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On Multivariate Quantile Regression: Directional Approach and Application with Growth Charts Open Access

Descriptions

Other title
Subject/Keyword
bivariate growth charts
multivariate quantile regression
multivariate quantile
halfspace depth
directional approach
Type of item
Thesis
Degree grantor
University of Alberta
Author or creator
Kong, Linglong
Supervisor and department
Mizera, Ivan (Mathematical and Statistical Sciences)
Examining committee member and department
Schmuland, Byron (Mathematical and Statistical Sciences)
Wei, Ying (Biostatistics, Columbia University)
Cui, Ying (Educational Psychology)
Prasad, Narasimha (Mathematical and Statistical Sciences)
Wiens, Douglas (Mathematical and Statistical Sciences)
Department
Department of Mathematical and Statistical Sciences
Specialization

Date accepted
2009-07-09T21:07:17Z
Graduation date
2009-11
Degree
Doctoral of Philosophy
Degree level
Doctoral
Abstract
In this thesis, we introduce a concept of directional quantile envelopes, the intersection of the halfspaces determined by directional quantiles, and show that they allow for explicit probabilistic interpretation, compared to other multivariate quantile concepts. Directional quantile envelopes provide a way to perform multivariate quantile regression: to ``regress contours'' on covariates. We also develop theory and algorithms for an important application of multivariate quantile regression in biometry: bivariate growth charts. We prove that directional quantiles are continuous and derive their closed-form expression for elliptically symmetric distributions. We provide probabilistic interpretations of directional quantile envelopes and establish that directional quantile envelopes are essentially halfspace depth contours. We show that distributions with smooth directional quantile envelopes are uniquely determined by their envelopes. We describe an estimation scheme of directional quantile envelopes and prove its affine equivariance. We establish the consistency of the estimates of directional quantile envelopes and describe their accuracy. The results are applied to estimation of bivariate extreme quantiles. One of the main contributions of this thesis is the construction of bivariate growth charts, an important application of multivariate quantile regression. We discuss the computation of our multivariate quantile regression by developing a fast elimination algorithm. The algorithm constructs the set of active halfspaces to form a directional quantile envelope. Applying this algorithm to a large number of quantile halfspaces, we can construct an arbitrary exact approximation of the direction quantile envelope. In the remainder of the thesis, we exhibit the connection between depth contours and directional regression quantiles (Laine, 2001), stated without proof in Koenker (2005). Our proof uses the duality theory of primal-dual linear programming. Aiming at interpreting halfspace depth contours, we explore their properties for empirical distributions, absolutely continuous distributions and certain general distributions. Finally, we propose a generalized quantile concept, depth quantile, inspired by halfspace depth (Tukey, 1975) and regression depth (Rousseeuw and Hubert, 1999). We study its properties in various data-analytic situations: multivariate and univariate locations, regression with and without intercept. In the end, we show an example that while the quantile regression of Koenker and Bassett (1978) fails, our concept provides sensible answers.
Language
English
DOI
doi:10.7939/R3G354
Rights
License granted by Linglong Kong (lkong@math.ualberta.ca) on 2009-07-02T18:15:32Z (GMT): Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of the above terms. The author reserves all other publication and other rights in association with the copyright in the thesis, and except as herein provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
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