63 definition let θ p be a parameter vector and let

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DEFINITION :Let Θ p be a parameter vector and let ̂ n : n 1,2,. .. and ̃ n : n be two n -asymptotically normal estimators of with Avar n ̂ n  C 1 and Avar n ̃ n  C 2 . Then ̂ n : n is asymptotically efficient relative to ̃ n : n if C 2 C 1 is positive semidefinite for all Θ . 64
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EXAMPLE : Consider random sampling from the Poisson distribution where we want to estimate exp P X 0 . Consider two estimators ̂ n exp X ̄ n ̃ n n 1 i 1 n 1 X i 0 Note that ̃ n is simply the fraction of zeros we observe in the sample. 65
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If we define a Bernoulli random variable as Y i 1 X i 0 then P Y i 1 . Since ̃ n Y ̄ n we know that ̃ n is actually unbiased with variance 1 / n exp  1 exp  / n . By the CLT, Avar n ̃ n  1 exp  1 exp  . Using the delta method, we derived Avar n ̂ n  exp 2 . 66
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We can show Avar n ̃ n  Avar n ̂ n  by noting it is true if and only if exp exp 2 exp 2 or, multplying through by exp 2 , exp 1 . This inequality is easily seen to be true because the function h exp 1 is zero when 0 and dh / d exp 1 0 for all 0. 67
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The previous example highlights a few useful points to carry forward. Without specifying anything about the population distribution (except that the event X 0 occurs with positive probability), ̃ n is consistent (and unbiased). Therefore, it is “robust” in the sense that it is valid for any count distribution. The estimator ̂ n exp X ̄ n is intrinsically tied to the Poisson distribution because it relies on the expression P X 0 exp , where E X . Therefore, ̂ n will be inconsistent for any distribution where P X 0 exp E X  (which is the vast majority). 68
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If the population is Poisson , we have shown that ̂ n is asymptotically more efficient than ̃ n , and ̃ n can be very inefficient. The ratio of the asymptotic variances is exp 1 , and this increases rapidly as increases.
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63 DEFINITION Let Θ p be a parameter vector and let n n 12...

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