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Unformatted text preview: 84 Solutions Manual for Statistical Inference b. The LRT statistic is (x) = sup (1/ n )ei xi / sup, ( n / n )(
i 1 i x / . i xi ) e ^ The numerator is maximized at 0 = x. For fixed , the denominator is maximized at ^ = xi /n. Thus i (x) = xn en ^ sup ( n / )(
n i 1 ^ i x / i = xi ) e xn n n / )( sup ( ^ i xi ) 1 . The denominator cannot be maximized in closed form. Numeric maximization could be used to compute the statistic for observed data x. 8.8 a. We will first find the MLEs of a and . We have
n L(a,  x) log L(a,  x) Thus log L a log L n =
i=1 n 2 1 e(xi ) /(2a) , 2a = 1 1 (xi  )2 .  log(2a)  2 2a i=1 =
i=1 n  
i=1 1 1 2 + (x  ) 2a 2a2 i =  n 1 + 2a 2a2 n (xi  )2
i=1 set = 0 = 1 1 1 2 + (x  ) + (xi  ) 2 2a2 i a
n n 1 =  + 2 2a2 (xi  )2 +
i=1 n  n x a set = 0. ^ We have to solve these two equations simultaneously to get MLEs of a and , say a and . ^ Solve the first equation for a in terms of to get a= 1 n
n (xi  )2 .
i=1 Substitute this into the second equation to get  ^ So we get = x, and a= ^ 1 n x
n n n n() x + + = 0. 2 2 a 2 ^ , x (xi  x)2 = i=1 the ratio of the usual MLEs of the mean and variance. (Verification that this is a maximum is lengthy. We omit it.) For a = 1, we just solve the second equation, which gives a quadratic in that leads to the restricted MLE ^ R = 1+ 1+4(^ +2 ) x 2 .
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This note was uploaded on 02/03/2012 for the course STA 1014 taught by Professor Dr.hackney during the Spring '12 term at UNF.
 Spring '12
 Dr.Hackney
 Statistics

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