Mathematical Sciences: Estimation in Semiparametric Models and Empirical Processes

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DMS 95-32039: Wellner This research involves the study of empirical processes, bootstrap methods in statistics, semiparametric models, and inverse problems and nonstandard asymptotics. The research will involve limit theory for infinite - dimensional M - estimates and related bootstrap methods, preservation theorems for uniform Donsker classes of functions, uniform in P bootstrap limit theorems, estimation of monotone and convex functions, convergence of iterative convex minorant algorithms, and the behavior of global functionals in models with interval censoring. %%% For the analysis of failure time data which are subject to right censoring and/or left truncation, many methods are now available. However, in many biomedical studies, including those arising from HIV/AIDS studies, failure times are censored or truncated in more complicated ways. One such complication is ``interval censoring'', in which one knows that the event of interest occurs within a random time interval, but not the exact timing: for example, HIV positivity or the drop of the CD4 count below a certain threshold may only be determined to be between two clinical visits, but not the exact time may remain unknown. This research will involve developing new statistical method for data arising from these and other more complicated censoring mechanisms ranging from animal carcinogenesis experiments to data in sociology and econometrics modelling. A detailed study and evaluation of the properties of methods recently proposed by other statisticians will be completed. ***