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Suppose that independent observations r1, 22,...,rn are available from the log-normal distribution with pdf 1 1 (log a S(r|0) = exp (27)\/20x on r> 0, where 0>0. It can be shown that the log-likelihood is e(0) = C – n log 0 - 202 i=1 (log a),

Suppose that independent observations r1, 22,...,rn are available from the log-normal distribution with pdf 1 1 (log a S(r|0) = exp (27)\/20x on r> 0, where 0>0. It can be shown that the log-likelihood is e(0) = C – n log 0 - 202 i=1 (log a),

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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Suppose that independent observations r1, 22,..., rn are available from the log-normal distribution with pdf 1 1 (log r f(r|0) : exp (2m)'/20x on r> 0, where 0 > 0. It can be shown that the log-likelihood is e(0) = C – n log 0 – 202 L (log x;)?, where C does not depend on 0 (you do not need to prove this result). Let ô be the MLE of 0. It also turns out to be the case that 2n E{-l"(0)} = 02 (Again, you do not need to prove this result.) In this question, interest lies in testing Họ : 0 = 60 against Hị : 0 # 0o. (a) Show that 1 1 log (LR) = –n log (log r:)². (b) What is the asymptotic null distribution of 2 log(LR)? Explain your answer. (c) How would you use your answer to part (b) to perform a fixed-level test of Ho of size a = 0,05? (No more calculations are needed.) (d) If 0o = 1, obtain formulae for both Wald statistics, W1 and W2. %3D
Transcribed Image Text:Suppose that independent observations r1, 22,..., rn are available from the log-normal distribution with pdf 1 1 (log r f(r|0) : exp (2m)'/20x on r> 0, where 0 > 0. It can be shown that the log-likelihood is e(0) = C – n log 0 – 202 L (log x;)?, where C does not depend on 0 (you do not need to prove this result). Let ô be the MLE of 0. It also turns out to be the case that 2n E{-l"(0)} = 02 (Again, you do not need to prove this result.) In this question, interest lies in testing Họ : 0 = 60 against Hị : 0 # 0o. (a) Show that 1 1 log (LR) = –n log (log r:)². (b) What is the asymptotic null distribution of 2 log(LR)? Explain your answer. (c) How would you use your answer to part (b) to perform a fixed-level test of Ho of size a = 0,05? (No more calculations are needed.) (d) If 0o = 1, obtain formulae for both Wald statistics, W1 and W2. %3D
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