Midterm 3: Due 03/8/99

Your first midterm of the spring semester is to construct a hypothesis testing theory for the family of Gamma densities. Recall that the family of gamma densities, $\{f_{a,b}(x), a \gt 0, b \gt 0\}$ are given by

$$f_{a,b}(x) = \left\{\begin{array} {lll} 0 & {\rm if} & x \le... ...^a}{\Gamma(a)}\exp(-bx)} & {\rm if} & x \gt 0\end{array}\right.$$

Things you should consider:

1.

Testing for a when b is known.

2.

Testing for b when a is known.

3.

Testing when a and b are unknown.

4.

Is this an exponential family? If so, how can this be used?

5.

Does this family have the monotone likelihood ration property?

6.

In the case of simple alternative versus simple null, what are the admissible tests? What does the risk set look like?

7.

Minimax, Bayes, and most powerful tests.

8.

Small and large sample tests.

9.

Effect of sample size.

10.

Changing parameters so that the new parameters represent mean and variance.

11.

Generating data sets on which to try your tests.

12.

Graphs of power for composite alternatives.

As this is an exam, you are to work separately. As usual, you may consult references and me. I will be happy to show you how to generate random data sets using MAPLE, and give you other technical advice. This is intended to be open ended, so be careful to manage your time carefully. I will try to limit your other homework during the next couple of weeks so that it doesn't cut into the time you put into this.