The differentiation approach to derive the maximum likelihood estimator (mle) is notappropriate in all the cases. Let X₁, X2,,X₁ be a random sample of size n from thepopulation of X. Consider the probability function of Xf(x; 0) =Je-(2-0), if 0

The probability density functionf(x)=e-(x-θ)We have to finda..Argue that the mle of is the first…

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 Je-(2-0), if 0

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 Je-(2-0), if 0

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

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.
(a) Argue that the mle of 0 is the first order statistic (Y₁). Denote it by ê.
(b) Find E(), and show that the mle of 0 is a biased estimator.

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Step 1: Given that

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Step 2: Finding Maximum likelihood estimator

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Step 3: Checking biasness of mle

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Follow-up Questions

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Follow-up Question

(e) Show that the mme of 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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Follow-up Question

(e) Show that the mme of is an unbiased estimator of 0.
(f) Derive the mean square error (mse) of the mme.

Solution

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Follow-up Question

(c) Derive the mean square error (mse) of ê.
(d) Find the method of moments estimator (mme) of 0. Denote it by 0.

Solution

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