Bayes nonparametrics for biased sampling and density estimation.

Abstract

A common assumption in statistics is that a random sample from a target distribution is available. Biased sampling models arise in those situations, common in observational studies, in which this assumption has to be relaxed. For the researcher can observe realizations not of the random variables of direct interest, but rather of some transformed random variables, produced by a biasing mechanism. The incorporation of the bias into the statistical model typically results in a modification of the likelihood function. Truncation and selection biases are of central interest in the first part of this dissertation. Nonparametric Bayes estimators of the target distribution function on the basis of biased samples are found and compared to the corresponding nonparametric maximum likelihood estimators. Algorithms and examples are presented. In a nonparametric Bayesian approach, the need arises for priors that have a large infinite-dimensional support contained in the space of all probability distributions on a given sample space. These prior processes should at the same time enjoy analytical tractability. Since Dirichlet process priors are a central tool for this kind of problems, special emphasis is put on them for the analysis of the biased sampling models. Dirichlet processes are also the starting point to construct a new nonparametric prior for the problem of estimating a density function, the focus of the second part of the thesis. The rationale for the prior introduced is that any measurable function, such as a density, can be approximated by step functions up to any desired level of accuracy. Therefore, an approximation mechanism is used to give probability 1 to an appropriate simple space of step functions. The step functions are taken to be stepwise constant with random heights over intervals of equal length, the bin width. The bin width is in turn considered random. The framework provides the Bayes estimator with enough flexibility to allow for its consistency, an important property shown to hold. Nontrivial computational issues arise.