stat5102 HW4 Each problem worths 20 points. Problem 1. Consider X1 , . . . , Xn are iid random variables from Exp(1/), (a). Show that mle is an unbiased estimator of and compute the MSE of mle . (b). Find the maximum likelihood estimate of the MSE of mle . Problem 2. Suppose x1 , . . . , xn are samples from a distribution with mean and variance 2 . We consider a class of estimators of using the weighted average of the samples n (c1 , c2 , . . . , cn ) = ci Xi . i=1 where ci are constants and sum to 1 n ci = 1. i=1 1 The sample mean X is ( n , . . . , n ). 1 (a). Show that (c1 , c2 , . . . , cn ) is an unbiased estimator of . (b). Compute the following ratio R= MSE((c1 , c2 , . . . , cn )) . MSE(X) (c). From (b) we see R only depends on c1 , c2 , . . . , cn . Consider R as a function of (c1 , c2 , . . . , cn ) and nd the choice of (c1 , c2 , . . . , cn ) that minimizes R. What is your conclusion? [hint: you may want to use the following result (see handout 1, page 9) 1 n n x2 ( i i=1 1 n n xi )2 = i=1 n1 2 S 0. n thus for our problem, take ci = xi 1 n n c2 ( i i=1 1 1 n n ci ) 2 = i=1 1 . n2 Problem 3. Consider two estimators of in N (, 1) based on 10 samples 1 = 0 and 2 = X. Note that 1 does not use the data and 2 is the MLE of . (a). Compute the bias, variance and MSE of 1 . (b). Suppose the true value of is 0.1. Which estimator has a smaller MSE? Now if we can increase the sample size to 500 by doing more experiments, which estimator has a smaller MSE? (c). Repeat (b) if the true value of is 0.5. Problem 4. Suppose x1 , . . . , xn are samples from Ber(p). The MLE of p is X p(1p) X(1X) and its MSE is n . By the invariance principle the MLE of MSE is . n Calculate the bias p(1 p) X(1 X) . E[ n n [hint: use the fact n i=1 Xi is Bin(n, p).] Problem 5. X1 , . . . , X2 , . . . , Xn are iid Gam(5, ). (a). Take = c . X Find the constant c to make an unbiased estimate of . (b). Compute the MSE of of the unbiased estimator in (a). [hint: use the fact n i=1 Xi Gam(n, ) if X1 , . . . , Xn are iid Gam(, )]. 2