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slides_10_asymptotic

# 57 this looks just like the variance of a linear

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This looks just like the variance of a linear combination a 1 W 1 a 2 W 2 with a 1 1 , a 2  2 , Var W 1 c 11 , Var W 2 c 22 , and Cov W 1 , W 2 c 12 . 58
EXERCISE :Let be a p 1 parameter vector with n ̂ n d Normal 0 , C  Define exp 1 2 ... p ̂ n exp ̂ n 1 ̂ n 2 ̂ np Show that Avar n ̂ n  2 j 1 p h 1 p c jh . 59

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5 . Asymptotic Equivalence and Asymptotic Efficiency We need a way to choose among consistent estimators. Two estimators can be consistent but one can have a distribution more tightly centered about the parameter it is estimating. Unlike finite sample analysis, where we compare variances of unbiased estimators, we have to settle for comparing asymptotic variances. Here we focus only on estimators that are n -consistent. 60
DEFINITION : Two sequences of n -consistent estimators for Θ , ̂ n : n 1,2,. .. and ̃ n : n are asymptotically equivalent (more precisely, n - asymptotically equivalent )if n ̃ n ̂ n n ̃ n n ̂ n o p 1 . By the asymptotic equivalence lemma, estimators that are n -asymptotically equivalent have the same asymptotic distribution (when scaled by n ). When n ̂ n and n ̃ n are asymptotically normal – the case we usually consider – asymptotic equivalance means n ̂ n and n ̃ n have the same asymptotic variances. 61

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EXAMPLE : Consider estimating the mean from a population with mean , variance 2 , using random sampling. Take ̂ n X ̄ n and ̃ n a n X ̄ n where a n : n 1,2,. .. is a nonrandom sequence with a n 1. Then n ̃ n X ̄ n n a n 1  X ̄ n Because X ̄ n p , asymptotic equivalence between the two estimators is ensured by n a n 1 o 1 which shows that a n must converge to unity sufficiently quickly. 62
In some previous examples, a n 1 c / n for some constant c , and so n a n 1 c / n 0. But, say, a n 1 cn 3/4 would work, too. There are uncountably many other choices. These kinds of examples again illustrate how one can abuse asymptotic analysis. For a given sample size n , we could choose a n very far from one – by choosing c large in magnitude – and just assert that, as n , a n will approach one. If possible – and it rarely is in complicated situations – we can use unbiasedness, or look at mean squared error, to limit abuse of asymptotics.

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57 This looks just like the variance of a linear...

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