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2. Consider a random sample X₁, X2,..., Xn of size n from the Uniform (0, 0) distri- bution, where @ > 0. (i) Show that the p.d.f. q(t; 0) of the sufficient statistic is given by T(X)= max X₁ 1

2. Consider a random sample X₁, X2,..., Xn of size n from the Uniform (0, 0) distri- bution, where @ > 0. (i) Show that the p.d.f. q(t; 0) of the sufficient statistic is given by T(X)= max X₁ 1

MATLAB: An Introduction with Applications
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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2. Consider a random sample X₁, X2,..., Xn of size n from the Uniform (0, 0) distri- bution, where @ > 0. (i) Show that the p.d.f. q(t; 0) of the sufficient statistic is given by T(X)= max X₁ 1<i<n ntn-1 0≤t ≤0. An Deduce expressions for the mean and variance of T. (ii) Show that the method of moments gives the estimator 0* = 2X of 0. Comment on this estimator with regard to the sufficiency principle and show that it can yield estimates of that are incompatible with the sample data, i.e. that there are cases in which it is not possible for * to attain the value 0. q(t; 0) = = (iii) Write down the maximum likelihood estimator of and show that it is not unbiased. (iv) Find the constant a such that aT is an unbiased estimator of 0. (v) Find the mean squared errors of the estimators of parts (ii), (iii) and (iv), respectively, and comment on their relative values.
Transcribed Image Text:2. Consider a random sample X₁, X2,..., Xn of size n from the Uniform (0, 0) distri- bution, where @ > 0. (i) Show that the p.d.f. q(t; 0) of the sufficient statistic is given by T(X)= max X₁ 1<i<n ntn-1 0≤t ≤0. An Deduce expressions for the mean and variance of T. (ii) Show that the method of moments gives the estimator 0* = 2X of 0. Comment on this estimator with regard to the sufficiency principle and show that it can yield estimates of that are incompatible with the sample data, i.e. that there are cases in which it is not possible for * to attain the value 0. q(t; 0) = = (iii) Write down the maximum likelihood estimator of and show that it is not unbiased. (iv) Find the constant a such that aT is an unbiased estimator of 0. (v) Find the mean squared errors of the estimators of parts (ii), (iii) and (iv), respectively, and comment on their relative values.
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