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Suppose that independent observations a1, r2,...,n are available from the log-normal distribution with pdf 1 f(r|0) = 1 (log r on r> 0, exp (27)!/20x where 0> 0. It can be shown that the log-likelihood is e(0) = C – n log 0 1 (log r.)², 202

Suppose that independent observations a1, r2,...,n are available from the log-normal distribution with pdf 1 f(r|0) = 1 (log r on r> 0, exp (27)!/20x where 0> 0. It can be shown that the log-likelihood is e(0) = C – n log 0 1 (log r.)², 202

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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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, 2 where 0 > 0. It can be shown that the log-likelihood is e(0) = C – n log 0 – 202 L (log x;)², 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 0 = 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, 2 where 0 > 0. It can be shown that the log-likelihood is e(0) = C – n log 0 – 202 L (log x;)², 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 0 = 1, obtain formulae for both Wald statistics, W1 and W2. %3D
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