slides_10_asymptotic

# Special case is a 0 which gives consistency 21 we

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special case is a 0, which gives consistency.) 21

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We already showed in the convergence slides that the sample average is generally n -consistent, and we will use convergence in distribution to show it for very general classes of estimators. Some estimators that are consistent are not n -consistent: They converge in probability but more slowly than the rate 1/ n . This is rare for so-called “parametric estimators” but happens for nonparametric estimators. Occasionally, estimators converge at a rate faster than n . 22
EXAMPLE : Consider estimating in the Uniform 0, distribution under random sampling. One possible estimator is to use the maximum of the observed outcomes: ̃ n max 1 i n X i ̃ n can be shown to be consistent, and it can be shown that E ̃ n n / n 1  . In fact, it can generally be shown that for any positive integer k , E ̃ n k n n k k 23

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An unbiased and consistent estimator is ̂ n n 1 n ̃ n . Using the previous expression for k 2, it can also be shown that Var ̂ n 2 n n 2 2 n 2 2 n , which is on the order of 1/ n 2 , rather than the typical 1/ n . 24