HW2sol - SOLUTIONS FOR HWK 2 Problem A (a) and (b) follow...

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SOLUTIONS FOR HWK 2 Problem A (a) and (b) follow from theorems proved in class since μ = θ,σ 2 = θ 2 . (d) E ( ˆ Θ) = 1 15 μ + 2 15 μ + 3 15 μ + 1 15 μ + 8 15 μ = μ . Hence ˆ θ is unbiased. (f) ) = ± 1 15 2 θ 2 + ± 2 15 2 θ 2 + ± 3 15 2 θ 2 + ± 1 15 2 θ 2 + ± 8 15 2 θ 2 = 0 . 351 θ 2 . (g) Since n = 5, we have V ar ( ¯ X ) = θ 2 / 5 = 0 . 2 θ 2 . Therefore: re ( ¯ X : ˆ Θ) = 0 . 351 θ 2 0 . 2 θ 2 = 1 . 755 > 1. h) ¯ X is much more efficient than ˆ Θ. Problem B Let r > 0 and let X be a random variable. Assume that E ( | X | r ) < . The moment of order r , μ r def = E ( X r ) (of course, the moment of order 1 μ 1 = μ = E ( X ), is the expectation of X ). Consider a random sample X 1 ,...,X n where X k ’s have the same distribution as X . The empirical moment of order r M r def = 1 n n k =1 X r k . 1) Prove that M r is an unbiased estimator of μ r where r > 0 . E ( M r ) = E ( 1 n n X k =1 X r k ) = 1 n E ( n X k =1 X r k ) = 1 n n X k =1 E ( X r k ) = 1 n n X k =1 μ r = 1 n r = μ r . 2) Prove that the sequence of the empirical moments M ( n ) r = 1 n n k =1 X r k is a strongly consistent sequence of estimators of the moment μ r . We apply the Strong Law of Large Numbers:
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This note was uploaded on 02/24/2009 for the course STAT 415 taught by Professor Senturk,damla during the Spring '06 term at Pennsylvania State University, University Park.

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HW2sol - SOLUTIONS FOR HWK 2 Problem A (a) and (b) follow...

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