Question

- Suppose a random sample of size n is taken from the following distribution with a known positive parameter a. f(x;0,-) = a20 V 27797z exp 0; ; 0<x<00,0< < 0,0 < 8 < 00 elsewhere For this distruttore, the formats for mye or and x-a are respectively, Myo (1) = exp v{(1 - V1 –24*70)} for 1 < 2112 and exp{}(-VT - 2/0)} My-- (1) for 1 < ✓1 - 2t/0 2 Find the maximum likelihood estimators, 0 and ji for 0 and , respectively. What is the distribution of pi? Check whether the estimator j is unbiased, consistent and sufficient. Derive the formula for expected (Fisher) information matrix, 7(0,1). Evaluate the efficiency of this estimator, i, using the Cramer-Rao lower bound. Find the maximum likelihood estimator of Ex-). Using the Della method, show that the variance of this estimator is given by GO n0

Answer 1

Solution: In this example, we have two unknown parameters, u and o, therefore the pa- rameter 6 = (u, a) is a vector. We first write out the log likelihood function as (,0 - 10g 27 – 262(X; – 1)] 2.2(X; – 1)2) = -n logo n n 1 5 log 27 (x - x) 202 i=1 i=1 Setting the partial derivative to be 0, we have n2 alu, o ди (X; - x) = 0 -2 0 i=1 n n a(,0) to-"X(X; - u)2 = 0 до Solving these equations will give us the MLE for p and o: o i=1 n À = X and ô = (X-X) n This time the MLE is the same as the result of method of moment. From these examples, we can see that the maximum likelihood result may or may not be the same as the result of method of moment.