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29 definition the sequence of estimators n n 12 is n

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DEFINITION : The sequence of estimators ̂ n : n 1,2,. .. is n - asymptotically normal if n ̂ n d Normal 0 , C  ,all Θ where C is a p p positive semidefinite matrix for all . The matrix C is usually called the asymptotic variance of n ̂ n . Because n ̂ n converges in distribution it is also O p 1 . Therefore, any n -aysmptotically normal estimator is necessarily n -consistent. 30
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We can loosely write ̂ n a ~ Normal , C / n which, as we will see later, gives us the correct answer for constructing hypotheses tests and interval estimators. But it is the asymptotic distribution of n ̂ n that is nondegenerate. 31
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While there is some confusion across (and within) literatures, it is natural to define Avar ̂ n C / n ; that is, the asymptotic variance of ̂ n is defined to be the asymptotic variance of n ̂ n divided by n . 32
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Even if Var ̂ n exists, it is not necessarily true that Var ̂ n C / n . However, in “obvious” cases where Var ̂ n is easily computed, Var ̂ n usually equals Avar ̂ n . If Var ̂ n does not exist, or is not known to exist, Avar ̂ n is still well defined by n ̂ n d Normal 0 , C  Avar ̂ n C / n 33
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For a single parameter estimator, we can define the asymptotic standard deviation as the square root of the asymptotic variance. So, for example, Asd ̂ nj c jj / n c jj n where c jj is the j th diagonal element of C . We will use estimates of these asymptotic standard deviations later for large-sample statistical inference. 34
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EXAMPLE : For estimating the mean from a population with finite second moment, under random sampling, we know n X ̄ n d Normal 0, 2 and so Avar n X ̄ n  2 , which means we write Avar X ̄ n 2 / n . In this case, the asymptotic variance of X ̄ n is the same as the actual variance, Var X ̄ n . 35
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EXAMPLE : In the previous setting consider a sequence of estimators ̂ n a n X ̄ n , where a n is a nonrandom sequence with a n 1as n . With a n 1 this estimator is generally biased, but it is consistent (and actually asymptotically unbiased). We can compute the asymptotic distribution of n ̂ n provided n a n 1 0. It is not enough just to assume a n 1; the convergence has to happen at rate n ,so a n 1 O n 1/2 .
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29 DEFINITION The sequence of estimators n n 12 is n...

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