36 write n n n a n x n n x n n a n 1 x n n x n o 1 o

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Write n ̂ n n a n X ̄ n n X ̄ n n a n 1 X ̄ n n X ̄ n o 1 O p 1 n X ̄ n o p 1 because n a n 1 o 1 and X ̄ n O p 1 . It follows by the asymptotic equivalence lemma that n ̂ n and n X ̄ n have the same asymptotic distribution, so n ̂ n d Normal 0, 2 even though E ̂ n a n and 2 Var n ̂ n  a n 2 2 . 37
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Establishing the asymptotic normality of estimators n ̂ n is often called first - order asymptotic theory . It is by far the easiest asymptotic theory. There are other refinements that attempt to improve the approximation, but they are generally difficult. 38
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DEFINITION :Let ̂ n ̂ n , ̂ n be an estimator of , , where ̂ n is q 1 and ̂ n is r 1. Assume n ̂ n d Normal 0 , C  , where we partition C as C C 11 C 12 C 12 C 22 and C 11 is q q ; C 12 is q r ; C 22 is r r . 39
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Then n ̂ n and n ̂ n are asymptotically uncorrelated if C 12 0 . In other words, the asymptotic covariance matrix is block diagonal, and looks like C C 11 0 0C 22 where Avar n ̂ n  C 11 Avar n ̂ n  C 22 40
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As a shorthand, we sometimes say ̂ n and ̂ n are asymptotically uncorrelated. But remember we might not even be able to compute the covariance between ̂ n and ̂ n , let alone show it is zero for any n . Later we will show that, for populations with finite fourth moments, the sample average and the sample variance are asymptotically uncorrelated whenever the distribution is symmetric. 41
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4 . The Delta Method We can use the various properties of convergence in probability and convergence in distribution to obtain the asymptotic distribution of a large class of estimators. For motivation, consider the problem of estimating exp P X 0 using a random sample from the Poisson distribution. 42
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We know a consistent estimator of is ̂ n exp X ̄ n . But its mean is complicated and its variance even moreso. Its full distribution is difficult to obtain. Question: Can we easily approximate the distribution of ̂ n ,atleastin “large” samples? Yes.
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36 Write n n n a n X n n X n n a n 1 X n n X n o 1 O p 1 n...

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