Mathematical Sciences: Expanison and Likelihood Methods forMartingales and Martingale Inference

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This research is concerned with inference for longitudinal data, such as of survival analysis and time series type. Tests and confidence intervals in such problems are typically based on the asymptotic behavior of martingales. This project will focus on more accurate ways of setting critical points in such inference. One approach to this problem is to transform the martingales to create statistics more accurately approximated by the asymptotic distribution. Another approach is to look for better approximations to the law of the martingale, via Edgeworth or saddlepoint expansions, or bootstrapping. The research is intended to improve the way statisticians interpret data from processes that evolve over time. Main cases are (1) long term medical data, where patients are followed over a period of time, and (2) economic data. These quite different types of settings can be studied with a relatively unified technology, which relates to the theory of fair games. Substantial gains in how data are interpreted can be obtained by further exploiting this connection.