hw5_solution-1

hw5_solution-1 - 131B HW#5 solution 7.8 Jointly Sucient...

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131B HW#5 solution 7.8 Jointly Suﬃcient Statistics 10. The p.d.f. of the uniform distribution is f ( x | θ ) = 1 θ 1 { 0 x θ } , so the likelihood is f n ( x | θ ) = n i =1 f ( x i | θ ) = 1 θ n 1 { max { x 1 ,...,x n }≤ θ } 1 { 0 min { x 1 ,...,x n }} It implies that the M.L.E. ˆ θ = max { X 1 , . . . , X n } . By the factorization criterion, ˆ θ is the suﬃcient statistic, so it is a minimal suﬃcient statistic. 12. The likelihood is f n ( x | θ ) = n i =1 f ( x | θ ) = 2 n i x i θ 2 n 1 { max { x 1 ,...,x n }≤ θ } 1 { 0 min { x 1 ,...,x n }} The M.L.E. of θ is ˆ θ = max { X 1 , . . . , X n } . ˆ θ is also a suﬃcient statistic by the factorization criterion. The median of the distribution is the value m such that m 0 f ( x | θ ) dx = 1 / 2, it implies that m = θ/ 2. By the invariance property, ˆ m = ˆ θ/ 2 is the M.L.E. of m . Note that ˆ m is also a suﬃcient statistic, so it is a minimal suﬃcient statistic. 16. It follows from Theorem 7.3.2 that the Bayes estimator of λ is ( α + n i =1 X i ) / ( β + n ).

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