STOR 655 May 5 2016 Name Final Exam All problem parts have equal weight In budgeting your time expect that some problems will take longer than others You can use results of earlier parts in later parts Remember answers without proper justi cation will not receive full credit 1 Assume are i i d bivariate normal cid 18 X1 cid 19 Y1 0 1 cid 19 cid 18 Xn cid 80 n Yn a Prove that the Fisher information I 1 b Consider Rn 1 n i 1 XiYi and Vn 1 n sistent estimators of cid 18 cid 18 0 cid 19 cid 18 cid 19 cid 19 0 1 cid 80 n 2 2 1 i 1 X 2 2 1 2 i Are they con c What is the asymptotic distribution of these estimators Are they asymptotically normal d Is R3 n V 2 n distribution a strongly consistent estimator of What is its asymptotic e Is any of the three estimators above asymptotically e cient If not improve at least one of them by scoring 2 Let X1 Xn i i d Uniform 0 0 a Find n the MLE Is it a consistent estimator of b What is the asymptotic distribution of n is it asymptotically normal Hint Consider convergence of n 0 n c Find the Bayes posterior distribution using an improper prior 1 d Consider rescaling the posterior distribution by the change of vari able n n What does the rescaled posterior distribution converges to as n Is this convergence in L1 e Consider parametric bootstrap estimator of the Tn 0 n n n cid 63 n where cid 63 cid 63 n n is based on bootstrap samples of X cid 63 T cid 63 n i i d Uniform 0 n Does it lead to correct inference Would the answer change if we used non parametric bootstrap i e the bootstrap samples to be resamples of the original data by 1 X cid 63 n 3 Let Z N 0 I be standard multivariate normal random variable r distribution if and only if P is a projection Show that Z cid 62 P Z has 2 matrix of rank r
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