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Unformatted text preview: Chapter 10 Asymptotic Evaluations 10.1 First calculate some moments for this distribution. E X = θ/ 3 , E X 2 = 1 / 3 , Var X = 1 3 θ 2 9 . So 3 ¯ X n is an unbiased estimator of θ with variance Var(3 ¯ X n ) = 9(Var X ) /n = (3 θ 2 ) /n → 0 as n → ∞ . So by Theorem 10.1.3, 3 ¯ X n is a consistent estimator of θ . 10 . 3 a. The log likelihood is n 2 log (2 πθ ) 1 2 X ( x i θ ) /θ. Differentiate and set equal to zero, and a little algebra will show that the MLE is the root of θ 2 + θ W = 0. The roots of this equation are ( 1 ± √ 1 + 4 W ) / 2, and the MLE is the root with the plus sign, as it has to be nonnegative. b. The second derivative of the log likelihood is ( 2 ∑ x 2 i + nθ ) / (2 θ 3 ), yielding an expected Fisher information of I ( θ ) = E θ 2 ∑ X 2 i + nθ 2 θ 3 = 2 nθ + n 2 θ 2 , and by Theorem 10.1.12 the variance of the MLE is 1 /I ( θ ). 10 . 4 a. Write ∑ X i Y i ∑ X 2 i = ∑ X i ( X i + i ) ∑ X 2 i = 1 + ∑ X i i ∑ X 2 i . From normality and independence E X i i = 0 , Var X i i = σ 2 ( μ 2 + τ 2 ) , E X 2 i = μ 2 + τ 2 , Var X 2 i = 2 τ 2 (2 μ 2 + τ 2 ) , and Cov( X i ,X i i ) = 0. Applying the formulas of Example 5.5.27, the asymptotic mean and variance are E ∑ X i Y i ∑ X 2 i ≈ 1 and Var ∑ X i Y i ∑ X 2 i ≈ nσ 2 ( μ 2 + τ 2 ) [ n ( μ 2 + τ 2 )] 2 = σ 2 n ( μ 2 + τ 2 ) b. ∑ Y i ∑ X i = β + ∑ i ∑ X i with approximate mean β and variance σ 2 / ( nμ 2 ). 102 Solutions Manual for Statistical Inference c. 1 n X Y i X i = β + 1 n X i X i with approximate mean β and variance σ 2 / ( nμ 2 ). 10 . 5 a. The integral of E T 2 n is unbounded near zero. We have E T 2 n > r n 2 πσ 2 Z 1 1 x 2 e ( x μ ) 2 / 2 σ 2 dx > r n 2 πσ 2 K Z 1 1 x 2 dx = ∞ , where K = max ≤ x ≤ 1 e ( x μ ) 2 / 2 σ 2 b. If we delete the interval ( δ,δ ), then the integrand is bounded, that is, over the range of integration 1 /x 2 < 1 /δ 2 . c. Assume μ > 0. A similar argument works for μ < 0. Then P ( δ < X < δ ) = P [ √ n ( δ μ ) < √ n ( X μ ) < √ n ( δ μ )] < P [ Z < √ n ( δ μ )] , where Z ∼ n(0 , 1). For δ < μ , the probability goes to 0 as n → ∞ . 10 . 7 We need to assume that τ ( θ ) is differentiable at θ = θ , the true value of the parameter. Then we apply Theorem 5.5.24 to Theorem 10.1.12. 10 . 9 We will do a more general problem that includes a ) and b ) as special cases. Suppose we want to estimate λ t e λ /t ! = P ( X = t ). Let T = T ( X 1 ,...,X n ) = 1 if X 1 = t if X 1 6 = t. Then E T = P ( T = 1) = P ( X 1 = t ), so T is an unbiased estimator. Since ∑ X i is a complete sufficient statistic for λ , E( T  ∑ X i ) is UMVUE. The UMVUE is 0 for y = ∑ X i < t , and for y ≥ t , E( T  y ) = P ( X 1 = t  X X i = y ) = P ( X 1 = t, ∑ X i = y ) P ( ∑ X i = y ) = P ( X 1 = t ) P ( ∑ n i =2 X i = y t ) P ( ∑ X i = y ) = { λ t e λ /t ! }{ [( n 1) λ ] y t e ( n 1) λ / ( y t )! } ( nλ...
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This note was uploaded on 04/18/2010 for the course STAT 622 taught by Professor Peruggia,m during the Spring '08 term at Ohio State.
 Spring '08
 Peruggia,M
 Variance

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