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(a) Suppose the random variable, X, follows a geometric distribution with parameter 0 (0 < < 1). Let X₁, X2,..., X₁, be a random sample of size n from the population of X. Justify if == is an unbiased estimator for 0. 2 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 1 (1:0) = {6-(0-0), if 0≤x≤00 for-00<0 <0 otherwise. (g) Find the efficiencies of the mle and mme, and discuss under what condition the two estimators, and , are equally efficient.

(a) Suppose the random variable, X, follows a geometric distribution with parameter 0 (0 < < 1). Let X₁, X2,..., X₁, be a random sample of size n from the population of X. Justify if == is an unbiased estimator for 0. 2 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 1 (1:0) = {6-(0-0), if 0≤x≤00 for-00<0 <0 otherwise. (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)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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(a) Suppose the random variable, X, follows a geometric distribution with parameter 0 (0 < 0 <1). Let X₁, X2,..., Xn be a random sample of size n from the population of X. Justify if ô = i is an unbiased estimator for 0. 2 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 f(x; 0) = Je-(2-0), if 0<x<∞ for -∞<<∞ 0, otherwise. (g) Find the efficiencies of the mle and mme, and discuss under what condition the two estimators, and , are equally efficient.
Transcribed Image Text:(a) Suppose the random variable, X, follows a geometric distribution with parameter 0 (0 < 0 <1). Let X₁, X2,..., Xn be a random sample of size n from the population of X. Justify if ô = i is an unbiased estimator for 0. 2 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 f(x; 0) = Je-(2-0), if 0<x<∞ for -∞<<∞ 0, otherwise. (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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