On Nonparametric Bayesian Inference for Tukey Depth

Abstract

The Dirichlet process is perhaps the most popular prior used in the nonparametric Bayesian inference. This prior which is placed on the space of probability distributions has conjugacy property and asymptotic consistency. In this thesis, our concentration is on applying this nonparametric Bayesian inference on the Tukey depth and Tukey median. Due to the complexity of the distribution of Tukey median, we use this nonparametric Bayesian inference, namely the Lo’s bootstrap, to approximate the distribution of the Tukey median. We also compare our results with the Efron’s bootstrap and Rubin’s bootstrap. Furthermore, the existing asymptotic theory for the Tukey median is reviewed. Based on these existing results, we conjecture that the bootstrap sample Tukey median converges to the same asymp- totic distribution and our simulation supports the conjecture that the asymptotic consistency holds.