Dr. Hackney STA Solutions pg 165

Dr. Hackney STA Solutions pg 165 - 10-8 Solutions Manual...

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10-8 Solutions Manual for Statistical Inference 10 . 23 a. The ARE is [2 σf ( μ )] 2 . We have Distribution Parameters variance f ( μ ) ARE normal μ = 0 = 1 1 . 3989 . 64 logistic μ = 0 = 1 π 2 / 3 . 25 . 82 double exp. μ = 0 = 1 2 . 5 2 b. If X 1 ,X 2 ,...,X n are iid f X with E X 1 = μ and Var X 1 = σ 2 , the ARE is σ 2 [2 * f X ( μ )] 2 . If we transform to Y i = ( X i - μ ) , the pdf of Y i is f Y ( y ) = σf X ( σy + μ ) with ARE [2 * f Y (0)] 2 = σ 2 [2 * f X ( μ )] 2 c. The median is more efficient for smaller ν , the distributions with heavier tails. ν Var X f (0) ARE 3 3 . 367 1 . 62 5 5 / 3 . 379 . 960 10 5 / 4 . 389 . 757 25 25 / 23 . 395 . 678 50 25 / 24 . 397 . 657 1 . 399 . 637 d. Again the heavier tails favor the median. δ σ ARE . 01 2 . 649 . 1 2 . 747 . 5 2 . 895 . 01 5 . 777 . 1 5 1 . 83 . 5 5 2 . 98 10 . 25 By transforming y = x - θ , Z -∞ ψ ( x - θ ) f ( x - θ ) dx = Z -∞ ψ ( y ) f ( y ) dy. Since ψ is an odd function, ψ ( y ) = - ψ ( - y ), and Z -∞ ψ ( y ) f ( y ) dy = Z 0 -∞ ψ ( y ) f ( y ) dy + Z 0 ψ ( y ) f ( y ) dy = Z 0 -∞ - ψ ( - y ) f ( y ) dy + Z 0 ψ
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This note was uploaded on 02/03/2012 for the course STA 1014 taught by Professor Dr.hackney during the Spring '12 term at UNF.

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