Differentiable Statistical Functionals and Bayes Asymptotics

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The investigator and co-workers are studying differentiability of functionals of empirical measures and distribution functions. In one dimension, p-variation norms work well for Frechet differentiability. Extensions to several dimensions are being pursued. In Bayes asymptotics, normal approximations with small relative errors are being found even for small probabilities of intermediate deviations.In Bayes asymptotics, one of the goals is to choose the best of several statistical models, possibly for multiple data sets. For example, multiple clinical trials may be done of a treatment for a disease. Three models are that the treatment is helpful, is harmful, or makes no difference. Further models incorporate the possibility that the treatment may have substantially different effects in different study populations. The procedure is to begin with a noninformative prior probability distribution on each model, then adjust it based on the likelihoods from each data set. Improved approximations of the updated probabilities are being investigated. For differentiable statistical functionals, a data set gives an approximation to a probability distribution. If a nonlinear transformation is applied both to the true distribution function and to its approximation, one looks for a linear transformation that approximates the nonlinear one as well as possible in the neighborhood of the true distribution. One is also looking for effective ways of bounding the discrepancy between the true and approximate distributions and their transformations.