Bootstrapping general empirical measures
نویسنده
چکیده
It is proved that the bootstrapped central limit theorem for empirical processes indexed by a class of functions F and based on a probability measure P holds a.s. if and only if F CLT (P ) and ∫ F dP < ∞, where F = supf F |f | and it holds in probability if and only if F ∈ CLT (P ). Thus, for a large class of statistics, no local uniformity of the CLT (about P ) is needed for the bootstrap to work. Consistency of the bootstrap (the bootstrapped law of large numbers) is also characterized. These results are proved under some mild measurability assumptions of F for P . AMS 1980 subject classifications. Primary: 60F17, 62E20; secondary: 60B12. Key works and phrases: bootstrapping, empirical processes, central limit theorem. 1 Research partially supported by National Science Foundation Grant No. DMS8619411 2 Research partially supported by National Science Foundation Grant No. DMS8601250 0 Introduction. B. Efron (1979) introduced the “bootstrap”, a resampling method for approximating the distribution functions of statistics Hn(X1, ..., Xn; P ), where the random variables Xi are independent, identically distributed with common law P (i.i.d.(P )). Since the empirical measure
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