Math 6010 • Linear Models

Let \(X_1,X_2,\ldots,X_n\) denote a random sample from a Poisson distribution with parameter \(\theta>0\).
\begin{parts}
\item Show that \((-1)^{X_1}\) is unbiased for \(e^{-2\theta}\), and use the Lehmann-Scheffe theorem to deduce a UMVUE of \(e^{-2\theta}\) based on the sample. (Hint: The conditional distribution of \(X_1\) given \(X_1+\cdots+X_n\) has a simple form.) (5 points)
\item Argue that the UMVUE in part (a) is a consistent estimator of \(e^{-2\theta}\). (5 points)
\end{parts}