥^{2})^{−k} when A ≥ 0 and 0 ≤ k ≤ p/2 − 1. Note that the case where A = 0 and k = p/2 − 1 corresponds to the fundamental harmonic prior. Note also that this class of priors arises as a scale mixture of normals.

More generally, the mixture of normals with respect to the inverse of the variance

π(∥θ∥^{2}) = ∫_{0}^{∞} (\frac{u}{2 π} )^{p/2} exp (−\frac{u}{2} ∥θ∥^{2}) h(u) du

satisfy the required conditions as soon as

u \frac{h′(u)}{h(u)} ≤ −2 .

Other examples can be found in [3] including priors which are non necessarily scale mixture of normals.

4 Concluding Remarks

In this work, we have proved minimaxity of generalized Bayes estimators corresponding to superharmonic priors of the form π(∥θ∥)^{2} with non decreasing Laplacian, for a broad class of spherically symmetric distributions when a residual vector is present. This subclass contains certain variance mixtures of normal distributions.

In this paper and in Fourdrinier and Strawderman [3] the minimax generalized Bayes estimators depend on the form of the underlying density. On the other hand, when the scale is unknown and a residual vector is available, the form of the generalized Bayes estimators is independent of the form of the density provided the prior distribution has a certain form, see Fourdrinier and Strawderman [5] and Maruyama [10]. For the model of the current paper however, the form of the Bayes estimator generally depends on the density f(·). In terms of distributional robustness of the generalized Bayes minimax estimators, the results of this paper fall closer to those in [3] then to those in [5] and [10]. In [3] the scale parameter is known and there is no residual vector, wheras in [5] and [10] the scale parameter is unknown but a residual vector is available. Here, of course, we are treating the intermediate case of a known scale parameter, but with the supplemental sample information of a residual vector.

Maruyama [9] and Fourdrinier et al. [2] gave classes of priors including some proper priors for which the resulting Bayes estimators are minimax. In our setting, due to the superharmonicity supposition, our priors cannot be proper (see [6]). Furthermore, in the case of mixtures of normal sampling distributions, our mixing distributions are not required to have monotone likelihood ratio as in the above papers.

References

[1] M. Abramowitz and Irene A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, ninth dover printing, tenth gpo printing edition, 1964.

[2] D. Fourdrinier, O. Kortbi, and W. E. Strawderman. Bayes minimax estimators of the mean of a scale mixture of multivariate normal distributions. J. Multivariate Anal., 99(1):74–93, 2008.

[3] D. Fourdrinier and W E. Strawderman. Generalized bayes minimax estimators of location vectors for spherically symmetric distributions. Journal of Multivariate Analysis, 99:735–750, 2008.

[4] D. Fourdrinier and W. E. Strawderman. A unified and generalized set of shrinkage bounds on minimax stein estimates. Journal of Multivariate Analysis, 99(10):2221–2233, 2008.

[5] D. Fourdrinier and W. E. Strawderman. Robust generalized bayes estimator of location vector for spherically symmetric distribution with unknown scale. 2009.

[6] D. Fourdrinier, W. E. Strawderman, and M. T. Wells. On the construction of bayes minimax estimators. Ann. Statis, 26:660–671, 1998.

[7] D. Fourdrinier, W. E. Strawderman, and M. T. Wells. Robust shrinkage for elliptically symmetric distributions with unknown covariance matrix. Journal of Multivariate Analysis, 85:24–39, 2003.

[8] D. Fourdrinier, W. E. Strawderman, and M. T. Wells. Estimation of a location parameter with restrictions of “vague information” for spherically symmetric distributions. The Annals of the Institute of Statistical Mathematics, 58:73–92, 2006.

[9] Y. Maruyama. Admissible minimax estimators of a mean vector of scale mixtures of multivariate normal distributions. Journal of Multivariate Analysis, 84:274–283, 2003.

[10] Y. Maruyama. A robust generalized bayes estimator improving on the james-stein estimator for spherically symmetric distributions. Statistics and Decisions, 21:69–78, 2003.

[11] W. E. Strawderman. Proper bayes minimax estimators of the multivariate normal mean vector for the case of common unknown variances. The Annals of Statistics, 1:1189–1194, 1973.

[12] W. E. Strawderman. Minimax estimation of location parameters for certain spherically symmetric distribution. Journal of Multivariate Analysis, 4:255–264, 1974.