Dr. Hackney STA Solutions pg 130

Dr. Hackney STA Solutions pg 130 - Second Edition 8-9...

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Second Edition 8-9 Maximizing, by differentiating the log-likelihood, yields the MLEs ˆ μ = - n i log x i and ˆ θ = - m j log y j . Under H 0 , the likelihood is L ( θ | x , y ) = θ n + m i x i j y j θ - 1 , and maximizing as above yields the restricted MLE, ˆ θ 0 = - n + m i log x i + j log y j . The LRT statistic is λ ( x , y ) = ˆ θ m + n 0 ˆ μ n ˆ θ m i x i ˆ θ 0 - ˆ μ j y j ˆ θ 0 - ˆ θ . b. Substituting in the formulas for ˆ θ , ˆ μ and ˆ θ 0 yields ( i x i ) ˆ θ 0 - ˆ μ j y j ˆ θ 0 - ˆ θ = 1 and λ ( x , y ) = ˆ θ m + n 0 ˆ μ n ˆ θ m = ˆ θ n 0 ˆ μ n ˆ θ m 0 ˆ θ m = m + n m m m + n n n (1 - T ) m T n . This is a unimodal function of T . So rejecting if λ ( x , y ) c is equivalent to rejecting if T c 1 or T c 2 , where c 1 and c 2 are appropriately chosen constants. c. Simple transformations yield - log X i exponential(1 ) and - log Y i exponential(1 ). Therefore, T = W/ ( W + V ) where W and V are independent, W gamma( n, 1 ) and V gamma( m, 1 ). Under H 0 , the scale parameters of W and V are equal. Then, a simple generalization of Exercise 4.19b yields
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