The investigator plans to concentrate his efforts on large sample theory problems connected with self-normalized sums and functionals of the empirical distribution. His interest in self-normalized sums arose largely out of the question: When is the Student-t statistic asymptotically normal? Among his goals in this area, he hopes to complete the theory of the asymptotic distribution of self-normalized sums. The particular functionals of the empirical distribution that he plans to investigate include statistics and estimators based on distance measures such as the empirical Hellinger distance and a class of integral functionals of density estimators, which are used to estimate entropy and the integral of the squared density and its derivative. This will likely lead to the development of a theory of local U-statistics processes. This theory would be analogous to the local empirical process theory that he has worked out with a number of his long-term collaborators to study the large sample properties of kernel-type function estimators. These include kernel-type estimators of the density function, the regression function and the conditional distribution function. Local empirical process theory has shown itself to be a very powerful tool to determine exact rates of point-wise and uniform consistency of kernel-type nonparametric function estimators. A local U-statistics process theory should be at least as applicable. The research problems outlined in this proposal should lead to the creation of a number of new and useful methods to treat large sample theory problems.
The investigator works in the borderline area between probability theory and mathematical statistics. He is delighted when he uncovers unexpected probabilistic phenomena that occasionally appear on this frontier. The problems that he is interested in are frequently motivated by questions about the limit behavior of statistical tests and estimators. They arise whenever one wants to make a decision on the basis of limited knowledge. This is especially the case when it is not feasible to sample an entire population to settle a question about that population. Some typical examples are when a medical investigator is interested in deciding whether or not a proposed new drug or medical treatment is effective or a fisheries scientist wants to estimate the number of salmon that will return this season to spawn.
