stat210a_2007_hw7_solutions

stat210a_2007_hw7_solutions - UC Berkeley Department of...

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Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 7- Solutions Fall 2007 Issued: Thursday, October 18 Due: Thursday, October 25 Problem 7.1 (a) E ( | X i | ) = 2 integraldisplay x 1 2 e x 2 2 2 dx = integraldisplay 2 e t 2 dt parenleftbigg t = x 2 2 parenrightbigg = radicalbigg 2 Thus, by WLLN(Weak Law of Large Number), C n n summationdisplay i =1 | X i | p C radicalbigg 2 Therefore, ( X ) is a consistent estimator of if and only if C = radicalbig / 2. (b) Note that I n ( ) = E parenleftbigg 2 l 2 ( X ) parenrightbigg = E parenleftBigg 3 4 n summationdisplay i =1 X 2 i n 2 parenrightBigg = 2 n 2 , V ar parenleftBigg C n n summationdisplay i =1 | X i | parenrightBigg = 2 2 n 2 Thus, 2 ( MLE ) = 2 2 and 2 ( ) = ( 2) 2 2 . 2 ( ) 2 ( MLE ) = 2 Problem 7.2 (a) First, E X ( n ( X )) = E X bracketleftbigg P Z parenleftbigg Z radicalbigg n n 1 ( a X n ) vextendsingle vextendsingle vextendsingle vextendsingle X parenrightbiggbracketrightbigg ( Z, X n : independent, Z N (0 , 1)) = P bracketleftbigg Z radicalbigg n n 1 ( a X n ) bracketrightbigg = P bracketleftBigg radicalbigg n 1 n Z + X n a bracketrightBigg = P ( X 1 a ) ( radicalbigg n 1 n Z + X n N ( , 1)) 1 Thus, n ( X ) is an unbiased estimator of g a ( ). Now, observe that g a ( ) = P ( X 1 a ) = P ( X 1 a ) = ( a ). Now, radicalbigg n n 1 ( a X n ) ( a ) = (( a X n ( a ) + parenleftbiggradicalbigg n n 1 1 parenrightbigg ( a X n ). Note that n parenleftbiggradicalbigg n n 1 1 parenrightbigg = n n 1( n + n 1) 0 and a X n a as n . Thus, n parenleftBigradicalBig n n 1 1 parenrightBig ( a X n ) p o Additionally, by the CLT, we have that: n ( X n ) = n ( ( a X n ) ( a ) ) d N (0 , 1) Thus, by slutsky theorem, n parenleftbiggradicalbigg n n 1 ( a X n ) ( a ) parenrightbigg d N (0 , 1) Letting h ( . ) = ( . ) and using the delta method yields: n parenleftbigg parenleftbiggradicalbigg n n 1 ( a X n ) parenrightbigg ( a )...
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stat210a_2007_hw7_solutions - UC Berkeley Department of...

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