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# HW6 - with mean μ and variance σ 2(a Use Slutzky’s...

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STA131C, Spring 2007 Prof. W. Polonik HW # 6 due: May 30, 2007 in class 1. Let X 1 , . . . , X n iid Poisson( λ ) , λ > 0 , and define hatwide λ = X . Show that both statistics T 1 = n ( hatwide λ - λ ) λ and T 2 = n ( hatwide λ - λ ) hatwide λ can be used to construct confidence intervals of asymptotic level 1 - α. (HINTS: First find the large sample (asymptotic) distribution of the two test statistics. Then use this large sample approximation to construct confidence intervals. Note that the case T 1 requires some special consideration to show that the resulting confidence set actually is a confidence interval.) 2. Let X
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Unformatted text preview: with mean μ and variance σ 2 . (a) Use Slutzky’s theorem to show that √ n ( X-μ ) 2 p → 0 as n → ∞ . (b) Use (a) to show that √ n ( S 2 n-σ 2 ) D → N (0 , γ 4 ) as n → ∞ , where γ 4 = E( X 1-μ ) 4-σ 4 . 3. Let X 1 , . . ., X n ∼ iid Beta( θ, 1) , θ > , i.e. their common pdF is f ( x ) = θ x θ-1 , x ∈ [0 , 1] . (a) Show that h θ = X 1-X is a method oF moment estimator oF θ . (b) ±ind the large sample distribution oF √ n ( θ-θ ) . (HINT: Use the δ-method.)...
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