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Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Solutions - Problem Set 9 Fall 2006 Issued: Thursday, November 2, 2006 Due: Thursday, November 9, 2006 Graded exercises Problem 9.1 (a) First, notice that g a ( ) = P ( X 1 a ) = P ( X 1- a- ) = ( a- ). Additionally, by the CLT, we have that: n ( - X n ) = n ( ( a- X n )- ( a- ) ) d N (0 , 1) Letting h ( . ) = ( . ) and using the delta method yields: n ( ( a- X n )- ( a- ) ) d N , ( a- ) 2 So: n ( ( a- X n )- g a ( ) ) d N , [ ( a- )] 2 To get to the result, it is enough to prove that n ( X n )- ( a- X n ) p 0. This follows from continuity of and the fact that: r n n- 1 ( a- X n )- ( a- X n ) = r n n- 1- 1 ( a- X n ) p (b) We know that I ( X i a ) is a Bernoulli variable with mean P ( X i a ) = F X ( a ) = g a ( ) and variance g a ( )(1- g a ( )). Using the central limit theorem: n ( ( X )- g a ( )) = n h P n i =1 I ( X i a )- g a n i d N (0 ,g a ( )(1- g a ( ))) Under normality, g a ( ) = ( a- ) and the result follows. (c) The asymptotic relative efficiency between n and n is: ARE ( , ) = ( a- )[1- ( a- )] 2 ( a- ) = ( a- )( - a ) 2 ( a- ) From the plots below, we can see that the non-parametric estimator is less efficient that the parametric one. Furthermore, the efficiency of the non-parametric estimate degrades exponentially fast as we move towards the tails of the distribution.degrades exponentially fast as we move towards the tails of the distribution....
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This note was uploaded on 10/17/2009 for the course STAT 210a taught by Professor Staff during the Fall '08 term at University of California, Berkeley.
- Fall '08