Ensemble minimaxity of James-Stein estimators

1 Introduction

Let $X \sim N_{p} (θ, Σ)$ where $p \geq 3$ , $θ = (θ_{1}, \dots, θ_{p})^{T}$ and $Σ = d i a g (σ_{1}^{2}, \dots, σ_{p}^{2})$ . Let us assume

σ_{1}^{2} > σ_{2}^{2} > \dots > σ_{p}^{2} .

(1.1)

We are interested in the estimation of $θ$

with respect to the ordinary squared error loss function

L (δ, θ) = ∥ δ - θ ∥^{2},

(1.2)

where the risk of an estimator $δ (X)$ is $R (δ, θ) = E [L (δ, θ)]$ . The MLE $X$ with constant risk $\sum σ_{i}^{2}$ is shown to be extended Bayes and hence minimax for any $p$ and any $Σ$ .

In the homoscedastic case $σ_{1}^{2} = \dots = σ_{p}^{2}$ , James and Stein (1961) showed that the shrinkage estimator

(1.3)

dominates the MLE $X$ for $p \geq 3$ . There is some literature discussing the minimax properties of shrinkage estimators under heteroscedasticity. Brown (1975) showed that the James-Stein estimator (1.3) is not necessarily minimax when the variances are not equal. Specifically, it is not minimax for any $c \in (0, 2 (p - 2))$ when $2 σ_{1}^{2} > \sum_{i = 1}^{p} σ_{i}^{2}$ . Berger (1976) showed that

(I - Σ^{- 1} \frac{c}{X^{T} Σ^{- 2} X}) X for c \in (0, 2 (p - 2))

(1.4)

is minimax for $p \geq 3$ and any $Σ$ . However, Casella (1980) argued that the James-Stein estimator (1.4) may not be desirable even if it is minimax. Ordinary minimax estimators, as in (1.4), typically shrink most on the coordinates with smaller variances. From Casella’s (1980) viewpoint, one of the most natural Jame-Stein variants is

(I - Σ \frac{c}{∥ X ∥^{2}}) X for c > 0,

(1.5)

which we are going to rescue, by providing some minimax properties related to Bayesian viewpoint.

In many applications, $θ_{i}$ are thought to follow some exchangeable prior distribution $π$ . It is then natural to consider the compound risk function which is then the Bayes risk with respect to the prior $π$

¯ R (δ, π) = \int_{R^{p}} R (θ, δ) π (d θ) .

(1.6)

Efron and Morris (1971); Efron and Morris (1972a, b, 1973) addressed this problem from both the Bayes and empirical Bayes perspective. In particular, they considered a prior distribution $θ \sim N_{p} (0, τ I_{p})$ with $τ \in (0, \infty)$ , and used the term “ensemble risk” for the compound risk. By introducing a set of ensemble risks

¯ R (δ, τ) = \int_{R^{p}} R (δ, θ) \frac{}{1} (2 π τ)^{p / 2} exp (- \frac{∥ θ ∥^{2}}{2 τ}) d θ,

(1.7)

we can define ensemble minimaxity with respect to a set of priors

P_{⋆} = {N_{p} (0, τ I_{p}) : τ \in (0, \infty)},

(1.8)

that is, an estimator $δ$ is said to be ensemble minimax with respect to $P_{⋆}$ if

sup τ \in (0, \infty) ¯ R (δ, τ) = inf δ^{'} sup τ \in (0, \infty) ¯ R (δ^{'}, τ) .

(1.9)

As a matter of fact, the second author in his unpublished manuscript, Brown, Nie and Xie (2011), has already introduced the concept of ensemble minimaxity. In this article, we follow their spirit but propose a simpler and clearer approach for establishing ensemble minimaxity of estimators.

Our article is organized as follows. In Section 2, we elaborate the definition of ensemble minimaxity and explain Casella’s (1980) viewpoint on the contradiction between minimaxity and well-conditioning. In Section 3, we show the ensemble minimaxity of various shrinkage estimators including a variant of the James-Stein estimator

(I - Σ \frac{p - 2}{(p - 2) σ_{1}^{2} + ∥ X ∥^{2}}) X

(1.10)

as well as the generalized Bayes estimator with respect to the hierarchical prior

θ | λ \sim N_{p} (0, (σ_{1}^{2} / λ) I - Σ), π (λ) \sim λ^{- 2} I_{(0, 1)} (λ)

(1.11)

which is a generalization of the harmonic prior $∥ θ ∥^{2 - p}$ for the heteroscedastic case.

2 Minimaxity, Ensemble Minimaxity and Casella’s viewpoint

If the prior $π (θ)$ were known, the resulting posterior mean $E [θ | x]$ would then be the optimal estimate under the sum of the squared error loss. However, it is typically not feasible to exactly specify the prior. One approach to avoid excessive dependence on the choice of prior, is to consider a set of priors $P$ on $Θ$ and study the properties of estimators based on the corresponding set of ensemble risks. As in classical decision theory, there rarely exists an estimator that achieves the minimum ensemble risk uniformly for all $π \in P$ . A more realistic goal as pursued in this paper is to study the ensemble minimaxity of James-Stein type estimators.

Recall that with ordinary risk $R (δ, θ)$ , $δ$ is said to be minimax if

sup θ \in Θ R (δ, θ) = inf δ^{'} sup θ \in Θ R (δ^{'}, θ) .

(2.1)

Similarly for the case of ensemble risk we have the following definition. Note the Bayes risk of $δ$ under the prior $π$ is given by (1.6). The estimator $δ$ is said to be ensemble minimax with respect to $P$ if

sup π \in P ¯ R (δ, π) = inf δ^{'} sup π \in P ¯ R (δ^{'}, π) .

(2.2)

The motivation for the above definitions comes from the use of the empirical Bayes method in simultaneous inference.

Efron and Morris (1972b), derived the James-Stein estimator through the parametric empirical Bayes model with

θ \sim N_{p} (0, τ I_{p})

. Note that in such an empirical Bayes model,

τ

is the unknown non-random parameter. Given the family

P_{⋆} = {N_{p} (0, τ I_{p}) : τ \in (0, \infty)}

, the Bayes risk is a function of

τ

as follows,

¯ R (δ, τ) = \int_{R^{p}} R (δ, θ) \frac{}{1} (2 π τ)^{p / 2} exp (- \frac{∥ θ ∥^{2}}{2 τ}) d θ .

(2.3)

Hence, with $¯ R (δ, τ)$ , the estimator $δ$ is said to be ensemble minimax with respect to $P_{⋆}$ if

sup τ \in (0, \infty) ¯ R (δ, τ) = inf δ^{'} sup τ \in (0, \infty) ¯ R (δ^{'}, τ),

(2.4)

which may be seen as the counterpart of ordinary minimaxity in the empirical Bayes model.

Clearly the usual estimator $X$ has constant risk, has constant Bayes risk and hence $X$ is ensemble minimax. Then the ensemble minimaxity of $δ$ follows if

¯ R (δ, τ) \leq p \sum i = 1 σ_{i}^{2}, \forall τ \in (0, \infty) .

Remark 2.1.

Note that ensemble minimaxity can also be interpreted as a particular case of Gamma minimaxity studied in the context of robust Bayes analysis by Good (1952); Berger (1979). However, in such studies, a “large” set consisting of many diffuse priors are usually included in the analysis. Since this is quite different from our formulation of the problem, we use the term ensemble minimaxity throughout our paper, following the Efron and Morris papers cited above.

A class of shrinkage estimators which we consider in this paper, is given by

δ_{ϕ} = (I - G \frac{ϕ (z)}{z}) x, % for z = x^{T} G Σ^{- 1} x = \sum \frac{g_{i} x_{i}^{2}}{σ_{i}^{2}},

(2.5)

where $G = diag (g_{1}, \dots, g_{p})$ with

0 < g_{i} \leq 1, \forall i .

Berger and Srinivasan (1978) showed, in their Corollary 2.7, that, given positive-definite $C$ and non-singular $B$ , a necessary condition for an estimator of the form

(I - B \frac{ϕ (x^{T} C x)}{x^{T} C x}) x

to be admissible is $B = k Σ C$ for some constant $k$ , which is satisfied by estimators among the class of (2.5).

A version of Baranchik’s (1964) sufficient condition for ordinary minimaxity is given in Appendix A; For given $G$ which satisfies

h (Σ, G) = 2 (\frac{\sum g_{i} σ_{i}^{2}}{max (g_{i} σ_{i}^{2})} - 2) > 0,

$δ_{ϕ}$ given by (2.5) is ordinary minimax if

ϕ (\cdot) is non-decreasing and 0 \leq ϕ \leq h (Σ, G) .

(2.6)

Berger (1976) showed that, for any given $Σ$ ,

max G h (Σ, G) = 2 (p - 2), a r g m a x G h (Σ, G) = σ_{p}^{2} Σ^{- 1} = d i a g (\frac{σ_{p}^{2}}{σ_{1}^{2}}, \dots, \frac{σ_{p}^{2}}{σ_{p - 1}^{2}}, 1)

which seems the right choice of $G$ . However, from the “conditioning” viewpoint of Casella (1980) which advocates more shrinkage on higher variance estimates, the descending order

g_{1} > \dots > g_{p}

(2.7)

is desirable, whereas $G = σ_{p}^{2} Σ^{- 1}$ corresponding to the ascending order $g_{1} < \dots < g_{p}$ under $Σ$ given by (1.1). As Casella (1980) pointed out, ordinary minimaxity cannot be enjoyed together with well-conditioning given by (2.7) when

h (Σ, c I) \leq 0 or equivalently \sum σ_{i}^{2} \leq 2 σ_{1}^{2}

for some $0 < c \leq 1$ . In fact, when $h (Σ, c I) \leq 0$ and $c = g_{1} > \dots > g_{p}$ , we have

c σ_{1}^{2} = g_{1} σ_{1}^{2}, c σ_{2}^{2} > g_{2} σ_{2}^{2}, \dots, c σ_{p}^{2} > g_{p} σ_{p}^{2}

and hence $h (Σ, G) < 0$ follows. The motivation of Casella (1980, 1985) seems to provide a better treatment for the case. Actually Brown (1975) pointed out essentially the same phenomenon from a slightly different viewpoint.

Ensemble minimaxity, based on ensemble risk given by (1.7), provides a way of saving shrinkage estimators with well-conditioning, estimators which are not necessarily ordinary minimax.

3 Ensemble minimaxity

3.1 A general theorem

We have the following theorem on ensemble minimaxity of $δ_{ϕ}$ with general $G$ , though we will eventually focus on $δ_{ϕ}$ with $G$ with the descending order $g_{1} > \dots > g_{p}$ as in (2.7).

Theorem 3.1.

Assume $ϕ (z)$ is non-negative, non-decreasing and concave. Also $ϕ (z) / z$ is assumed non-increasing. Then

δ_{ϕ} = (I - G \frac{ϕ (z)}{z}) x, % for z = x^{T} G Σ^{- 1} x = \sum \frac{g_{i} x_{i}^{2}}{σ_{i}^{2}}

is ensemble minimax if

ϕ (p min i g_{i} (1 + τ / σ_{i}^{2})) \leq 2 (p - 2) \frac{{min}_{i} g_{i} (1 + τ / σ_{i}^{2})}{{max}_{i} g_{i} (1 + τ / σ_{i}^{2})}, \forall τ \in (0, \infty) .

(3.1)

Proof.

Recall, for $i = 1, \dots, p$ ,

x_{i} | θ_{i} \sim N (θ_{i}, σ_{i}^{2}), and% θ_{i} \sim N (0, τ) .

Then the posterior and marginal are given by

θ_{i} | x_{i} \sim N (\frac{τ}{τ + σ_{i}^{2}} x_{i}, \frac{τ σ_{i}^{2}}{τ + σ_{i}^{2}}) and x_{i} \sim N (0, τ + σ_{i}^{2}),

respectively, where $θ_{1} | x_{1}, \dots, θ_{p} | x_{p}$ are mutually independent and $x_{1}, \dots, x_{p}$ are mutually independent. Then the Bayes risk is given by

	$¯ R (δ_{ϕ}, τ)$	$= p \sum i = 1 E_{θ} E_{x \| θ} ⎡ ⎣ {(1 - g_{i} \frac{ϕ (z)}{z}) x_{i} - θ_{i}}_{i}^{2} ⎤ ⎦$
		$= p \sum i = 1 E_{x} E_{θ \| x} ⎡ ⎣ {(1 - g_{i} \frac{ϕ (z)}{z}) x_{i} - θ_{i}}_{i}^{2} ⎤ ⎦$
		$= p \sum i = 1 E_{x} E_{θ \| x} ⎡ ⎣ {(1 - g_{i} \frac{ϕ (z)}{z}) x_{i} - E [θ_{i} \| x_{i}] + E [θ_{i} \| x_{i}] - θ_{i}}_{i}^{2} ⎤ ⎦$
		$= p \sum i = 1 E_{x} ⎡ ⎣ {(1 - g_{i} \frac{ϕ (z)}{} z) x_{i} - E [θ_{i} \| x_{i}]}_{i}^{2} ⎤ ⎦ + p \sum i = 1 V a r (θ_{i} \| x_{i})$
		$= p \sum i = 1 E_{x} ⎡ ⎣ {(\frac{σ_{i}^{2}}{τ + σ_{i}^{2}} x_{i} - g_{i} \frac{ϕ (z)}{z} x_{i})}_{i}^{2} ⎤ ⎦ + p \sum i = 1 \frac{τ σ_{i}^{2}}{τ + σ_{i}^{2}} .$

Since the first term of the r.h.s. of the above equality is rewritten as

	$p \sum i = 1 E_{x} ⎡ ⎣ {(\frac{σ_{i}^{2}}{τ + σ_{i}^{2}} x_{i} - g_{i} \frac{ϕ (z)}{z} x_{i})}_{i}^{2} ⎤ ⎦$
	$= p \sum i = 1 {(\frac{σ_{i}^{2}}{τ + σ_{i}^{2}})}^{2} E_{x} [x_{i}^{2}] - 2 E_{x} [p \sum i = 1 \frac{σ_{i}^{2} g_{i} x_{i}^{2}}{τ + σ_{i}^{2}} \frac{ϕ (z)}{z}] + E_{x} [p \sum i = 1 g_{i}^{2} x_{i}^{2} \frac{ϕ^{2} (z)}{z^{2}}]$
	$= p \sum i = 1 \frac{σ_{i}^{4}}{τ + σ_{i}^{2}} - 2 E_{x} [p \sum i = 1 \frac{σ_{i}^{2} g_{i} x_{i}^{2}}{τ + σ_{i}^{2}} \frac{ϕ (z)}{z}] + E_{x} [p \sum i = 1 g_{i}^{2} x_{i}^{2} \frac{ϕ^{2} (z)}{z^{2}}],$

we have

¯ R (δ_{ϕ}, τ) - \sum σ_{i}^{2} = - 2 E_{x} [p \sum i = 1 \frac{σ_{i}^{2} g_{i} x_{i}^{2}}{τ + σ_{i}^{2}} \frac{ϕ (z)}{z}] + E_{x} [p \sum i = 1 g_{i}^{2} x_{i}^{2} \frac{ϕ^{2} (z)}{z^{2}}] .

(3.2)

Let

w_{i} = \frac{x_{i}^{2}}{σ_{i}^{2} + τ}, w = p \sum i = 1 w_{i} and t_{i} = \frac{w_{i}}{w} for i = 1, \dots, p .

Then

w \sim χ_{p}^{2}, t = (t_{1}, \dots, t_{p})^{T} \sim D i r i c h l e t (1 / 2, \dots, 1 / 2),

and $w$ and $t$ are mutually independent. With the notation, we have

x_{i}^{2} = w t_{i} (σ_{i}^{2} + τ) and z = x^{T} G Σ^{- 1} x = p \sum i = 1 \frac{g_{i} x_{i}^{2}}{σ_{i}^{2}} = w p \sum i = 1 t_{i} g_{i} (1 + \frac{τ}{σ_{i}^{2}})

and hence

	$E_{x} [\sum g_{i}^{2} x_{i}^{2} \frac{ϕ^{2} (z)}{z^{2}}]$	$= E_{w, t} [\frac{\sum t_{i} g_{i}^{2} (σ_{i}^{2} + τ)}{\sum t_{i} g_{i} (1 + τ / σ_{i}^{2})} \frac{ϕ (w \sum t_{i} g_{i} (1 + τ / σ_{i}^{2}))^{2}}{w \sum t_{i} g_{i} (1 + τ / σ_{i}^{2})}]$
		$= E_{t} [\frac{\sum t_{i} g_{i}^{2} (σ_{i}^{2} + τ)}{\sum t_{i} g_{i} (1 + τ / σ_{i}^{2})} E_{w \| t} [\frac{ϕ (w \sum t_{i} g_{i} (1 + τ / σ_{i}^{2}))^{2}}{w \sum t_{i} g_{i} (1 + τ / σ_{i}^{2})}]] .$

Since $ϕ (w) / w$ is non-increasing and $ϕ (w)$ is non-decreasing, by the correlation inequality, we have

	$E_{w \| t} [\frac{ϕ (w \sum t_{i} g_{i} (1 + τ / σ_{i}^{2}))^{2}}{w \sum t_{i} g_{i} (1 + τ / σ_{i}^{2})}]$
	$\leq E_{w \| t} [ϕ (w \sum t_{i} g_{i} (1 + τ / σ_{i}^{2}))] E_{w \| t} [\frac{ϕ (w \sum t_{i} g_{i} (1 + τ / σ_{i}^{2}))}{w \sum t_{i} g_{i} (1 + τ / σ_{i}^{2})}]$
	$\leq E_{w \| t} [ϕ (w \sum t_{i} g_{i} (1 + τ / σ_{i}^{2}))] E_{w} [\frac{ϕ (w min g_{i} (1 + τ / σ_{i}^{2}))}{w min g_{i} (1 + τ / σ_{i}^{2})}],$

and hence

\begin{matrix} E_{x} [\sum g_{i}^{2} x_{i}^{2} \frac{ϕ^{2} (z)}{z^{2}}] \leq E_{w} [\frac{ϕ (w min g_{i} (1 + τ / σ_{i}^{2}))}{w min g_{i} (1 + τ / σ_{i}^{2})}] \times E_{t} [\frac{\sum t_{i} g_{i}^{2} (σ_{i}^{2} + τ)}{\sum t_{i} g_{i} (1 + τ / σ_{i}^{2})} E_{w | t} [ϕ (w \sum t_{i} g_{i} (1 + τ / σ_{i}^{2}))]] . \end{matrix}

(3.3)

In the first part of the r.h.s. of the inequality (3.3), we have

\begin{matrix} E_{w} [\frac{ϕ (w min g_{i} (1 + τ / σ_{i}^{2}))}{w}] & \leq E_{w} [1 / w] E_{w} [ϕ (w min i g_{i} (1 + τ / σ_{i}^{2}))] \leq \frac{ϕ (E_{w} [w] {min}_{i} g_{i} (1 + τ / σ_{i}^{2}))}{p - 2} = \frac{ϕ (p {min}_{i} g_{i} (1 + τ / σ_{i}^{2}))}{p - 2}, \end{matrix}

(3.4)

where the first and second inequality follow from the correlation inequality and Jensen’s inequality, respectively. In the second part of the r.h.s. of the inequality (3.3), by the inequality

\sum t_{i} g_{i}^{2} (σ_{i}^{2} + τ) \leq max i g_{i} (1 + τ / σ_{i}^{2}) \sum t_{i} g_{i} σ_{i}^{2},

we have

\begin{matrix} E_{t} [\frac{\sum t_{i} g_{i}^{2} (σ_{i}^{2} + τ)}{\sum t_{i} g_{i} (1 + τ / σ_{i}^{2})} E_{w | t} [ϕ (w \sum t_{i} g_{i} (1 + τ / σ_{i}^{2}))]] = E_{w, t} [\frac{\sum t_{i} g_{i}^{2} (σ_{i}^{2} + τ)}{\sum t_{i} g_{i} (1 + τ / σ_{i}^{2})} ϕ (w \sum t_{i} g_{i} (1 + τ / σ_{i}^{2}))] \leq max i g_{i} (1 + τ / σ_{i}^{2}) E_{w, t} [\frac{\sum t_{i} g_{i} σ_{i}^{2}}{\sum t_{i} g_{i} (1 + τ / σ_{i}^{2})} ϕ (w \sum t_{i} g_{i} (1 + τ / σ_{i}^{2}))] = max i g_{i} (1 + τ / σ_{i}^{2}) E_{x} [\sum \frac{σ_{i}^{2} g_{i} x_{i}^{2}}{τ + σ_{i}^{2}} \frac{ϕ (z)}{z}] . \end{matrix}

(3.5)

By (3.3), (3.4) and (3.5), we have

\begin{matrix} E_{x} [\sum g_{i}^{2} x_{i}^{2} \frac{ϕ^{2} (z)}{z^{2}}] \leq \frac{ϕ (p {min}_{i} g_{i} (1 + τ / σ_{i}^{2}))}{p - 2} \frac{}{{max}_{i} g_{i} (1 + τ / σ_{i}^{2})} min i g_{i} (1 + τ / σ_{i}^{2}) E_{x} [\sum \frac{σ_{i}^{2} g_{i} x_{i}^{2}}{τ + σ_{i}^{2}} \frac{ϕ (z)}{z}], \end{matrix}

(3.6)

and, by (3.2) and (3.6),

	$¯ R (δ_{ϕ}, τ) - \sum σ_{i}^{2}$
	$\leq (\frac{ϕ (p {min}_{i} g_{i} (1 + τ / σ_{i}^{2}))}{p - 2} \frac{{max}_{i} g_{i} (1 + τ / σ_{i}^{2})}{{min}_{i} g_{i} (1 + τ / σ_{i}^{2})} - 2) E_{x} [\sum \frac{σ_{i}^{2} g_{i} x_{i}^{2}}{τ + σ_{i}^{2}} \frac{ϕ (z)}{z}],$

which guarantees $¯ R (δ_{ϕ}, τ) - \sum σ_{i}^{2} \leq 0$ for all $τ \in (0, \infty)$ under the condition (3.1). ∎

Given $Σ$ , the choice $G = Σ / σ_{1}^{2}$ with descending order $g_{1} > \dots > g_{p}$ , is one of the most natural choice of $G$ from Casella’s (1980) viewpoint. In this case, we have

	$\frac{{min}_{i} g_{i} (1 + τ / σ_{i}^{2})}{{max}_{i} g_{i} (1 + τ / σ_{i}^{2})} = \frac{{min}_{i} {σ_{i}^{2} + τ}}{{max}_{i} {σ_{i}^{2} + τ}} = \frac{σ_{p}^{2} + τ}{σ_{1}^{2} + τ},$
	$p min i g_{i} (1 + τ / σ_{i}^{2}) = \frac{p}{σ_{1}^{2}} min i (σ_{i}^{2} + τ) = p \frac{σ_{p}^{2} + τ}{σ_{1}^{2}},$

and hence a following corollary.

Corollary 3.1.

Assume that $ϕ (z)$ is non-negative, non-decreasing and concave. Also $ϕ (z) / z$ is assumed non-increasing. Then

δ_{ϕ} = (I - Σ \frac{ϕ (∥ x ∥^{2} / σ_{1}^{2})}{∥ x ∥^{2}}) x

is ensemble minimax if

ϕ (p (σ_{p}^{2} + τ) / σ_{1}^{2}) \leq 2 (p - 2) \frac{σ_{p}^{2} + τ}{σ_{1}^{2} + τ}, \forall τ \in (0, \infty) .

(3.7)

3.2 An ensemble minimax James-Stein variant

As an example of Corollary 3.1, we consider

ϕ (z) = \frac{c_{1} z}{c_{2} + z}

(3.8)

for $c_{1} > 0$ and $c_{2} \geq 0$ , which is motivated by Stein (1956) and James and Stein (1961). Under $Σ = I_{p}$ , Stein (1956) suggested that there exist estimators dominating the usual estimator $x$ among a class of estimators $δ_{ϕ}$ with $ϕ$ given by (3.8) for small $c_{1}$ and large $c_{2}$ . Following Stein (1956), James and Stein (1961) showed that $δ_{ϕ}$ with $0 < c_{1} < 2 (p - 2)$ and $c_{2} = 0$ is ordinary minimax. The choice $c_{2} = 0$ is, however, not good since, by Corollary 3.1, $c_{1}$ cannot be larger than $2 (p - 2) σ_{p}^{2} / σ_{1}^{2}$ . With positive $c_{2}$ , we can see that $c_{1}$ can be much larger as follows.

Note that $ϕ (z)$ given by (3.8) is non-negative, increasing and concave and that $ϕ (z) / z$ is decreasing. Then the sufficient condition in (3.7) is

\frac{c_{1} p (σ_{p}^{2} + τ) / σ_{1}^{2}}{c_{2} + p (σ_{p}^{2} + τ) / σ_{1}^{2}} \leq 2 (p - 2) \frac{σ_{p}^{2} + τ}{σ_{1}^{2} + τ} \forall τ \in (0, \infty),

which is equivalent to

2 (p - 2) {σ_{1}^{2} c_{2} + p (σ_{p}^{2} + τ)} - c_{1} p (σ_{1}^{2} + τ) \geq 0 \forall τ \in (0, \infty)

p τ {2 (p - 2) - c_{1}} + 2 (p - 2) σ_{1}^{2} {c_{2} - p (\frac{c_{1}}{2 (p - 2)} - \frac{σ_{p}^{2}}{σ_{1}^{2}})} \geq 0 \forall τ \in (0, \infty) .

Hence we have a following result.

Theorem 3.2.

When

$0 < c_{1} \leq 2 (p - 2) and c_{2} \geq max (0, p (\frac{c_{1}}{2 (p - 2)} - \frac{σ_{p}^{2}}{σ_{1}^{2}})),$ (3.9)

the shrinkage estimator

$(I - Σ \frac{c_{1}}{c_{2} σ_{1}^{2} + ∥ x ∥^{2}}) x$

is ensemble minimax.
It is ordinary minimax if

$2 (\sum σ_{i}^{4} / σ_{1}^{4} - 2) \geq c_{1} .$

Part 2 above follows from Theorem A.1.

It seems to us that one of the most interesting estimators with ensemble minimaxity from Part 1 is

(I - Σ \frac{p - 2}{(p - 2) σ_{1}^{2} + ∥ x ∥^{2}}) x

(3.10)

with the choice $c_{1} = c_{2} = p - 2$ satisfying (3.9). It is clear that the $i$ -th shrinkage factor

1 - \frac{(p - 2) σ_{i}^{2}}{(p - 2) σ_{1}^{2} + ∥ x ∥^{2}}

is nonnegative for any $x$ and any $Σ$ , which is a nice property.

3.3 A generalized Bayes ensemble minimax estimator

In this subsection, we provide a generalized Bayes ensemble minimax estimator. Following Strawderman (1971), Berger (1976) and Maruyama and Strawderman (2005), we consider the generalized harmonic prior

θ | λ \sim N_{p} (0, λ^{- 1} Σ G^{- 1} - Σ), π (λ) \sim λ^{- 2} I_{(0, 1)} (λ)

(3.11)

where $G = diag (g_{1}, \dots, g_{p})$ satisfies $0 < \forall g_{i} \leq 1$ . Note that for $Σ = G = I_{p}$ , the density of $θ$ is exactly $π (θ) = ∥ θ ∥^{2 - p}$ , since $λ^{- 1} Σ G^{- 1} - Σ = {(1 - λ) / λ} I_{p}$ and

	$\frac{1}{(2 π)^{p / 2}} \int_{0}^{1} {(\frac{λ}{1 - λ})}^{p / 2} exp (- \frac{λ ∥ θ ∥^{2}}{2 (1 - λ)}) λ^{- 2} d λ$
	$= \frac{1}{(2 π)^{p / 2}} \int_{0}^{\infty} g^{p / 2 - 2} exp (- g ∥ θ ∥^{2} / 2) d g$
	$= \frac{Γ (p / 2 - 1) 2^{p / 2 - 1}}{(2 π)^{p / 2}} ∥ θ ∥^{2 - p} .$

The prior $π (θ) = ∥ θ ∥^{2 - p}$ is called the harmonic prior and was originally investigated by Baranchik (1964) and Stein (1974). Berger (1980) and Berger and Strawderman (1996) recommended the use of the prior (3.11) mainly because it is on the boundary of admissibility.

By the way of Strawderman (1971), the generalized Bayes estimator with respect to the prior is given by

δ_{*} = (I - G \frac{ϕ_{*} (z)}{z}) x, % for z = x^{T} G Σ^{- 1} x

(3.12)

with

ϕ_{*} (z) = z \frac{\int_{0}^{1} λ^{p / 2 - 1} exp (- z λ / 2) d λ}{\int_{0}^{1} λ^{p / 2 - 2} exp (- z λ / 2) d λ},

where $ϕ_{*} (z)$ satisfies the following properties

[label= H0]
$ϕ_{*} (z)$ is increasing in $z$ .
$ϕ_{*} (z)$ is concave.
${lim}_{z \to \infty} ϕ_{*} (z) = p - 2$ .
$ϕ_{*} (z) / z$ is decreasing in $z$ .
The derivative of $ϕ_{*} (z)$ at $z = 0$ is $(p - 2) / p$ .

Under the choice $G = Σ / σ_{1}^{2}$ and with the condition of Corollary 3.1, we have a following result.

Theorem 3.3.

The estimator $δ_{*}$ is ensemble minimax.
The estimator $δ_{*}$ is ordinary minimax when

$2 (\sum σ_{i}^{4} / σ_{1}^{4} - 2) \geq p - 2.$
The estimator $δ_{*}$ is conventional admissible.

Proof.

[Part 1] Recall that the sufficient condition for ensemble minimaxity is given by Corollary 3.1. By 1–5, we have only to check (3.7) in Corollary 3.1.

For $τ \geq max (0, σ_{1}^{2} - 2 σ_{p}^{2})$ , we have

2 (p - 2) \frac{σ_{p}^{2} + τ}{σ_{1}^{2} + τ} \geq p - 2.

By the properties 1 and 3,

ϕ_{*} (p (σ_{p}^{2} + τ) / σ_{1}^{2}) \leq p - 2.

for $τ \in (0, \infty)$ . Hence for $τ \geq max (0, σ_{1}^{2} - 2 σ_{p}^{2})$ , it follows that

ϕ_{*} (p (σ_{p}^{2} + τ) / σ_{1}^{2}) \leq 2 (p - 2) \frac{σ_{p}^{2} + τ}{σ_{1}^{2} + τ} .

So it suffices to show

ϕ_{*} (p (σ_{p}^{2} + τ) / σ_{1}^{2}) \leq 2 (p - 2) \frac{σ_{p}^{2} + τ}{σ_{1}^{2} + τ}

when $σ_{1}^{2} - 2 σ_{p}^{2} > 0$ and $0 < τ < σ_{1}^{2} - 2 σ_{p}^{2}$ . By the properties 2 and 5, we have $ϕ_{*} (z) \leq {(p - 2) / p} z$ for all $z \geq 0$ . Then

	$¯ R (δ_{ϕ}, τ)$	$= p \sum i = 1 E_{θ} E_{x \| θ} ⎡ ⎣ {(1 - g_{i} \frac{ϕ (z)}{z}) x_{i} - θ_{i}}_{i}^{2} ⎤ ⎦$
		$= p \sum i = 1 E_{x} E_{θ \| x} ⎡ ⎣ {(1 - g_{i} \frac{ϕ (z)}{z}) x_{i} - θ_{i}}_{i}^{2} ⎤ ⎦$
		$= p \sum i = 1 E_{x} E_{θ \| x} ⎡ ⎣ {(1 - g_{i} \frac{ϕ (z)}{z}) x_{i} - E [θ_{i} \| x_{i}] + E [θ_{i} \| x_{i}] - θ_{i}}_{i}^{2} ⎤ ⎦$
		$= p \sum i = 1 E_{x} ⎡ ⎣ {(1 - g_{i} \frac{ϕ (z)}{} z) x_{i} - E [θ_{i} \| x_{i}]}_{i}^{2} ⎤ ⎦ + p \sum i = 1 V a r (θ_{i} \| x_{i})$
		$= p \sum i = 1 E_{x} ⎡ ⎣ {(\frac{σ_{i}^{2}}{τ + σ_{i}^{2}} x_{i} - g_{i} \frac{ϕ (z)}{z} x_{i})}_{i}^{2} ⎤ ⎦ + p \sum i = 1 \frac{τ σ_{i}^{2}}{τ + σ_{i}^{2}} .$

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Enhanced Balancing of Bias-Variance Tradeoff in Stochastic Estimation: A Minimax Perspective

1 Introduction

2 Minimaxity, Ensemble Minimaxity and Casella’s viewpoint

Remark 2.1.

3 Ensemble minimaxity

3.1 A general theorem

Theorem 3.1.

Proof.

Corollary 3.1.

3.2 An ensemble minimax James-Stein variant

Theorem 3.2.

3.3 A generalized Bayes ensemble minimax estimator

Theorem 3.3.

Proof.

Ensemble minimaxity of James-Stein estimators

Related Research

Admissible estimators of a multivariate normal mean vector when the scale is unknown

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Enhanced Balancing of Bias-Variance Tradeoff in Stochastic Estimation: A Minimax Perspective

1 Introduction

2 Minimaxity, Ensemble Minimaxity and Casella’s viewpoint

Remark 2.1.

3 Ensemble minimaxity

3.1 A general theorem

Theorem 3.1.

Proof.

Corollary 3.1.

3.2 An ensemble minimax James-Stein variant

Theorem 3.2.

3.3 A generalized Bayes ensemble minimax estimator

Theorem 3.3.

Proof.