UC Riverside

Markov Chain Monte Carlo (MCMC) methods are widely used and preferred when the sampling distribution is intractable. In estimation problems with Monte Carlo samples, it is critical to quantify uncertainty of estimators on some intervals since the true function could not be obtained in most cases. Traditional pointwise confidence intervals could provide certain coverage probability in a single point but fail to provide simultaneous coverage for the whole function without a multiplicity correction. The Bonferroni method corrects for multiplicity, but these conservative intervals do not achieve the desired nominal level. This dissertation focuses on providing and quantifying the uncertainty of estimators in the form of a confidence band (CB) to increase the reliability of the resulting inferences.

We begin with MCMC basics and point estimation methods. Then we provide estimators for densities and general functions separately. We discuss the covariance matrix and Central Limit Theorem as preliminary settings. Afterwards, we review pointwise and Bonferroni methods to construct CBs. We propose three methods in calculating simultaneous CBs with theories and algorithms which are followed by examples to compare the coverage probabilities and the band widths. To provide more intuitive results, we compared the bands with three simulation examples: AR(1) model, mixed normal distribution, and a general function case. Then we used four real data examples: Michigan survey example, Telescope data example, time varying model, and a Bayesian reliability model to explain our proposed simultaneous bands.