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hw9_stat210a_solutions

hw9_stat210a_solutions - UC Berkeley Department of...

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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 ( θ ) = P ( X 1 a ) = P ( X 1 - θ a - θ ) = Φ( a - θ ). Additionally, by the CLT, we have that: n ( θ - ¯ X n ) = n ( ( a - ¯ X n ) - ( a - θ ) ) d N (0 , 1) Letting h ( . ) = Φ( . ) and using the delta method yields: n ( Φ( a - ¯ X n ) - Φ( a - θ ) ) d N 0 , Φ ( a - θ ) 2 So: n ( Φ( a - ¯ X n ) - g a ( θ ) ) d N 0 , [ φ ( a - θ )] 2 To get to the result, it is enough to prove that δ n ( ¯ X n ) - Φ( a - ¯ X n ) p 0. This follows from continuity of Φ and the fact that: n n - 1 ( a - ¯ X n ) - ( a - ¯ X n ) = n n - 1 - 1 ( a - ¯ X n ) p 0 (b) We know that I ( X i a ) is a Bernoulli variable with mean P ( X i a ) = F X ( a ) = g a ( θ ) and variance g a ( θ )(1 - g a ( θ )). Using the central limit theorem: n ( δ ( X ) - g a ( θ )) = n P n i =1 I ( X i a ) - g a n d → N (0 , g a ( θ )(1 - g a ( θ ))) Under normality, g a ( θ ) = Φ( a - θ ) and the result follows. (c) The asymptotic relative efficiency between δ n and δ n is: ARE ( δ , δ ) = Φ( a - θ ) [1 - Φ( a - θ )] φ 2 ( a - θ ) = Φ( a - θ )Φ( θ - a ) φ 2 ( a - θ ) 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. 1

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(a) ARE ( δ , δ ) vs.a - θ (b) [ ARE ( δ , δ )] - 1 vs.a - θ Problem 9.2 (a) We have that Y 0 Poisson( θ ) and given Y j - 1 , Y j Poisson( θy j - 1 ). It follows that the
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hw9_stat210a_solutions - UC Berkeley Department of...

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