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Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 11 Fall 2007 Issued: Monday, November 19 Due: Thursday, November 29 Reading: Keener: Chapter 22; B & D: Chapter 5 Problem 11.1 Consider the simple binary hypothesis test H : X ∼ N ( μ ,σ 2 ) versus H 1 : X ∼ N ( μ 1 ,σ 2 1 ), μ < μ 1 . (a) Suppose that ( X 1 ,...,X n ) are i.i.d. samples under H . Show that q n ( ² ) := P " 1 n n X i =1 X i- μ ≥ ² # ≤ exp(- n² 2 2 σ 2 ) . Hint: By Markov’s inequality, we have q ≤ inf t> E [exp( t P n i =1 X i )] exp( tn ( ² + μ )) . Optimize in t . (b) Thinking of a Bayesian formulation, suppose that the two hypotheses are equally likely a priori. Consider the LRT that returns H 1 if 1 n ∑ n i =1 X i ≥ γ , and H otherwise. Use the result from part (a) (and an analogous statement for an error probability under H 1 ) to determine the threshold γ that minimizes the Chernoff exponent lim n → + ∞ 1 n log ( 1 2 P " 1 n n X i =1...
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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