stat5102 HW5 Each problem worths 20 points. Problem 1. Let Y1 Bin(2007, p). (a) Compute the Fisher information I1 (p). (b) Let W (Y1 ) represent an unbiased estimator of p. Use the Information Inequality to show that V ar(W (Y1 )) is no less than p(1p) . 2007 Y1 (c) Show that p = 2007 is an unbiased estimator of p and has the smallest variance among all unbiased estimators of p. Problem 2. Let X1 , . . . , Xn are iid random variables and their density function is 1 x f (x|) = e , x > 0. (a) Compute the Fisher information In (). (b) Suppose W (X1 , . . . , Xn ) is an unbiased estimator of 2 . Then use the 4 Information Inequality to show that V ar(W ) is no less than 4 . n Problem 3. Let x1 , . . . , xn be n samples from a continuous distribution with density function 1 x2 f (x|) = e 2 . 2 1 (note that X1 , . . . , Xn are iid N (0, ).) (a) Compute the Fisher information In (). (b) Show W = 1 n n i=1 1 x2 is an unbiased estimator of . Calculate V ar(W ). i (c) Use the information Inequality to show that W has the smallest variance 1 among all unbiased estimators of . Problem 4. Suppose x1 , . . . , xn are n samples from the following distribution f (x|) = x1 , 0 < x < 1 > 0. (a) Compute the Fisher information In (). (b) Use the information Inequality to show that the variance of any unbiased 1 1 estimator of must be at least n2 . 1 Problem 5. Let x1 , . . . , xn be n samples from a discrete distribution with probability function given by P (X = 1) = P (X = 1) = p , 2 P (X = 0) = 1 p. 0 < p < 1. Note that we can write the probability function as f (x|p) = p 2 |x| (1 p)1|x| . (a) Compute the Fisher information I(p). (b) Use the information Inequality to show that the variance of any unbiased estimator of p must be at least p(1p) . n (c) Find an unbiased estimator of p such that its variance equals the information lower bound. 2