homework10 - Homework 10 36-325/725 due Friday Nov 16 (1)...

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Unformatted text preview: Homework 10 36-325/725 due Friday Nov 16 (1) Let where and are unknown parameters and . (1a) Find the mle and . (1b) Let . Find the mle of . (1c) Let be the mle from (1b). Let be the nonparametric plug-in estimator of . Suppose that , and . Find the MSE of by simulation. Find the MSE of analytically. Compare. . Find the Fisher information matrix (2) Let the inverse Fisher informatiom matrix . and find (3) Let . Let be the .95 percentile, i.e. . Find the mle of . Find an expression for a confidence interval for . Let 4 g l" ' e q4 g 5 i ' 4 g k" ' @ 4 g 5 5 j 5e e g 5 i g i ' h@ 4 g 5 ' g 5 f d ISe eq`G% @ 4 ' %# $h Y (&$"! u6&vvq (4) Let . Show that the mle is consistent. Hint: . For any , . (5) Let . Find the Fisher information . Continued On Next Page ... 1 9 @ 4 9 ' 5 89 4 g shfed' r 4 u0s' r YT @ `XW 2 T ) P9 HFD B @ VR@ 2 U@ ) T 4 D ' 8 9 SGERQ9 P9 9 HFD B @ 4 D ' IGECA9 28 8 ) 2 5 76) 42 ' %# 310)(&$"! 9 t ur 9 4i g ' a qyfedcxw6&vvq 4 ' !q ! n Gtsr$&q $pom 4i g ' a qphfedcb (6) Let Let (a) Find the maximum likelihood estimate of . (b) Find an approximate 95 per cent confidence interval for (c) Define . Show that is a consistent estimator of . (d) Compute the asymptotic relative efficiency of to . Hint: Use the delta method to get the standard error of the mle. Then compute the standard error (i.e. the standard deviation) of . (e) Suppose that the data are not really normal. Show that is not consistent. What, if anything, does converge to? 2 } } 8} 8} P} } { Y|rw Y @ w Y z r 7rw r T ux y 8} P} 4 T v0cxw eeq ' a w W @ 1 w 14 T ' ~P } P} 8} . Define @ T 4 @ ' j~} ...
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