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($\lambda$).

(a)

Find the exact confidence levels of 95% intervals based on normal approximations to the distributions of the pivots $T_1 = {\bar X}/ \lambda$, T₂ =1/T₁, and $T_3 = \log(T_1)$ for n=10, 20 and 40.

(b)

Find the shortest exact 95% confidence interval based on T₁; get numerical values for n=10, 20 and 40.

(c)

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 $R_X=\{0,5\}$.

(a)

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.

(b)

Now suppose that $Y_1,\ldots,Y_5$ are iid Bernouilli(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.

(c)

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 $\phi(X)$.