STAT 801

Problems: Assignment 5

Postscript version of these questions

1. Suppose $X_1,\ldots,X_m$ are iid $N( \mu, \sigma^2 )$ and $Y_1,\ldots,Y_n$ are iid $N( \chi, \tau^2)$. Assume the $X$s are independent of the $Y$s.
   1. Find complete and sufficient statistics.

   2. Find UMVUE's of $\mu - \chi$ and $\sigma^2/\tau^2$.

   3. Now suppose you know that $\sigma = \tau$. Find UMVUE's of $\chi-\mu$ and of $( \chi-\mu )/ \sigma$. (You have already found the UMVUE for $\sigma^2$.)

   4. Now suppose $\sigma$ and $\tau$ are unknown but that you know that $\mu = \chi$. Prove there is no UMVUE for $\mu$. (Hint: Find the UMVUE if you knew $\sigma / \tau =a$ with $a$ known. Use the fact that the solution depends on $a$ to finish the proof.)

   5. Why doesn't the Lehmann-Scheffé theorem apply?

2. Suppose $X_1, \ldots ,X_n$ iid Poisson( $\lambda$ ). Find the UMVUE for $\lambda$ and for $1-\exp(- \lambda ) = P(X_1 \ne 0)$.

3. Suppose $X_1, \ldots ,X_n$ iid with
   
   $$P(X_1=k) = {\rm Prob}({\rm Poisson}( \lambda )=k \vert {\rm Poisson}( \lambda )> 0)$$
   
   
   for $k=1,2,3, \ldots$ . For $n=1$ and 2 find the UMVUE of $1 - \exp ( - \lambda )$. (Hint: The expected value of any function of $X$ is a power series in $\lambda$ divided by $e^\lambda- 1$. Set this equal to $1 - \exp ( - \lambda )$ and deduce that two power series are equal. Since this implies their coefficients are the same you can see what the estimate must be. )

4. Exponential families: Suppose $X_1, \ldots ,X_n$ are iid with density
   
   $$f(x; \theta_1,\ldots,\theta_p ) = c(\theta) \exp(\sum_1^pT_i(x)\theta_i ) h(x).$$
   
   
   1. Find minimal sufficient statistics.

   2. If $S_1,\ldots,S_p$ are the minimal sufficient statistics show that setting $S_i = {\rm E}_\theta(S_i)$ and solving gives the likelihood equations. (Note the connection to the method of moments.)