Question

3. [20 marks Consider the multinomial distribution with 3 categories, where the random variables Xi, X2 and Xs have the joint probability function (a) [4 marks] Find the maximum likelihood estimator θ of θ. (b) [4 marks Find that the Fisher information matrix I(0). (c) [4 marks] Show that θ is an MVUE. (d) 4 marks Find the approximate distribution of Y 2X-X2, when the sample size n is large (e) [4 marks] Assume that X-(253, 234, 513). Find the Wald 95% confidence interval for θ,-&z.

Answer 1

a) To find the maximum likelihood estimator, we nod to find a θ for θ such that 0e Frorn this we get two equations, one for θ1 and θ2 respectively. Therefore from (1), we get 2rs 23 6, θ2 1-6,-292 2x3 +x1 + 2 Hence θ = (AN3 KL 2X2, 24, 1 X. 2X2) b) Osve that, 凸 2x3 6, 1-6,-2 and 082 Hence the fisher information matrix would be c) Observe that (X1, X2, Xa) is an sufficient estimator of θ. Since θ is a func- tion of (Xi, X2. X3), hence we can say that it is MVUE d) Observe that,