Two new algorithms for nonparametric analysis given incomplete data.

摘要

The nonparametric Bayesian analysis for incomplete data has been limited due to its computational difficulties, because the posterior distribution under a Dirichlet process prior is a very complicated mixture over possible partitions of the data. Even the latest Markov chain Monte Carlo simulation methods share this problem, especially when sample size is moderate or large.;In this thesis, as a computationally simple alternative, we study a simple recursive method called partial predictive recursion (PPR) for mixture models and interval censored data. This algorithm calculates approximate Bayes estimates under a Dirichlet process prior. The consistency of the algorithm provides theoretical support for the approximation. We also show that PPR has a potentially useful non-Bayesian application which can be used to obtain solutions of the self-consistency equations that are known to be satisfied by the nonparametric maximum likelihood estimate. So, PPR is an alternative to the EM algorithm and other methods that are used to solve such equations. Moreover, for finite mixture models and the interval censored problem, PPR can be directly applied to calculate the nonparametric maximum likelihood estimator. To numerically investigate the performance of this approximation, PPR was compared to the Markov chain Monte Carlo methods for real examples.;Also in this thesis, we propose a new algorithm called the pool-monotone-groups algorithm (PMGA) for calculating the nonparametric maximum likelihood estimator given interval censored data. PMGA has advantages over existing methods such as the pool-adjacent-violators algorithm (PAVA) and the greatest-convex-minorant (GCM) algorithm. Finally, we propose a survival function estimator for the double censored problem. A variance estimator for the nonparametric maximum likelihood estimator for double censored data is also proposed.