Due 3/31/99

1.

Compute the Fisher Information number $I(\theta)$ in each of the following cases. Use a sample of size 1.

(a)

The Poisson distribution, where $\theta = \lambda$.

(b)

The Normal distribution, where $\theta = \mu$.

(c)

The Normal distribution, where $\theta = \sigma$.

2.

Suppose that $(X_1,\dots,X_n)$ is a random sample from a normal population with mean $\mu$ and variance $\sigma^2$.

(a)

Show that if $\sigma^2$ is known and $\mu$ is not known then

$$\frac{X_1 + \cdots +X_n}{n}$$

is a uniform minimum variance unbiased estimate of $\mu$.

(b)

Show that if $\mu = \mu_0$ and $\sigma^2$ is unknown, then

$$n^{-1}\sum_{k=1}^n (X_k-\mu_0)^2$$

is a uniform minimum variance unbiased estimate of $\sigma^2$.