Admissibility and Minimaxity in Exponential Families

Lemma 2.1 has been extended by Berger (1990a) to include the case where the estimand need not be restricted to a finite interval, but, instead, attains a maximum or minimum at a finite parameter value.

Lemma 8.1 Let the estimand $g(\theta )$ be nonconstant with global maximum or minimum at a point $\theta^{*} \in \Omega$ for which $f\left(x \mid \theta^{*}\right)>0$ a.e. (with respect to a dominating measure $\mu )$, and let the loss $L(\theta , d)$ satisfy the assumptions of Lemma 2.1. Then, any estimator $\delta$ taking values above the maximum of $g(\theta )$, or below the minimum, is inadmissible.

Determine the Bayes risk of the estimator (2.4) when $\theta$ has the prior distribution $N\left(\mu , \tau^{2}\right)$.

Prove part (d) in the second proof of Example 2.8, that there exists a sequence of values $\theta_{i} \rightarrow -\infty$ with $b^{\prime }\left(\theta_{i}\right) \rightarrow 0$.

Show that an estimator $a X+b(0 \leq a \leq 1)$ of $\mathbb {E}_{\theta }\left[X\right]$ is inadmissible (with squared error loss) under each of the following conditions:

Show that an estimator $[1 /(1+\lambda )+\varepsilon ] X$ of $\mathbb {E}_{\theta }\left[X\right]$ is inadmissible (with squared error loss) under each of the following conditions:

Show that if $\operatorname {var}_{\theta }(X) / E_{\theta }^{2}(X)>\lambda >0$, an estimator $[1 /(1+\lambda )+\varepsilon ] X+b$ is inadmissible (with squared error loss) under each of the following conditions:

Brown (1986a) points out a connection between the information inequality and the unbiased estimator of the risk of Stein-type estimators.

A density function $f(x \mid \theta )$ is variation reducing of order $n+1\left(V R_{n+1}\right)$ if, for any function $g(x)$ with $k(k \leq n)$ sign changes (ignoring zeros), the expectation $\mathbb {E}_{\theta }\left[g(X)\right]= \int g(x) f(x \mid \theta ) d x$ has at most $k$ sign changes. If $\mathbb {E}_{\theta }\left[g(X)\right]$ has exactly $k$ sign changes, they are in the same order. Show that $f(x \mid \theta )$ is $V R_{2}$ if and only if it has monotone likelihood ratio. (See TSH2, Lemma 2, Section 3.3 for the "if" implication). Brown et al. (1981) provide a thorough introduction to this topic, including $V R$ characterizations of many families of distributions (the exponential family is $V R_{\infty }$, as is the $\chi_{v}^{2}$ with $\nu$ the parameter, and the noncentral $\chi_{\nu }^{2}(\lambda )$ in $\lambda$ ). There is an equivalence between $V R_{n}$ and $T P_{n}$, Karlin's (1968) total positivity of order $n$, in that $V R_{n}=T P_{n}$.

For the situation of Example 2.9, show that:

For the situation of Example 2.10, show that:

Suppose $X$ has distribution $F_{\xi }$ and $Y$ has distribution $G_{\eta }$, where $\xi$ and $\eta$ vary independently. If it is known that $\eta =\eta_{0}$, then any estimator $\delta (X, Y)$ can be improved upon by

In Example 2.13, prove that the estimator $a Y+b$ is inadmissible when $a>1 /(r+1)$.

Let $X_{1}, \ldots , X_{n}$ be iid according to a $N\left(0, \sigma^{2}\right)$ density, and let $S^{2}=\sum X_{i}^{2}$. We are interested in estimating $\sigma^{2}$ under squared error loss using linear estimators $c S^{2}+d$, where $c$ and $d$ are constants. Show that:

For the situation of Example 2.15, let $Z=\bar{X} / S$.

Show the equivalence of the following relationships:

In Example 2.17, show that the estimator $a X / n+b$ is inadmissible for all $(a, b)$ outside the triangle (2.39).

Prove admissibility of the estimators corresponding to the interior of the triangle (2.39), by applying Theorem 2.4 and using the results of Example 4.1.5.

Use Theorem 2.14 to provide an alternative proof for the admissibility of the estimator $a \bar{X}+b$ satisfying (2.6), in Example 2.5.

Determine which estimators $a X+b$ are admissible for estimating $\mathbb {E}\left[X\right]$ in the following situations, for squared error loss:

Let $X$ have the $\operatorname {Poisson}(\lambda )$ distribution, and consider the estimation of $\lambda$ under the loss $(d-\lambda )^{2} / \lambda$ with the restriction $0 \leq \lambda \leq m$, where $m$ is known.

Show that the conditions (2.41) and (2.42) of Example 2.22 are not only sufficient but also necessary for admissibility of (2.40).

Let $X$ and $Y$ be independently distributed according to Poisson distributions with $\mathbb {E}\left[X\right]=\xi$ and $\mathbb {E}\left[Y\right]=\eta$, respectively. Show that $a X+b Y+c$ is admissible for estimating $\xi$ with squared error loss if and only if either $0 \leq a<1, b \geq 0, c \geq 0$ or $a=1, b=c=0$ (Makani 1972).

Let $X$ be distributed with density $\frac{1}{2} \beta (\theta ) e^{\theta x} e^{-\left|x\right|},\left|\theta \right|<1$.

Let $X$ be distributed as $N(\theta , 1)$ and let $\theta$ have the improper prior density $\pi (\theta )=e^{\theta }$ $(-\infty <\theta <\infty )$. For squared error loss, the formal Bayes estimator of $\theta$ is $X+1$, which is neither minimax nor admissible. (See also Problem 2.15.) Conditions under which the formal Bayes estimator corresponding to an improper prior distribution for $\theta$ in Example 3.4 is admissible are given by Zidek (1970).

Show that the natural parameter space of the family (2.16) is $(-\infty , \infty )$ for the normal (variance known), binomial, and Poisson distribution but not in the gamma or negative binomial case.