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Transformations and Bayesian Estimation of Skewed and Heavy-Tailed Densities

Bean, Andrew Taylor
http://rave.ohiolink.edu/etdc/view?acc_num=osu1503015935192212

Abstract Details

2017, Doctor of Philosophy, Ohio State University, Statistics.
In data analysis applications characterized by large and possibly irregular data sets, nonparametric statistical techniques aim to ensure that, as the sample size grows, all unusual features of the data generating process can be captured. Good large-sample performance can be guaranteed in broad classes of problems. Yet within these broad classes, some problems may be substantially more difficult than others. This fact, long recognized in classical nonparametrics, also holds in the growing field of Bayesian nonparametrics, where flexible prior distributions are developed to allow for an infinite-dimensional set of possible truths. This dissertation studies the Bayesian approach to the classic problem of nonparametric density estimation, in the presence of specific irregularities such as heavy tails and skew. The problem of estimating an unknown probability density is recognized as being harder when the density is skewed or heavy tailed than when it is symmetric and light-tailed. It is more challenging problem for classical kernel density estimators, where the expected squared-error loss is higher for heavier tailed densities. It is also a more challenging problem in Bayesian density estimation, where heavy tails preclude the analytical treatment required to establish a large-sample convergence rate for the popular Dirichlet-Process (DP) mixture model. Our proposed approach addresses these features by incorporating a low-dimensional parametric transformation of the sample, estimated from the data, with the aim of setting up an easier density estimation problem on the transformed scale. This strategy was proposed earlier in combination with kernel density estimators, and we illustrate its usefulness in the Bayesian context. Further, we develop a set of transformations estimated in a way to ensure that the fastest proven convergence rate for the DP mixture is applicable to the transformed problem. The transformation-density estimation technique makes advantageous use of a parametric pre-analysis to address specific irregularities in the data generating process. Since the parametric stage is low-dimensional, and governed by a faster convergence rate, the asymptotic performance of the model is enhanced without slowing down the overall convergence rate. We consider other settings where this recipe for semiparametric analysis --- with parametric sub-analyses designed to address specific irregularities, or to simplify the main nonparametric component of the analysis --- might be beneficial.
Xinyi Xu (Advisor)
Steven MacEachern (Advisor)
Yoonkyung Lee (Committee Member)
Matthew Pratola (Committee Member)
146 p.
Statistics
density estimation; transformations; nonparametric statistics; Dirichlet process mixtures; heavy tails; regular variation

Recommended Citations

Citations

  • Bean, A. T. (2017). Transformations and Bayesian Estimation of Skewed and Heavy-Tailed Densities [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1503015935192212

    APA Style (7th edition)

  • Bean, Andrew. Transformations and Bayesian Estimation of Skewed and Heavy-Tailed Densities. 2017. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1503015935192212.

    MLA Style (8th edition)

  • Bean, Andrew. "Transformations and Bayesian Estimation of Skewed and Heavy-Tailed Densities." Doctoral dissertation, Ohio State University, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=osu1503015935192212

    Chicago Manual of Style (17th edition)

osu1503015935192212
914
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This open access ETD is published by The Ohio State University and OhioLINK.