STAT 801

Problems: Assignment 6

Postscript version of these questions

1. Suppose $X_1,\ldots,X_n$ are independent Poisson($\lambda$) variables. Find the UMP level $\alpha$ test of $\lambda \le 1$ versus $\lambda > 1$ and evaluate the constants for the case $n=3$ and $\alpha=0.05$.

2. Suppose $X$ has a Gamma($\theta,\phi$) distribution with shape parameter $\theta$ known. Find the UMPU test of $\phi=\phi_o$ and evaluate the constants for the case $\alpha=0.05$ and $\theta=2$.

3. Suppose $X_1,\ldots,X_n$ are iid exponential($\lambda$).
   1. Find the exact confidence levels of 95% intervals based on normal approximations to the distributions of the pivots $T_1 = {\bar X}/ \lambda$, $T_2 =1/T_1$, and $T_3 = \log(T_1)$ for $n$=10, 20 and 40.

   2. Find the shortest exact 95% confidence interval based on $T_1$; get numerical values for $n$=10, 20 and 40.

   3. Find the exact confidence level of 95% confidence intervals based on the chi-squared approximation to the distribution of deviance drop. Compare the results with the previous question based on length and coverage probabilities. Figure out how to make a convincing comparison. Which method is better?

4. In the course notes I discussed, for the Binomial$(5,p)$ problem, a test of $p=1/2$ against $p=3/4$ based on the rejection region $R_X=\{0,5\}$.
   1. Show that this test is uniformly most powerful among non-randomized tests at the level $\alpha=1/16$ for testing $p=1/2$ against $p>1/2$.

   2. Now suppose that $Y_1,\ldots,Y_5$ are iid Bernoulli$(p)$. Show that the region $R_Y = \{(1,1,1,1,1), (1,1,1,1,0)\}$ has level $1/16$ and is more powerful than the test based on $R_X$ for each $p>1/2$.

   3. If $\phi_Y= 1((Y_1,\ldots,Y_5) \in R_Y)$ show that
      
      $$\phi(X) = E_{1/2}(\Phi(Y_1,\ldots,Y_5)\vert X)$$
      
      
      is a test function, evaluate its power and level.

5. Suppose $\phi(X)$ is a test function and $S(X)$ is a sufficient statistic for some model. Show that
   
   $$E(\phi(X)\vert S)$$
   
   
   is a test function and compare its power and level to that of $\phi(X)$.