ABSTRACT
Exact or closed-form solutions, if available, to statistical problems are favored in statistical theory and practice for obvious reasons. For example, its simplicity facilitates the communication of statistical analysis to a wider audience and thus make the contribution of statistics visible. Unfortunately, it may not be possible to obtain exact solutions in many practical statistical models. However, surprisingly the IBF approach allows such solutions in a variety of nontrivial statistical problems (Ng, 1997a; Tian, 2000; Tian & Tan, 2003). In this chapter, we first demonstrate that the IBF can be used to derive closed-form solutions to various Bayesian missing data problems including sample surveys with nonresponse, misclassified multinomial model, genetic linkage model, Weibull process with missing data, prediction problem, bivariate normal model, the 2×2 crossover trial with missing data and hierarchical models. We will show how to derive the explicit posterior density of the parameter of interest given the observed data. With this, subsequent statistical inference is straightforward. Then we extend the IBF in product measurable space (PMS) to nonproduct measurable space (NPMS) and focus on their applications in obtaining exact solutions in multivariate distributions.
