Maximum Likelihood Estimation

1. Find the MLE of based on a random sample 1,2,..., from each of the following p.d.f.'s.

(a) (|) = 1 where 0 < <1, 0 <, and 0 otherwise.

(b) (|) = ( +1)2 where 1 < , 0 < , and 0 otherwise.

(c) where 0 < , 0 < , and 0 otherwise.

(d) (|) = 2 (|) = (1 )1 for = 1,2,..., 0 < < 1, and 0 otherwise

2. Find the asymptotic variance of the MLE in each part of question 1.

3. Consider two independent random samples 1,2,..., (,2 1) and 1,2,..., (,2 2).

(a) Using the data from the two random samples find the m.l.e. of , 1 2 , and 2 2.

(b) Find the asymptotic variance of .