Single-index quantile regression

This is the post-print version of the final paper published in Journal of Multivariate Analysis. The published article is available from the link below. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. Copyright @ 2010 Elsevier B.V.Nonparametric quantile regression with multivariate covariates is a difficult estimation problem due to the “curse of dimensionality”. To reduce the dimensionality while still retaining the flexibility of a nonparametric model, we propose modeling the conditional quantile by a single-index function View the MathML sourceg0(xTγ0), where a univariate link function g0(⋅)g0(⋅) is applied to a linear combination of covariates View the MathML sourcexTγ0, often called the single-index. We introduce a practical algorithm where the unknown link function g0(⋅)g0(⋅) is estimated by local linear quantile regression and the parametric index is estimated through linear quantile regression. Large sample properties of estimators are studied, which facilitate further inference. Both the modeling and estimation approaches are demonstrated by simulation studies and real data applications

Penalized single-index quantile regression

This article is made available through the Brunel Open Access Publishing Fund. Copyright for this article is retained by the author(s), with first publication rights granted to the journal. This is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).The single-index (SI) regression and single-index quantile (SIQ) estimation methods product linear combinations of all the original predictors. However, it is possible that there are many unimportant predictors within the original predictors. Thus, the precision of parameter estimation as well as the accuracy of prediction will be effected by the existence of those unimportant predictors when the previous methods are used. In this article, an extension of the SIQ method of Wu et al. (2010) has been proposed, which considers Lasso and Adaptive Lasso for estimation and variable selection. Computational algorithms have been developed in order to calculate the penalized SIQ estimates. A simulation study and a real data application have been used to assess the performance of the methods under consideration

Characterization of the asymptotic distribution of semiparametric M-estimators

This paper develops a concrete formula for the asymptotic distribution of two-step, possibly non-smooth semiparametric M-estimators under general misspecification. Our regularity conditions are relatively straightforward to verify and also weaker than those available in the literature. The first-stage nonparametric estimation may depend on finite dimensional parameters. We characterize: (1) conditions under which the first-stage estimation of nonparametric components do not affect the asymptotic distribution, (2) conditions under which the asymptotic distribution is affected by the derivatives of the first-stage nonparametric estimator with respect to the finite-dimensional parameters, and (3) conditions under which one can allow non-smooth objective functions. Our framework is illustrated by applying it to three examples: (1) profiled estimation of a single index quantile regression model, (2) semiparametric least squares estimation under model misspecification, and (3) a smoothed matching estimator. © 2010 Elsevier B.V. All rights reserved

Characterization of the asymptotic distribution of semiparametric M-estimators

This paper develops a concrete formula for the asymptotic distribution of two-step, possibly non-smooth semiparametric M-estimators under general misspecification. Our regularity conditions are relatively straightforward to verify and also weaker than those available in the literature. The first-stage nonparametric estimation may depend on finite dimensional parameters. We characterize: (1) conditions under which the first-stage estimation of nonparametric components do not affect the asymptotic distribution, (2) conditions under which the asymptotic distribution is affected by the derivatives of the first-stage nonparametric estimator with respect to the finite-dimensional parameters, and (3) conditions under which one can allow non-smooth objective functions. Our framework is illustrated by applying it to three examples: (1) profiled estimation of a single index quantile regression model, (2) semiparametric least squares estimation under model misspecification, and (3) a smoothed matching estimator.

"Characterization of the Asymptotic Distribution of Semiparametric M-Estimators"

This paper develops a concrete formula for the asymptotic distribution of two-step, possibly non-smooth semiparametric M-estimators under general misspecification. Our regularity conditions are relatively straightforward to verify and also weaker than those available in the literature. The first-stage nonparametric estimation may depend on finite dimensional parameters. We characterize: (1) conditions under which the first-stage estimation of nonparametric components do not affect the asymptotic distribution, (2) conditions under which the asymptotic distribution is affected by the derivatives of the first-stage nonparametric estimator with respect to the finite-dimensional parameters, and (3) conditions under which one can allow non-smooth objective functions. Our framework is illustrated by applying it to three examples: (1) profiled estimation of a single index quantile regression model, (2) semiparametric least squares estimation under model misspecification, and (3) a smoothed matching estimator.

Bayesian Quantile Regression for Single-Index Models

Using an asymmetric Laplace distribution, which provides a mechanism for Bayesian inference of quantile regression models, we develop a fully Bayesian approach to fitting single-index models in conditional quantile regression. In this work, we use a Gaussian process prior for the unknown nonparametric link function and a Laplace distribution on the index vector, with the latter motivated by the recent popularity of the Bayesian lasso idea. We design a Markov chain Monte Carlo algorithm for posterior inference. Careful consideration of the singularity of the kernel matrix, and tractability of some of the full conditional distributions leads to a partially collapsed approach where the nonparametric link function is integrated out in some of the sampling steps. Our simulations demonstrate the superior performance of the Bayesian method versus the frequentist approach. The method is further illustrated by an application to the hurricane data.Comment: 26 pages, 8 figures, 10 table

Some statistical methods for dimension reduction

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel UniversityThe aim of the work in this thesis is to carry out dimension reduction (DR) for high dimensional (HD) data by using statistical methods for variable selection, feature extraction and a combination of the two. In Chapter 2, the DR is carried out through robust feature extraction. Robust canonical correlation (RCCA) methods have been proposed. In the correlation matrix of canonical correlation analysis (CCA), we suggest that the Pearson correlation should be substituted by robust correlation measures in order to obtain robust correlation matrices. These matrices have been employed for producing RCCA. Moreover, the classical covariance matrix has been substituted by robust estimators for multivariate location and dispersion in order to get RCCA. In Chapter 3 and 4, the DR is carried out by combining the ideas of variable selection using regularisation methods with feature extraction, through the minimum average variance estimator (MAVE) and single index quantile regression (SIQ) methods, respectively. In particular, we extend the sparse MAVE (SMAVE) reported in (Wang and Yin, 2008) by combining the MAVE loss function with different regularisation penalties in Chapter 3. An extension of the SIQ of Wu et al. (2010) by considering different regularisation penalties is proposed in Chapter 4. In Chapter 5, the DR is done through variable selection under Bayesian framework. A flexible Bayesian framework for regularisation in quantile regression (QR) model has been proposed. This work is different from Bayesian Lasso quantile regression (BLQR), employing the asymmetric Laplace error distribution (ALD). The error distribution is assumed to be an infinite mixture of Gaussian (IMG) densities