hw8 - γ An unbiased estimator of γ is ˆ γ = n-1 ∑ n i...

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APPM 4/5520 Problem Set Eight (Due Wednesday, November 16th) 1. Let X 1 , X 2 , . . . X n be a random sample from the binomial distribution with parameters m and p . Find the MLE of P ( X = 0). 2. Let X 1 , X 2 , . . . , X n be a random sample from the Poisson ( λ ) distribution. (a) Find the Cram´ er-Rao lower bound (CRLB) for the variance of all unbiased estimators of λ . (b) Find the Cram´ er-Rao lower bound (CRLB) for the variance of all unbiased estimators of P ( X 1 = 0). (c) Find the Cram´ er-Rao lower bound (CRLB) for the variance of all unbiased estimators of P ( X (1) = 0) where X (1) = min( X 1 , X 2 , . . . , X n ). 3. Let X 1 , X 2 , . . . , X n be a random sample from the exponential distribution with rate λ . Recall that the MLE for λ is ˆ λ = ˆ λ n = 1 / X . Show directly, without quoting properties of MLEs, that (a) ˆ λ n is asymptotically unbiased (b) ˆ λ n is asymptotically e±cient 4. Let X 1 , X 2 , . . . , X n be a random sample from the Pareto distribution with parameter
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Unformatted text preview: γ . An unbiased estimator of γ is ˆ γ = n-1 ∑ n i =1 ln(1 + X i ) . (You do not have to show that it is unbiased.) Does the variance of this estimator achieve the CRLB? 5. Required for 5520 students only The quantity U ( v X, θ ) := ∂ ∂θ ln f ( v X ; θ ) given in the CRLB is known as the “score statistic”. (a) Show that the variance of the score statistic is the Fisher information. (b) Suppose there exists an unbiased estimator T ( v X ) of τ ( θ ). Show that T ( v X ) attains the CRLB if and only if the score statistic can be expressed in the form U ( v X ; θ ) = g ( θ )[ T ( v X )-τ ( θ )] for some function g ( θ ). ( Hint: Check out the proof of the CRLB on the website. )...
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