Assignment 11

Problem: Let $X_1,\dots,X_n$ be a random sample from a Pareto$(1,\beta)$ distribution with density $f(x\vert\beta) = \beta/x^{\beta+1}$ for $x \ge 1$. Find the asymptotic relative efficiency of the method of moments estimator of $\beta$ to the MLE of $\beta$.

Due Friday, April 18, 2003.

Problem: Let $X_1,\dots,X_n$ be $i.i.d.$ Poisson$(\lambda)$ and let $W = e^{-\overline{X}}$. Find the parametric bootstrap variance Var$^*(W)$ and show that Var$^*(W)/$Var$(W) \overset{P}{\rightarrow} 1$ as $n \rightarrow \infty$.

Due Friday, April 18, 2003.

1. Let $X_1,\dots,X_n$ be a random sample that may come from a Poisson distribution with mean $\lambda$. Find the sandwich estimator of the asymptotic variance of the MLE $\widehat{\lambda} = \overline{X}$.
2. Let $g(x) = e^{-x}$ for $x > 0$ be an exponential density with mean one and let $f(x\vert\theta)$ be a $N(\theta,1)$ density. Find the value $\theta^*$ corresponding to the density of the form $f(x\vert\theta)$ that is closest to $g$ in Kullback-Liebler divergence.

Due Friday, April 18, 2003.