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Unformatted text preview: Second Edition 8-9 Maximizing, by differentiating the log-likelihood, yields the MLEs =- ^ Under H0 , the likelihood is L(|x, y) = n+m i n i log xi ^ and = - m . j log yj -1 xi
j yj , and maximizing as above yields the restricted MLE, ^ 0 = - The LRT statistic is ^ m+n (x, y) = 0 n m ^ ^
^ 0 -^ n+m . i log xi + j log yj j 0 - ^ ^ yj . xi
i ^ ^ ^ b. Substituting in the formulas for , and 0 yields ( (x, y) = ^ ^ ^m+n 0 n m = 0 0 = n m ^ ^ n m ^ ^ m+n m
m i xi ) ^ 0 -^ ^ ^ 0 - j yj n = 1 and m+n n (1 - T )m T n . This is a unimodal function of T . So rejecting if (x, y) c is equivalent to rejecting if T c1 or T c2 , where c1 and c2 are appropriately chosen constants. c. Simple transformations yield - log Xi exponential(1/) and - log Yi exponential(1/). Therefore, T = W/(W + V ) where W and V are independent, W gamma(n, 1/) and V gamma(m, 1/). Under H0 , the scale parameters of W and V are equal. Then, a simple generalization of Exercise 4.19b yields T beta(n, m). The constants c1 and c2 are determined by the two equations P (T c1 ) + P (T c2 ) = 8.18 a. () |X-0 | |X-0 | >c c = 1 - P / n / n c c = 1 - P - X-0 n n -c/ n + 0 - X- c/ n + 0 - = 1 - P / n / n / n 0 - 0 - = 1 - P -c + Z c + / n / n 0 - 0 - = 1 + -c + - c+ , / n / n = P and (1 - c1 )m cn = (1 - c2 )m cn . 1 2 where Z n(0, 1) and is the standard normal cdf. ...
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- Spring '12