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    188宝金博页面版: statisticalinference课件9

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    内容提示： 9Lecture 9: Higher order asymptotics9.1MotivationGeneral results in estimation and in other inference procedures can usually be defendedonly on asymptotic grounds (when the sample size n tends to infinity). An inferenceprocedure that is optimal for a finite sample size is very rarely possible to constructand it depends heavily on the specific distributional assumptions about the sample thathas given rise to the data. Hence such a procedure is too much individually tailoredand can not be offered as a genera...

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    9Lecture 9: Higher order asymptotics9.1MotivationGeneral results in estimation and in other inference procedures can usually be defendedonly on asymptotic grounds (when the sample size n tends to infinity). An inferenceprocedure that is optimal for a finite sample size is very rarely possible to constructand it depends heavily on the specific distributional assumptions about the sample thathas given rise to the data. Hence such a procedure is too much individually tailoredand can not be offered as a general methodological tool to be used in situations wherethe distributional assumptions have been changed. General inference procedures couldbe offered on asymptotic grounds only and we have seen how useful they can be whendiscussing the MLE, for example.Most useful in statistical practice are the so-called first-order asymptotic results.They state that under some regularity conditions asymptotic normality of certain esti-mator holds (or that under regularity conditions, the null distribution of a certain test-statistic is asymptotically normal or chi-squared etc.) We have already discussed manysuch results in Lectures 5 and 6.The techniques used to show such results are usually a combination of central limittheorem (CLT) and Taylor expansions. Again, in our Lecture 5 about properties of MLEwe have seen these techniques demonstrated at work. The final product of such resultsis the statement about asymptotic normality of a suitably normalized statistic. However,sometimes, the sample sizes used in practice are not large enough to warrant that theaccuracy achieved by the asymptotic normal approximation is precise enough. Then is isworth trying to include higher order expansions for the distribution of the statistic ofinterest hoping that these more complex expressions will bring about a better approxi-mation for the distribution when the sample size is not as large.To take an example to illustrate our point, assume that we are dealing with thedistribution of the sample mean¯X taken from a (not necessarily normal) population withfinite mean µ and variance σ2. Let Zn=σsays that Fn(z) →n→∞Φ(z) for every z ∈ R. If we also assume that a third finite momentγ exists then the famous Berry-Esseen theorem states that√n(¯X−µ)and Fn(z) = P(Zn≤ z). Then the CLT|Fn(z) − Φ(z)| = O(1√n)uniformly in z where the bound on the right hand side depends on the absolute thirdmoment. More precise statement can be obtained in the form|Fn(z) − Φ(z) −C1(F)p1(z)φ(z)√n| = O(1n) = o(1√n)uniformly in z where φ(z) is the density of the standard normal, C1is a suitably chosenconstant and p1(z) is certain first degree polynomial. Expansions in the formFn(z) = Φ(z) +ks=1qs(z)ns/2+ o(n−k/2)63

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