Asymptotic Efficiency

Let $X_{1}, \ldots , X_{n}$ be iid as $N(0,1)$. Consider the two estimators

where $S_{n}=\Sigma \left(X_{i}-\bar{X}\right)^{2}, \mathbb {P}\left(S_{n}>a_{n}\right)=1 / n$, and $T_{n}^{\prime }=\left(X_{1}+\cdots +X_{k_{n}}\right) / k_{n}$ with $k_{n}$ the largest integer $\leq \sqrt{n}$.

If $k_{n}\left[\delta_{n}-g(\theta )\right] \xrightarrow {\mathcal{L}} H$ for some sequence $k_{n}$, show that the same result holds if $k_{n}$ is replaced by $k_{n}^{\prime }$, where $k_{n} / k_{n}^{\prime } \rightarrow 1$.

Assume that the distribution of $Y_{n}=\sqrt{n}\left(\delta_{n}-g(\theta )\right)$ converges to a distribution with mean 0 and variance $v(\theta )$. Use Fatou's lemma (Lemma 1.2.6) to establish that $\operatorname {Var}_{\theta }\left[\delta_{n}\right] \rightarrow 0$ for all $\theta$.

If $X_{1}, \ldots , X_{n}$ are a sample from a one-parameter exponential family (1.5.2), then $\Sigma T\left(X_{i}\right)$ is minimal sufficient and $\mathbb {E}\left[(1 / n) \Sigma T\left(X_{i}\right)\right]=(\partial / \partial \eta ) A(\eta )=\tau$. Show that for any function $g(\cdot )$ for which Theorem 1.8.12 holds, $g\left((1 / n) \Sigma T\left(X_{i}\right)\right)$ is asymptotically unbiased for $g(\tau )$.

If $X_{1}, \ldots , X_{n}$ are iid $n\left(\mu , \sigma^{2}\right)$, show that $S^{r}=\left[1 /(n-1) \Sigma \left(x_{i}-\bar{x}\right)^{2}\right]^{r / 2}$ is an asymptotically unbiased estimator of $\sigma^{r}$.

Let $X_{1}, \ldots , X_{n}$ be iid as $U(0, \theta )$. From Example 2.1.14, $\delta_{n}=(n+1) X_{(n)} / n$ is the UMVU estimator of $\theta$, whereas the MLE is $X_{(n)}$. Determine the limit distribution of

For the situation of Problem 2.6:

Verify the asymptotic distribution claimed for $\delta_{n}$ in Example 2.5.

Let $\delta_{n}$ be any estimator satisfying (2.2) with $g(\theta )=\theta$. Construct a sequence $\delta_{n}^{\prime }$ such that $\sqrt{n}\left(\delta_{n}^{\prime }-\theta \right) \xrightarrow {\mathcal{L}} N\left[0, w^{2}(\theta )\right]$ with $w(\theta )=v(\theta )$ for $\theta \neq \theta_{0}$ and $w\left(\theta_{0}\right)=0$.

In the preceding problem, construct $\delta_{n}^{\prime }$ such that $w(\theta )=v(\theta )$ for all $\theta \neq \theta_{0}$ and $\theta_{1}$ and $<v(\theta )$ for $\theta =\theta_{0}$ and $\theta_{1}$.

Construct a sequence $\left\{ \delta_{n}\right\}$ satisfying (2.2) but for which the bias $b_{n}(\theta )$ does not tend to zero.

In Example 2.7 with $R_{n}(\theta )$ given by (2.11), show that $R_{n}(\theta ) \rightarrow 1$ for $\theta \neq 0$ and that $R_{n}(0) \rightarrow a^{2}$.

Let $b_{n}(\theta )=\mathbb {E}_{\theta }\left[\delta_{n}\right]-\theta$ be the bias of the estimator $\delta_{n}$ of Example 2.5.

In Example 2.7, show that if $\theta_{n}=c / \sqrt{n}$, then $R_{n}\left(\theta_{n}\right) \rightarrow a^{2}+c^{2}(1-a)^{2}$.