2 STAT 330: Mathematical Statistics Winter 2016 Assignment 3 Name: ID: Due on March 28th (Monday) 11:30am in drop boxes across the hall from MC 4065/4066. 1 1. [6 marks] Suppose X1 , X2 , . . . are independent r.v.'s with Xi U N IF (0, 1), i = 1, 2, . . . Let Un = max(X1 , . . . , Xn ) for n = 1, 2, . . . Show the following results: p (a) eUn e. d (b) en(1Un ) U, where U U N IF (0, 1). 2. [5 marks] Suppose X1 , . . . , Xn , . . . are i.i.d. r.v.'s, and Xi EXP(1) with pdf f (x) = ex for x > 0. Let X(n) = max(X1 , . . . , Xn ). Find a sequence of constants an such that X(n) an converges in distribution. 3. [9 marks] (a) Let Y1 , . . . , Yn . . . be i.i.d. r.v.'s, and Zn = n(Yn m)/c for n = 1, 2, . . . , where c and m are constants. Suppose that d Zn Z, where Z N (0, 1). Show that p Yn m. (b) Let X1 , X2 , . . . be independent Bernoulli(p) r.v.'s, where p is a constant between 0 and 1. Let 1 Xn = n (i) Show that n Xi for n = 1, 2, . . . , i=1 d n(Xn p) X, where X N [0, p(1 p)]. d (ii) Show that for p = 1/2, n[Xn (1 Xn ) p(1 p)] Y , where Y N [0, (1 2p)2 p(1 p)]. 2 4. [9 marks] Suppose (X1 , X2 , , Xn ) is a random sample from the distribution with cdf F (x; ) = 1 2 /x2 , x > , > 0, and zero otherwise. Let Yn = min(X1 , X2 , , Xn ), and Un = n(Yn / 1). (a) Find the limiting distribution of Yn . (b) Is Yn an unbiased estimator of ? Justify your answer. (c) Find the limiting distribution of Un . 5. [11 marks] Suppose X1 , . . . , Xn are i.i.d. r.v.'s following GEO(p) distribution with pmf f (x) = (1 p)x p for x = 0, 1, . . . , where p is a constant between 0 and 1. (a) Find the score function and the ML estimator of p. (b) Find the observed information and the Fisher information. (c) Find the ML estimator of E(Xi ). (d) If n = 20 and 20 i=1 xi = 40, nd the ML estimate of p. 3