Math Problems Vault

Problem 5.4.1

In Example 4.2, show that an estimator $\delta$ is equivariant if and only if it satisfies (4.11) and (4.12).

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Problem 5.4.2

Show that a function $\mu$ satisfies (4.12) if and only if it depends only on $\Sigma X_{i}^{2}$ .

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Problem 5.4.3

Verify the Bayes estimator (4.15).

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Problem 5.4.4

Let $X_{i}$ be independent with binomial distribution $b\left(p_{i}, n_{i}\right), i=1, \ldots , r$ . For estimating $p=\left(p_{1}, \ldots , p_{r}\right)$ with average squared error loss (4.17), find the minimax estimator of $p$ , and determine whether it is admissible.

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Problem 5.4.5

(a)(b)(c)

Establishing the admissibility of the normal mean in two dimensions is quite difficult, made so by the fact that the conjugate priors fail in the limiting Bayes method. Let

X \sim N_{2}(\boldsymbol {\theta }, I) \quad \text{ and } \quad L(\boldsymbol {\theta }, \delta )=|\boldsymbol {\theta }-\delta |^{2}.

The conjugate priors are $\boldsymbol {\theta } \sim N_{2}\left(0, \tau^{2} I\right), \tau^{2}>0$ .

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(a)

For this sequence of priors, verify that the limiting Bayes argument, as in Example 2.8, results in inequality (4.18), which does not establish admissibility.

(b)

Stein (in James and Stein 1961), proposed the sequence of priors that works to prove $X$ is admissible by the limiting Bayes method. A version of these priors, given by Brown and Hwang (1982), is

g_{n}(\boldsymbol {\theta })= \begin{cases} 1 & \text{ if }|\boldsymbol {\theta }| \leq 1 \\ 1-\frac{\log |\boldsymbol {\theta }|}{\log n} & \text{ if } 1 \leq |\boldsymbol {\theta }| \leq n \\ 0 & \text{ if }|\boldsymbol {\theta }| \geq n\end{cases}

for $n=2,3, \ldots$ . Show that $\delta^{g_{n}}(x) \rightarrow x$ a.e. as $n \rightarrow \infty$ .

(c)

A special case of the very general results of Brown and Hwang (1982) state that for the prior $\pi_{n}(\boldsymbol {\theta })^{2} g(\boldsymbol {\theta })$ , the limiting Bayes method (Blyth's method) will establish the admissibility of the estimator $\delta^{g}(x)$ [the generalized Bayes estimator against $g(\boldsymbol {\theta })$ ] if

\int _{\left\{ \boldsymbol {\theta }:\left|\boldsymbol {\theta }\right|>1\right\} } \frac{g(\boldsymbol {\theta }) d \boldsymbol {\theta }}{|\boldsymbol {\theta }|^{2}[\max \left\{ \log \left|\boldsymbol {\theta }\right|, \log 2\right\} ]^{2}}<\infty .

Show that this holds for $g(\theta )=1$ and that $\delta^{g}(x)=x$ , so $x$ is admissible.

Remark.

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Stein (1956b) originally established the admissibility of $X$ in two dimensions using an argument based on the information inequality. His proof was complicated by the fact that he needed some additional invariance arguments to establish the result. See Theorem 7.19 and Problem 7.19 for more general statements of the Brown/Hwang result.

Problem 5.4.6

(a)(b)(c)(d)

Let $X_{1}, X_{2}, \ldots , X_{r}$ be independent with $X_{i} \sim N\left(\theta_{i}, 1\right)$ . The following heuristic argument, due to Stein (1956b), suggests that it should be possible, at least for large $r$ and hence large $\left|\theta \right|$ , to improve on the estimator $\mathbf{X}=\left(X_{1}, X_{2}, \ldots , X_{r}\right)$ .

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(a)

Use a Taylor series argument to show

|\mathbf{x}|^{2}=r+|\boldsymbol {\theta }|^{2}+O_{p}\left[\left(r+|\boldsymbol {\theta }|^{2}\right)^{1 / 2}\right],

so, with high probability, the true $\boldsymbol {\theta }$ is in the sphere $\left\{ \boldsymbol {\theta }:|\boldsymbol {\theta }|^{2} \leq |\mathbf{x}|^{2}-r\right\}$ . The usual estimator $\mathbf{X}$ is approximately the same size as $\boldsymbol {\theta }$ and will almost certainly be outside of this sphere.

(b)

Part (a) suggested to Stein an estimator of the form $\delta (\mathbf{x})=\left[1-h\left(|\mathbf{x}|^{2}\right)\right] \mathbf{x}$ . Show that

|\boldsymbol {\theta }-\delta (\mathbf{x})|^{2}=(1-h)^{2}|\mathbf{x}-\boldsymbol {\theta }|^{2}-2 h(1-h) \boldsymbol {\theta }^{\prime }(\mathbf{x}-\boldsymbol {\theta })+h^{2}|\boldsymbol {\theta }|^{2}.

(c)

Establish that $\boldsymbol {\theta }^{\prime }(\mathbf{x}-\boldsymbol {\theta }) /\left|\boldsymbol {\theta }\right|=Z \sim N(0,1)$ , and $|\mathbf{x}-\boldsymbol {\theta }|^{2} \approx r$ , and, hence,

|\boldsymbol {\theta }-\delta (\mathbf{x})|^{2} \approx (1-h)^{2} r+h^{2}|\boldsymbol {\theta }|^{2}+O_{p}\left[\left(r+|\boldsymbol {\theta }|^{2}\right)^{1 / 2}\right].

(d)

Show that the leading term in part (c) is minimized at $h=r /\left(r+|\boldsymbol {\theta }|^{2}\right)$ , and since $|\mathbf{x}|^{2} \approx r+|\boldsymbol {\theta }|^{2}$ , this leads to the estimator $\delta (\mathbf{x})=\left(1-\frac{r}{|\mathbf{x}|^{2}}\right) \mathbf{x}$ of (4.20).

Problem 5.4.7

If $S^{2}$ is distributed as $\chi_{r}^{2}$ , use (2.2.5) to show that $\mathbb {E}\left[S^{-2}\right]=1 /(r-2)$ .

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Problem 5.4.8

In Example 4.7, show that $\mathcal{R}$ is nonsingular for $\rho_{1}$ and $\rho_{2}$ and singular for $\rho_{3}$ and $\rho_{4}$ .

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Problem 5.4.9

Show that the function $\rho_{2}$ of Example 4.7 is convex.

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Problem 5.4.10

(a)(b)

In Example 4.7, show that $X$ is admissible for

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(a)

$\rho_{3}$ and

(b)

$\rho_{4}$ .

Problem 5.4.11

In Example 4.8, show that $\mathbf{X}$ is admissible under the assumptions (ii)(a).

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Problem 5.4.12

Let $\mathcal{L}$ be a family of loss functions and suppose there exists $L_{0} \in \mathcal{L}$ and a minimax estimator $\delta_{0}$ with respect to $L_{0}$ such that in the notation of (4.29),

\sup _{L, \theta } R_{L}\left(\theta , \delta _{0}\right)=\sup _{\theta } R_{L_{0}}\left(\theta , \delta _{0}\right).

Then, $\delta_{0}$ is minimax with respect to $\mathcal{L}$ ; that is, it minimizes $\sup_{L, \theta } R_{L}(\theta , \delta )$ .

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Problem 5.4.13

Assuming (4.25), show that $E=1-\left[(r-2)^{2} / r|\mathbf{X}-\mu |^{2}\right]$ is the unique unbiased estimator of the risk (5..4.25), and that $E$ is inadmissible.

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Remark.

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The estimator $E$ is also unbiased for estimation of the loss $L(\boldsymbol {\theta }, \delta )$ . See Note 9.5.

Problem 5.4.14

(a)(b)

A natural extension of risk domination under a particular loss is to risk domination under a class of losses. Hwang (1985) defines universal domination of $\delta$ by $\delta^{\prime }$ if the inequality

\mathbb {E}_{\theta }\left[L\left(\left|\boldsymbol {\theta }-\delta ^{\prime }(\mathbf{X})\right|\right)\right] \leq \mathbb {E}_{\theta }\left[L(\left|\boldsymbol {\theta }-\delta (\mathbf{X})\right|)\right] \text{ for all } \boldsymbol {\theta }

holds for all loss functions $L(\cdot )$ that are nondecreasing, with at least one loss function producing nonidentical risks.

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(a)

Show that $\delta^{\prime }$ universally dominates $\delta$ if and only if it stochastically dominates $\delta$ , that is, if and only if

\mathbb {P}_{\theta }\left(\left|\boldsymbol {\theta }-\delta ^{\prime }(\mathbf{X})\right|>k\right) \leq \mathbb {P}_{\theta }\left(\left|\boldsymbol {\theta }-\delta (\mathbf{X})\right|>k\right)

for all $k$ and $\boldsymbol {\theta }$ with strict inequality for some $\boldsymbol {\theta }$ .

(b)

For $X \sim N_{r}(\boldsymbol {\theta }, I)$ , show that the James-Stein estimator $\delta^{c}(\mathbf{x})=\left(1-c /|\mathbf{x}|^{2}\right) \mathbf{x}$ does not universally dominate x.

Hwang (1985) and Brown and Hwang (1989) explore many facets of universal domination. Hwang (1985) shows that even $\delta^{+}$ does not universally dominate $X$ unless the class of loss functions is restricted. We also note that although the inequality in part (a) may seem reminiscent of the "Pitman closeness" criterion, there is really no relation. The criterion of Pitman closeness suffers from a number of defects not shared by stochastic domination (see Robert et al. 1993).