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

Description

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 densities, which is absolutely continuous. We illustrate through examples that those nonparametric Bayes estimates based on the Bernstein-Dirichlet process are more robust to sample variation and tend to have smaller estimation errors than those based on the Dirichlet process. In certain settings, the new estimators can even outperform Stein's estimator and Efron and Morris's limited translation estimator. Chapter 3 examines the asymptotic behavior of the correlation pursuit stepwise variable selection procedure that has been proposed recently by (Zhong et al ., 2008). More specifically, we analyze the asymptotic distribution of the test statistics under the null hypothesis of no effect for selected predictors and the power of the test under the alternative hypothesis. We also compare the new procedure with the classical linear regression algorithm for linear models, and discuss the possibility of generalizing the method to multiple index models.

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