Large Sample Problems Arising on the Borderline between Probability and Mathematical Statistics

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The investigator plans to concentrate his efforts on large sample theoryproblems connected with self-normalized sums and functionals of the empiricaldistribution. His interest in self-normalized sums arose largely out of thequestion: When is the Student-t statistic asymptotically normal? Among hisgoals in this area, he hopes to complete the theory of the asymptoticdistribution of self-normalized sums. The particular functionals of theempirical distribution that he plans to investigate include statistics andestimators based on distance measures such as the empirical Hellinger distanceand a class of integral functionals of density estimators, which are used toestimate entropy and the integral of the squared density and its derivative.This will likely lead to the development of a theory of local U-statisticsprocesses. This theory would be analogous to the local empirical process theorythat he has worked out with a number of his long-term collaborators to studythe large sample properties of kernel-type function estimators. These includekernel-type estimators of the density function, the regression function and theconditional distribution function. Local empirical process theory has shownitself to be a very powerful tool to determine exact rates of point-wise anduniform consistency of kernel-type nonparametric function estimators. A localU-statistics process theory should be at least as applicable. The researchproblems outlined in this proposal should lead to the creation of a number ofnew and useful methods to treat large sample theory problems.The investigator works in the borderline area between probability theory andmathematical statistics. He is delighted when he uncovers unexpectedprobabilistic phenomena that occasionally appear on this frontier. The problemsthat he is interested in are frequently motivated by questions about the limitbehavior of statistical tests and estimators. They arise whenever one wants tomake a decision on the basis of limited knowledge. This is especially the casewhen it is not feasible to sample an entire population to settle a questionabout that population. Some typical examples are when a medical investigator isinterested in deciding whether or not a proposed new drug or medical treatmentis effective or a fisheries scientist wants to estimate the number of salmonthat will return this season to spawn.