Nonparametric studies of doubly stochastic Poisson processes, binomial data, and high dimension, low sample size data

This thesis presents three nonparametric methods respectively for (1) making inferences of doubly stochastic Poisson processes; (2) analyzing binomial data via Bernstein polynomial priors; (3) performing variable selection for high dimension, low sample size data. In Chapter 1 of this thesis, we analyze sequences of arrival data that follow doubly stochastic Poisson processes. Complementing existing parametric methods, we consider nonparametric inference for stochastic arrival rates, paying particular attention to their autocorrelation function (ACF). We introduce a kernel method to reconstruct the arrival rate from the Poisson data, and to estimate the autocorrelation function. We consider both practical implementation and theoretical properties of the method, illustrating through simulated examples as well as the analysis of real photon arrival data from single-molecule experiments in biophysics. In Chapter 2 of this thesis, we examine nonparametric hierarchical Bayes procedures that employ the Bernstein-Dirichlet processes as prior distributions for analyzing binomial data. We find that the predictive density of a future binomial observation can be expressed as a mixture of beta