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Unformatted text preview: Second Edition 103 and for t = 1 ARE n ^ n1 n1 n
^ n ^ , e = (  1)e
n n1 n1 n n 2 n1 n n . 1 + log Since [(n  1)/n]n e1 as n , both of these AREs are equal to 1 in the limit. ^ d. For these data, n = 15, Xi = y = 104 and the MLE of is = X = 6.9333. The estimates are MLE UMVUE P (X = 0) .000975 .000765 P (X = 1) .006758 .005684 10.11 a. It is easiest to use the Mathematica code in Example A.0.7. The second derivative of the log likelihood is 2 log 2 1 x1+/ ex/ [/] / = 1 (/), 2 where (z) = (z)/(z) is the digamma function. b. Estimation of does not affect the calculation. c. For = known, the MOM estimate of is x/. The MLE comes from differentiating the log likelihood d set n log  xi / = 0 = x/. d i d. The MOM estimate of comes from solving 1 n xi = and
i 1 n x2 = 2 + , i
i ~ ^ x which yields = 2 /. The approximate variance is quite a pain to calculate. Start from 1 2 , E^ 2 , Var^ 2 3 , n n where we used Exercise 5.8(b) for the variance of 2 . Now using Example 5.5.27 and (and ^ ~ 3 3 . The ARE is then assuming the covariance is zero), we have Var n EX = , VarX = ^ ~ ARE(, ) = 3 3 / E  d2 l(, X d 2 . Here is a small table of AREs. There are some entries that are less than one  this is due to using an approximation for the MOM variance. 1 2 3 4 5 6 7 8 9 10 1 1.878 4.238 6.816 9.509 12.27 15.075 17.913 20.774 23.653 26.546 3 0.547 1.179 1.878 2.629 3.419 4.238 5.08 5.941 6.816 7.704 6 0.262 0.547 0.853 1.179 1.521 1.878 2.248 2.629 3.02 3.419 10 0.154 0.317 0.488 0.667 0.853 1.046 1.246 1.451 1.662 1.878 ...
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 Spring '12
 Dr.Hackney
 Statistics

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