Let g (x) be a density function of a random variable with mean and variance 2.

Let g (x) be a density function of a random variable with mean ¹ and variance ¾2. Let X be a random variable with density function f (x j µ) Æ g (x)

¡

1ŵ

¡

x ¡¹

¢¢

.

Assume g (x), ¹ and ¾2 are known. The unknown parameter is µ. Assume that X has bounded support so that f (x j µ) ¸ 0 for all x. (Basically, don’t worry if f (x j µ) ¸ 0.)

(a) Verify that R 1

¡1 f (x j µ)dx Æ 1.

(b) Calculate E[X] .

(c) Find the information Iµ for µ. Write your expression as an expectation of some function of X.

(d) Find a simplified expression for Iµ when µ Æ 0.

(e) Given a randomsample {X1, ...,Xn} write the log-likelihood function for µ.

(f ) Find the first-order-condition for the MLE bµ for µ. You will not be able to solve for bµ.

(g) Using the known asymptotic distribution for maximumlikelihood estimators, find the asymptotic distribution for p

n

¡bµ¡µ

¢

as n!1.

(h) How does the asymptotic distribution simplify when µ Æ 0?