Stat 330 Mathematical Statistics - Assignment 3 You need to use the cover sheet provided in Learn. Due on March 24 (Friday) 12pm to the drop boxes located across the hall from MC 4065/4066. 1. [10 marks] Suppose X1 , . . . , Xn is a random sample from the GEO() distribution where X represents P the number of failures before the first success in a series of Bernoulli trials. Let Yn = ni=1 Xi . n = (a) [6 marks] Find the limiting distribution of X 1 ). Yn n , Vn = n n+Yn and Wn = n(Xn (b) [4 marks] Show n(Vn ) d Z N (0, 1) Zn = p Vn2 (1 Vn ) 2. [15 marks] The following model is proposed for the distribution of family size in a large population: P(k children in family; ) = k , P(0 children in family; ) = for k = 1, 2, . . . 12 1 . The parameter is unknown and 0 < < 21 . Fifty families were chosen at random from the population. The observed numbers of children are given in the following table: No. of children Frequency observed 0 17 1 22 2 7 3 3 4 1 Total 50 (a) [4 marks] Find the likelihood, log likelihood, score and information functions for . (b) [4 marks] Find the M.L. estimate of and the observed information. Check whether n is a maximum point or not. What is the estimated asymptotic variance of the M.L. estimate? (c) [3 marks] Find the M.L. estimate of P(0 children in family). Which property are you using here? (d) [2 marks] Derive the relative likelihood function. (e) [2 marks] A large study done 20 years earlier indicated that = 0.45. Is this value plausible for these data? 3. [15 marks] Suppose X1 , . . . , Xn is a random sample from Gamma( 12 , 1 ) distribution. (a) [3 marks] Find n , the M.L. estimator of . Check whether n is a maximum point or not. P (b) [4 marks] Justify the statement n 0 where 0 is the true value of . P P (c) [4 marks] Show that Q = 2 ni=1 Xi 2 (n). If n = 20 and 20 i=1 xi = 6, use the pivotal quantity Q to construct an exact 95% equal tail C.I. for . d (d) [4 marks] Verify that J(n )1/2 (n 0 ) N (0, 1). Use this asymptotic pivotal quantity to construct an approximate 95% C.I. for . Compare this interval with the exact C.I. from (c). 1