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Practice Final

Practice Final - 3 Let X 1 X n ∼ iid F with E X = μ and...

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STA131C, Spring 2007 Prof. W. Polonik Practice Exercises III 1. Let X 1 , . . . , X n iid Poisson( λ ) . (a) Find the Fisher-Information I ( λ ) (based on one observation) for this model. (b) Find the large sample distribution of the MLE hatwide λ . (c) Show that g ( x ) = 2 x is a variance stabilizing transformation for hatwide λ . (In other words, show that 2 n ( radicalbig hatwide λ - λ ) D N (0 , 1) by using the δ -method.) 2. Let X 1 , . . . , X n iid F with F = f. (a) Find a formula for the pdf of X (1) = min { X 1 , . . . , X n } in terms of F and f . Now consider the special case of X i iid U (0 , 1) . (b) Find E( X (1) ) and Var( X (1) ) . (c) Show that X (1) 0 in probability as n → ∞ . (Observe here that X (1) of course depends on the sample size n , although the used notation does not reflect this.)
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Unformatted text preview: 3. Let X 1 , . . ., X n ∼ iid F with E( X ) = μ and Var( X ) = σ 2 < ∞ . Consider the statistic T n = √ n ( X-μ ) S where X = 1 n ∑ n i =1 X i and S 2 = 1 n ∑ n i =1 ( X i-X ) 2 . Show that T n → D N (0 , 1) as n → ∞ . You can use the fact that S 2 n → σ 2 in probability as n → ∞ . 4. Suppose you know that √ n ( h θ n-θ ) → N (0 , σ 2 ) as n → ∞ where σ 2 < ∞ . (a) Show that this result implies that h θ → θ in probability as n → ∞ . (b) What is the limit (in probability) of n 1 / 4 ( h θ n-θ ) as n → ∞ ?...
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