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HW6sol - 10-705 Intermediate Statistics Fall 2012 Homework...

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10-705: Intermediate Statistics Fall 2012 Homework 6 Solutions Lecturer: Larry Wasserman TA: Wanjie Wang, Haijie Gu Problem 1 The mle for Binomial ( k,p ) is ˆ p mle = X n /k . By the asymptotic efficiency of the mle, we have: n p mle - p ) N (0 ,p (1 - p ) /k ) By delta method, n τ 1 - τ ( p )) N (0 , ( τ 0 ( p )) 2 p (1 - p ) /k ) where τ ( p ) = ( k 2 ) p 2 (1 - p ) k - 2 . For ˆ τ 2 = 1 n n i =1 I ( X i = 2), n τ 2 - τ ) N (0 ( p )(1 - τ ( p ))) Hence ARE ( ˆ τ 1 , ˆ τ 2 ) = τ ( p )(1 - τ ( p )) ( τ 0 ( p )) 2 p (1 - p ) /k Expand τ ( p ) and τ 0 ( p ) above, we get: ARE ( ˆ τ 1 , ˆ τ 2 ) = τ ( p )(1 - τ ( p )) ( τ 0 ( p )) 2 p (1 - p ) /k = k (1 - ( k 2 ) p 2 (1 - p ) k - 2 ) p (1 - p ) (2 - pk ) 2 1
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Problem 2[C&B] 8.13 α = . 05 take c =1 645. The power function is β ( μ )= P ° ¯ X μ σ/ n > 1 . 645 μ σ/ n ± = P ° Z> 1 . 645 σ ± . Note that the power will equal . 5when μ . 645 σ/ n . b. For H 0 : μ = 0 vs. H A : μ ° = 0 the LRT is to reject H 0 if | ¯ x | >cσ/ n (Example 8.2.2). For α = . 05 take c . 96. The power function is β ( μ P ² 1 . 96 nμ/σ Z 1 . 96 + nμ/σ ³ . In this case, μ = ± 1 . 96 σ/ n gives power of approximately . 5. 8.13 a. The size of φ 1 is α 1 = P ( X 1 >. 95 | θ = 0) = . 05. The size of φ 2 is α 2 = P ( X 1 + X 2 >C | θ = 0). If 1 C 2, this is α 2 = P ( X 1 + X 2 | θ =0)= ´ 1 1 C ´ 1 C x 1 1 dx 2 dx 1 = (2 C ) 2 2 . Setting this equal to α and solving for C gives C =2 2 α , and for α = . 05, we get C . 1 1 . 68. b. For the ±rst test we have the power function β 1 ( θ P θ ( X 1 95) = µ 0 if θ ≤− . 05 θ + . 05 if . 05 < θ . 95 1 if . 95 < θ . Using the distribution of Y = X 1 + X 2 , given by f Y ( y | θ µ y 2 θ if 2 θ y< 2 θ +1 2 θ +2 y if 2 θ +1 2 θ 0 otherwise,
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HW6sol - 10-705 Intermediate Statistics Fall 2012 Homework...

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