Download the full-sized PDF of On Multivariate Quantile Regression: Directional Approach and Application with Growth ChartsDownload the full-sized PDF



Permanent link (DOI):


Export to: EndNote  |  Zotero  |  Mendeley


This file is in the following communities:

Graduate Studies and Research, Faculty of


This file is in the following collections:

Theses and Dissertations

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


Other title
bivariate growth charts
multivariate quantile regression
multivariate quantile
halfspace depth
directional approach
Type of item
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 of Mathematical and Statistical Sciences

Date accepted
Graduation date
Doctoral of Philosophy
Degree level
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.
License granted by Linglong Kong ( 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.
Citation for previous publication

File Details

Date Uploaded
Date Modified
Audit Status
Audits have not yet been run on this file.
File format: pdf (Portable Document Format)
Mime type: application/pdf
File size: 2556195
Last modified: 2015:10:12 11:14:16-06:00
Filename: Kong_Linglong_Fall 2009.pdf
Original checksum: bf72c65bd8416e81039f48414bdb12d6
Well formed: true
Valid: true
Status message: Too many fonts to report; some fonts omitted. Total fonts = 1474
File title: Microsoft Word - Linglong_UAlibraryrelease.doc
File author: Linglong Bei
Page count: 198
Activity of users you follow
User Activity Date