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(a) Find a single statistic which is sufficient for the parameter 0. (b) Determine the method-of-moments estimator of 0, and call it 6. (c) Determine the maximum likelihood estimator of 0, and call it ê,.

(a) Find a single statistic which is sufficient for the parameter 0. (b) Determine the method-of-moments estimator of 0, and call it 6. (c) Determine the maximum likelihood estimator of 0, and call it ê,.

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 3CR
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6. Let Y1, Y2. ...,Y be a random sample of n iid continuous random variables drawn from a population with the pdf ( 2y fy (y) = { 02 y > 0 elsewhere Here, 0 > 0 is an unknown parameter. You may take as a given (you're welcome!) that the likelihood for this sample is 2" L(0\y1. 2.... Yn) %3! e j31 You may also assume that E(Y) = V Vo and E(Y*) = 20 3 %3D (a) Find a single statistic which is sufficient for the parameter 0. (b) Determine the method-of-moments estimator of 6, and call it 6. (c) Determine the maximum likelihood estimator of 6, and call it ô. (d) What is the maximum likelihood estimator of vo? Justify your answer by referring to an appropriate result from the notes/textbook. (e) Which of 6, or ô2 is the MVUE of 0? Both? Neither? Make sure to justify your answer. (f) Consider the problem of testing the simple hypotheses Ho: 0 = 1 against H,: 0 = 0a, where 0. > 1 is some known constant. Find the form of the rejection region of the most powerful a-level test. (g) Would the uniformly most powerful (UMP) level-a test of Ho: 0 = 1 against H,: 0 >1 have the same rejection region? Briefly explain why or why not.
Transcribed Image Text:6. Let Y1, Y2. ...,Y be a random sample of n iid continuous random variables drawn from a population with the pdf ( 2y fy (y) = { 02 y > 0 elsewhere Here, 0 > 0 is an unknown parameter. You may take as a given (you're welcome!) that the likelihood for this sample is 2" L(0\y1. 2.... Yn) %3! e j31 You may also assume that E(Y) = V Vo and E(Y*) = 20 3 %3D (a) Find a single statistic which is sufficient for the parameter 0. (b) Determine the method-of-moments estimator of 6, and call it 6. (c) Determine the maximum likelihood estimator of 6, and call it ô. (d) What is the maximum likelihood estimator of vo? Justify your answer by referring to an appropriate result from the notes/textbook. (e) Which of 6, or ô2 is the MVUE of 0? Both? Neither? Make sure to justify your answer. (f) Consider the problem of testing the simple hypotheses Ho: 0 = 1 against H,: 0 = 0a, where 0. > 1 is some known constant. Find the form of the rejection region of the most powerful a-level test. (g) Would the uniformly most powerful (UMP) level-a test of Ho: 0 = 1 against H,: 0 >1 have the same rejection region? Briefly explain why or why not.
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