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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

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 44CR
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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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