The differentiation approach to derive the maximum likelihood estimator (mle) is not appropriate in all the cases. Let X₁, X2, X₁ be a random sample of size n from the population of X. Consider the probability function of X -(2-0), if 0 < x <∞ for -x<0<∞ f(x; 0) = 10, otherwise. (d) Find the method of moments estimator (mme) of 8. Denote it by . (e) Show that the mme of 0 is an unbiased estimator of 0. (f) Derive the mean square error (mse) of the mme. (g) Find the efficiencies of the mle and mme, and discuss under what condition the two estimators, and , are equally efficient.
The differentiation approach to derive the maximum likelihood estimator (mle) is not appropriate in all the cases. Let X₁, X2, X₁ be a random sample of size n from the population of X. Consider the probability function of X -(2-0), if 0 < x <∞ for -x<0<∞ f(x; 0) = 10, otherwise. (d) Find the method of moments estimator (mme) of 8. Denote it by . (e) Show that the mme of 0 is an unbiased estimator of 0. (f) Derive the mean square error (mse) of the mme. (g) Find the efficiencies of the mle and mme, and discuss under what condition the two estimators, and , are equally efficient.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Transcribed Image Text:The differentiation approach to derive the maximum likelihood estimator (mle) is not
appropriate in all the cases. Let X₁, X2,,X, be a random sample of size n from the
population of X. Consider the probability function of X
f(x;0)
Se-(2-0), if 0<x<∞ for -∞ <<∞
10, otherwise.
(d) Find the method of moments estimator (mme) of 0. Denote it by 0.
(e) Show that the mme of 0 is an unbiased estimator of 0.
(f) Derive the mean square error (mse) of the mme.
(g) Find the efficiencies of the mle and mme, and discuss under what condition the two
estimators, and , are equally efficient.
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