Markov Chain Monte Carlo stochastic approximation algorithms, Smoothing Spline ANOVA frailty models and applications

This thesis contains three parts: (I) potential problems in the implementation of the Markov Chain Monte Carlo Stochastic Approximation Algorithms (MCMCSAA), remedies and new adaptive algorithms; (II) Smoothing Spline ANOVA (SS ANOVA) frailty models; and (III) application of methods in the first two parts to investigate hormone generating mechanisms.;Part I consists of Chapters one and two. Chapter one introduces MCMCSAA and new hybrid algorithms. These new algorithms are proposed to improve the speed and stability of the existing ones. Chapter two compares the performance of various MCMCSAA using simulations. We implement the existing algorithms and the new hybrid algorithms with three different stopping criterions. We explore the essential factors affecting the speed and precision of the algorithms, and find that MCMCSAA can be sensitive to the choices of the initial values, MCMC sample size, step-size and the form of the I-matrix. There is no ultimate best algorithm for both precision and efficiency. In general, the proposed hybrid algorithms are stable and fast.;Part II consists of Chapters three and four. Chapter three introduces new SS ANOVA frailty models for recurrent events data. The general estimation methods utilizing penalized likelihood are proposed for the SS ANOVA frailty models. We adapt and modify the MCMCSAA for the new frailty models. Chapter four carries out two sets of simulations for the new frailty models. One set is based on the Weibull distribution and the other one is based on Gumbel distribution. The bootstrap confidence intervals are constructed as well. From simulations, we conclude that all MCMCSAA with suitable stopping criteria provide sensible estimates. And the new hybrid algorithm G4 achieves the same precision in about half of the CPU time comparing to existing algorithms.;Part III consists of Chapter five. It applies the new frailty models with hybrid MCMCSAA algorithm to investigate hormone secretion generating mechanism. We estimate the hazard function of inter-arrival times. For our particular ACTH and cortisol data, frailty (unobserved heterogeneity) does not show up. The effects of amplitude and decay rate do not seem significant.