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a) b) c) Show that for any >0 and S, X₁. lim P(|S, -612 ) = 0. Show that a statistic S, in a) is the maximum likelihood estimator of the parameter 8. Let 8₁=X₁2X+2X-X₁ and 6₂ = (X₁ + X₂ + X₁ + X₂) be two unbiased estimators of 6. W one of the two estimators is more efficient?

a) b) c) Show that for any >0 and S, X₁. lim P(|S, -612 ) = 0. Show that a statistic S, in a) is the maximum likelihood estimator of the parameter 8. Let 8₁=X₁2X+2X-X₁ and 6₂ = (X₁ + X₂ + X₁ + X₂) be two unbiased estimators of 6. W one of the two estimators is more efficient?

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.CR: Chapter 13 Review
Problem 52CR
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Suppose that a sequence of mutually independent and identically distributed discrete random variables X₁, X₂, X3, X has the following probability density function f(x: 0) = a) b) c) 6e-0 x! for x = 0,1,2,... 0, elsewhere Show that for any e > 0 and S, X₁, lim P(|S, -01 ≥ £) = 0. Show that a statistic S, in a) is the maximum likelihood estimator of the parameter 8. Let 8₁ = X₁ +2x+2x₁-X and 6₂ = (X₁ + X₂ + X₁ + X₁) be two unbiased estimators of 8. Which one of the two estimators is more efficient? d) What is the Cramer-Rao lower bound for the variance of the unbiased estimator of the parameter @? e) Use the one-parameter regular exponential family definition to find the functions,
Transcribed Image Text:Suppose that a sequence of mutually independent and identically distributed discrete random variables X₁, X₂, X3, X has the following probability density function f(x: 0) = a) b) c) 6e-0 x! for x = 0,1,2,... 0, elsewhere Show that for any e > 0 and S, X₁, lim P(|S, -01 ≥ £) = 0. Show that a statistic S, in a) is the maximum likelihood estimator of the parameter 8. Let 8₁ = X₁ +2x+2x₁-X and 6₂ = (X₁ + X₂ + X₁ + X₁) be two unbiased estimators of 8. Which one of the two estimators is more efficient? d) What is the Cramer-Rao lower bound for the variance of the unbiased estimator of the parameter @? e) Use the one-parameter regular exponential family definition to find the functions,
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