Question

A random sample of size n, {XI, , X, from an exponential population with mean ?, is to be used to test Ho : ? ?? versus H1 : ??Bo for a given value of ?? (a) Show that the expression for likelihood ratio statistic is ? ( ) eT (b) Show that the critical region of the likelihood ratio test can be written as (c) Without referring to Wilks theorem (Theorem 9.1.4), show that -2log(A) is approximately dis- tributed as ?2(1) for large n under Ho. Hints: Use the second order Taylor expansion log(12/2; . and the central limit theorem for X; » and note that n +n -2 log(A)--2 log e- 0 d) Using Wilks theorem (or part (c)), show that for large n the critical region -2 log(A) > 3.841 has size 0.05, approximately. (e) Use the central limit theorem to show that for large n a critical region .-80>1.96 has size 0.05, approximately. (f) Discuss the relationship between two tests in (d) and (e)

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December D 팅 うー ㄣ Tafierss ThDecember-eNol 2. χ005, bo i = 3.84 of 0.0 25ラ7:00.5 x-9 2 ラズ>χα

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