∙We already showed in the convergence slides that the sample averageis generallyn-consistent, and we will use convergence in distributionto show it for very general classes of estimators.∙Some estimators that are consistent are notn-consistent: Theyconverge in probability but more slowly than the rate 1/n. This is rarefor so-called “parametric estimators” but happens for nonparametricestimators.∙Occasionally, estimators converge at a rate faster thann.22
EXAMPLE: Consider estimatingin theUniform0,distributionunder random sampling. One possible estimator is to use the maximumof the observed outcomes:̃nmax1≤i≤nXi∙̃ncan be shown to be consistent, and it can be shown thatẼnn/n1. In fact, it can generally be shown that for anypositive integerk,Ẽnknnkk23
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∙An unbiased and consistent estimator iŝnn1ñn.Using the previous expression fork2, it can also be shown thatVar̂n2nn22n22n,which is on the order of 1/n2, rather than the typical 1/n.24