First Examples

Prove the parts of Theorem 1.4 relating to

In model (1.9), suppose that $n=2$ and that $f$ satisfies $f\left(-x_{1},-x_{2}\right)=f\left(x_{2}, x_{1}\right)$. Show that the distribution of $\left(X_{1}+X_{2}\right) / 2$ given $X_{2}-X_{1}=y$ is symmetric about 0. Note that if $X_{1}$ and $X_{2}$ are iid according to a distribution which is symmetric about 0, the above equation holds.

If $X_{1}$ and $X_{2}$ are distributed according to (1.9) with $n=2$ and $f$ satisfying the assumptions of Problem 1.2, and if $\rho$ is convex and even, then the MRE estimator of $\xi$ is $\left(X_{1}+X_{2}\right) / 2$.

Under the assumptions of Example 1.18, show that

For each of the three loss functions of Example 1.18, compare the risk of the MRE estimator to that of the UMVU estimator.

If $T$ is a sufficient statistic for the family (1.9), show that the estimator (1.28) is a function of $T$ only.

Let $X_{i}(i=1,2,3)$ be independently distributed with density $f\left(x_{i}-\xi \right)$ and let $\delta =X_{1}$ if $X_{3}>0$ and $=X_{2}$ if $X_{3} \leq 0$. Show that the estimator $\delta$ of $\xi$ has constant risk for any invariant loss function, but $\delta$ is not location equivariant.

Consider a location family: $X$ has density $f(x-\xi )$, where $\xi$ is the location parameter to be estimated and the loss function is $\rho (d-\xi )$ for a general nonnegative loss $\rho$, not assumed convex. An MRE (minimum risk equivariant) estimator is one that minimizes the risk among all estimators equivariant under location shifts; this is the setting of Example 1.13, which shows that when $X$ is symmetric about $\xi$ and $\rho$ is convex and even, $X$ itself is MRE.

Corollary 1.14 states: suppose that $0 \leq \rho (t) \leq M$ for all values of $t$, that $\rho (t) \rightarrow M$ as $t \rightarrow \pm \infty$, and that the density $f$ of $X$ is continuous a.e. Then, an MRE estimator of $\xi$ exists.

Prove Corollary 1.14.

Let $X_{1}, \ldots , X_{n}$ be distributed as in Example 1.19 and let the loss function be that of Example 1.15. Determine the totality of MRE estimators and show that the midrange is one of them.

Consider the loss function

If $X$ is a random variable with density $f$ and distribution function $F$, show that $\mathbb {E}\left[\rho (X-v)\right]$ is minimized for any $v$ satisfying $F(v)=B /(A+B)$.

In Example 1.16, find the MRE estimator of $\xi$ when the loss function is given by Problem 1.10.

Show that an estimator $\delta (X)$ of $g(\theta )$ is risk-unbiased with respect to the loss function of Problem 1.10 if $F_{\theta }[g(\theta )]=B /(A+B)$, where $F_{\theta }$ is the cdf of $\delta (X)$ under $\theta$.

Suppose $X_{1}, \ldots , X_{m}$ and $Y_{1}, \ldots , Y_{n}$ have joint density $f\left(x_{1}-\xi , \ldots , x_{m}-\xi ; y_{1}-\eta , \ldots , y_{n}-\eta \right)$ and consider the problem of estimating $\Delta =\eta -\xi$. Explain why it is desirable for the loss function $L(\xi , \eta ; d)$ to be of the form $\rho (d-\Delta )$ and for an estimator $\delta$ of $\Delta$ to satisfy $\delta (\mathbf{x}+a, \mathbf{y}+b)=\delta (\mathbf{x}, \mathbf{y})+(b-a)$.

Under the assumptions of the preceding problem, prove the equivalents of Theorems 1.4-1.17 and Corollaries 1.11-1.14 for estimators satisfying the restriction.

In Problem 1.13, determine the totality of estimators satisfying the restriction when $m=n=1$.

In Problem 1.13, suppose the $X$ 's and $Y$ 's are independently normally distributed with known variances $\sigma^{2}$ and $\tau^{2}$. Find conditions on $\rho$ under which the MRE estimator is $\bar{Y}-\bar{X}$.

In Problem 1.13, suppose the $X$ 's and $Y$ 's are independently distributed as $E(\xi , 1)$ and $E(\eta , t)$, respectively, and that $m=n$. Find conditions on $\rho$ under which the MRE estimator of $\Delta$ is $Y_{(1)}-X_{(1)}$.

In Problem 1.13, suppose that X and Y are independent and that the loss function is squared error. If $\hat{\xi }$ and $\hat{\eta }$ are the MRE estimators of $\xi$ and $\eta$, respectively, the MRE estimator of $\Delta$ is $\hat{\eta }-\hat{\xi }$.

Suppose the $X$ 's and $Y$ 's are distributed as in Problem 1.17 but with $m \neq n$. Determine the MRE estimator of $\Delta$ when the loss is squared error.

For any density $f$ of $\mathbf{X}=\left(X_{1}, \ldots , X_{n}\right)$, the probability of the set $A=\left\{ \mathbf{x}: 0< \int_{-\infty }^{\infty } f(\mathbf{x}-u) d u<\infty \right\}$ is 1.

Under the assumptions of Theorem 1.10, if there exists an equivariant estimator $\delta_{0}$ of $\xi$ with finite expected squared error, show that

Let $\delta_{0}$ be location equivariant and let $\mathcal{U}$ be the class of all functions $u$ satisfying (1.20) and such that $u(X)$ is an unbiased estimator of zero. Then, $\delta_{0}$ is MRE if and only if $\operatorname {cov}\left[\delta_{0}, u(X)\right]=0$ for all $u \in \mathcal{U}.^{2}$ (Note the analogy with Theorem 2.1.7.)

${ }^{2}$ Communicated by P. Bickel.