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Dr. Hackney STA Solutions pg 123

Dr. Hackney STA Solutions pg 123 - 8-2 Solutions Manual for...

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8-2 Solutions Manual for Statistical Inference and d dy log λ ( y ) = log θ 0 - log(1 - θ 0 ) - log y m - y 1 y + log m - y m + ( m - y ) 1 m - y = log θ 0 y/m ( m - y m ) 1 - θ 0 . For y/m > θ 0 , 1 - y/m = ( m - y ) /m < 1 - θ 0 , so each fraction above is less than 1, and the log is less than 0. Thus d dy log λ < 0 which shows that λ is decreasing in y and λ ( y ) < c if and only if y > b . 8.4 For discrete random variables, L ( θ | x ) = f ( x | θ ) = P ( X = x | θ ). So the numerator and denomi- nator of λ ( x ) are the supremum of this probability over the indicated sets. 8.5 a. The log-likelihood is log L ( θ, ν | x ) = n log θ + log ν - ( θ + 1) log i x i , ν x (1) , where x (1) = min i x i . For any value of θ , this is an increasing function of ν for ν x (1) . So both the restricted and unrestricted MLEs of ν are ˆ ν = x (1) . To find the MLE of θ , set ∂θ log L ( θ, x (1) | x ) = n θ + n log x (1) - log i x i = 0 , and solve for θ yielding ˆ θ = n log( i x i /x n (1) ) = n T . ( 2 /∂θ 2 ) log L ( θ, x (1) | x ) = - n/θ 2 < 0, for all θ . So ˆ θ is a maximum.
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