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    188宝金博页面版: statisticalinference课件8

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    内容提示： 8Lecture 8: Robustness. Estimating statistical func-tionals.8.1Motivation. Basic idea of robustnessAlong this course, we have studied theories about how to construct optimal procedures(be they Likelihood-based or Bayesian) when certain parametric model F(X, θ) is given.These theories say nothing about the behaviour of the optimal procedures when themodels are only approximately valid. Going over in such cases directly to purely Non-parametric approach would also not address properly the situation since th...

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    8Lecture 8: Robustness. Estimating statistical func-tionals.8.1Motivation. Basic idea of robustnessAlong this course, we have studied theories about how to construct optimal procedures(be they Likelihood-based or Bayesian) when certain parametric model F(X, θ) is given.These theories say nothing about the behaviour of the optimal procedures when themodels are only approximately valid. Going over in such cases directly to purely Non-parametric approach would also not address properly the situation since the idea about(relatively small) deviation form a baseline parametric model would be lost. The properapproach would be the robustness approach where we still keep the idea about the idealparametric model but allow for deviations from it. Speaking loosely, nonparametric statis-tics allows ”all” possible probability distributions and reduces the ignorance about themonly by one or a few dimensions. Classical parametric statistics allows only a very ”thin”finite-dimensional subset of probability distributions, i.e., the ideal parametric model ofinterest for which usually optimal inferences are available. Robust statistics allows a full-dimensional neighbourhood of a parametric model, thus being more realistic and yet,at a price of a relatively small loss of efficiency at the ideal model, provides almost thesame advantages as a strict parametric model in a ”broader” neighbourhood of the idealparametric model.The problem in robustness is to construct estimators that are close to efficient ifthe parametric model holds but are at the same time less sensitive to small deviationsfrom the ideal model.8.1.1Simple exampleOne of the simple examples to start with, is estimating the location parameter of acontinuous symmetric distribution. Assume a sample x1, x2, . . . , xn is available from alocation parameter family F(x, θ) = F(x − θ), θ ∈ R1. Denote the density by f(x, θ) =f(x − θ). If F is a normal distribution then θ coincides with its mean, median and mode.As we know, in this case the estimator ¯ x is efficient for θ for any fixed sample size. Butassume now that F is Cauchy with a density f(x) =parameter θ in this model does not coincide with the mean of the distribution (in fact, theCauchy distribution does not have a finite mean) but coincides with its median and mode.It can be shown (see lecture) that ¯ x has the same distribution as the distribution ofa single observation from the Cauchy model! Therefore ¯ x is even not consistent for θ inthe Cauchy model! The reason for the good behaviour of ¯ x as an estimator of locationparameter θ in the normal family and for its ”bad” behaviour in the Cauchy family arethe heavy tails of the Cauchy distribution, i.e. it allows with a large probability for verylarge (in absolute value) realizations to occur. Because ofthis observation, we would decideto ignore the observations with a large absolute value and use the empirical medianinstead when estimating the location parameter of the Cauchy distribution. The empiricalmedian˜θnis not sensitive to large realizations in the tail of the distribution, hence it ismore robust as a location parameter estimator.1π11+x2. Then f(x, θ) =1π11+(x−θ)2. The56

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