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(Ultra-)High Dimensional Partially Linear Single Index Models for Quantile Regression

Zhang, Yuankun
http://orcid.org/0000-0002-7680-8597
http://rave.ohiolink.edu/etdc/view?acc_num=ucin1535703962712806

Abstract Details

2018, PhD, University of Cincinnati, Arts and Sciences: Mathematical Sciences.
Nonparametric modeling tends to capture the underlying structures in the data without imposing strong model assumptions. The nonparametric estimation provides powerful data-driven approaches to fit a flexible model to the data. Single-index models are useful and appealing tools to preserve the flexibility and interpretability but to overcome “curse of dimensionality” problems in nonparametric regression. In this dissertation, we consider partially linear single-index models for quantile regression. This set of semi-parametric models allow some of covariates in linear form and other covariates in nonparametric term to reflect the non-linear feature in modeling the conditional quantiles of the response variable. We first develop efficient estimation and variable selection for partially linear single-index quantile models in the fixed dimension. We use spline smoothing with B-spline basis to estimate the nonparametric component and adopt the non-convex penalties to select variables simultaneously. We study the theoretical properties of the resulting estimators and establish the “oracle property” for penalized estimation. With the rise of new technologies used in data collection and storage, high dimensional data spring up and become available in various scientific fields. Often researchers face the new challenge that the dimension of the explanatory variables, p, may increase with the sample size, n, or potentially become much larger than n. For those problems of high to ultra-high dimensionality, data are likely to be heterogeneous and the underlying model is prone to be nonlinear. Variable selection will also play a critical role in the dimension reduction and modeling process. Thus, we propose a penalized estimation under the sparsity assumption for partially linear single-index quantile models in high dimension. We further investigate ultra-high dimensional penalized partially linear single-index quantile models in which both linear components and single-index variables in nonparametric term can be in high to ultra-high dimension. Both the numerical studies and the real data applications demonstrate that the proposed models and variable selection procedure work very well, even for data in ultra-high dimensionality.
Dan Ralescu, Ph.D. (Committee Chair)
Yan Yu, Ph.D. (Committee Chair)
Emily Kang, Ph.D. (Committee Member)
Ju-Yi Yen (Committee Member)
107 p.
Statistics
Nonparametric modeling; Single-index models; Quantile regression; Splines; Variable selection; High dimensionality

Recommended Citations

Citations

  • Zhang, Y. (2018). (Ultra-)High Dimensional Partially Linear Single Index Models for Quantile Regression [Doctoral dissertation, University of Cincinnati]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1535703962712806

    APA Style (7th edition)

  • Zhang, Yuankun. (Ultra-)High Dimensional Partially Linear Single Index Models for Quantile Regression. 2018. University of Cincinnati, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ucin1535703962712806.

    MLA Style (8th edition)

  • Zhang, Yuankun. "(Ultra-)High Dimensional Partially Linear Single Index Models for Quantile Regression." Doctoral dissertation, University of Cincinnati, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1535703962712806

    Chicago Manual of Style (17th edition)

ucin1535703962712806
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This open access ETD is published by University of Cincinnati and OhioLINK.