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

  • Author / Creator
    Kong, Linglong
  • 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.

  • Subjects / Keywords
  • Graduation date
    2009-11
  • Type of Item
    Thesis
  • Degree
    Doctoral of Philosophy
  • DOI
    https://doi.org/10.7939/R3G354
  • License
    This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
  • Language
    English
  • Institution
    University of Alberta
  • Degree level
    Doctoral
  • Department
    • Department of Mathematical and Statistical Sciences
  • Supervisor / co-supervisor and their department(s)
    • Mizera, Ivan (Mathematical and Statistical Sciences)
  • Examining committee members and their departments
    • Wei, Ying (Biostatistics, Columbia University)
    • Cui, Ying (Educational Psychology)
    • Prasad, Narasimha (Mathematical and Statistical Sciences)
    • Schmuland, Byron (Mathematical and Statistical Sciences)
    • Wiens, Douglas (Mathematical and Statistical Sciences)