Semiparametric Bayesian analysis: Selection models and meteorological applications

Jaeyong Lee, Purdue University

Abstract

Selection models are appropriate when a datum x enters the sample only with probability or weight w(x). Examples include aerial survey data, oil discovery, and meta-analysis. It is typically assumed that the weight function w is monotone, but the precise form of the weight function is often unknown to the statistician. In this paper, the Dirichlet process prior, centered on a parametric form, is used as a prior distribution on the weight function. This allows for incorporation of knowledge about the weight function, without restricting it to some particular form. Because of the form of the likelihood and the prior, computation via straightforward Gibbs sampling is not possible; however, by introducing latent variables related to the selection mechanism, Gibbs sampling can be implemented in the case where the total number of selected and unselected observations, N, is known. When N is unknown, a reversible jump Markov chain sampler is needed to carry out the computations. An important difficulty of practical nonidentifiability is uncovered, even for selection models in which the weight functions are theoretically identifiable. The proposed solution to this problem depends on the existence of prior knowledge concerning the effective range of the weight function. On the theoretical side, it is well known that nonparametric Bayes procedures need not be consistent. We prove consistency of our procedure under certain regularity conditions. Meteorologists have been interested in ozone depletion for many years because of its significant effect on the earth's environment. Recently, there has been considerable interest among meteorologists in modelling ozone levels as a function of altitude. Not only are environmental effects highly altitude-dependent, but understanding ozone altitude profiles can be important in understanding the ozone depletion process. Ozonesonde data is herein studied and modeled. Interestingly, the vertical ozone profiles show 3–4 modes which dictate the shape of the ozone profiles. Based on this observation, we model ozone profiles with a four component mixture of normal density functions. This allows the entire ozone profile to be summarized by the means, standard deviations, and weights of the mixture components. These parameters are then modeled as time-dependent, via a Dynamic Linear Model.

Degree

Ph.D.

Advisors

Berger, Purdue University.

Subject Area

Statistics|Atmosphere

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