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 ( Xμ ) S where X = 1 n ∑ n i =1 X i and S 2 = 1 n ∑ n i =1 ( X iX ) 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 nθ ) → 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 nθ ) as n → ∞ ?...
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 Spring '07
 Polonik
 Statistics, Let X1, Xn, Prof. W. Polonik

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