# Research on Quantile Regression and Related Methods in Statistics

Project: Research

## Description

This is a two-part project. The first part considers quantile regression in varying coefficient models with censored data, and the second part develops an estimation procedure that combines the ideas of least squares and quantile regressions.It is well-known that regression quantiles have the advantage over conditional mean regression of being able to estimate directly the effects of covariates on quantiles other than the centre of the distribution. Quantile regression is also more robust than the method of least squares to outliers. There is a vast literature on quantile regression, and quantile regression methods, both parametric and non-parametric, have been developed for a wide range of problems. Of the various non-parametric approaches that have been proposed, the varying-coefficient approach has the appealing advantage that it avoids many of the curse of dimensionality problems commonly associated with other non-parametric methods because the non-parametric functions are restricted only to some of the variables. This notwithstanding, the varying-coefficient approach has not been considered in the context of censored quantile regression, in which some of the variables are not completely observed. Data censoring can arise in many contexts, and there is a sizeable literature on this topic. To deal with the complexity caused by data censoring when implementing the varying-coefficient approach in a quantile regression framework, we propose to modify the estimating functions by applying inverse probability weighting to the observations. However, the resultant estimating equations will be non-smooth, and to reconcile this difficulty we intend to draw on the majorize-minimize (MM) algorithm, adapting it to the context of the current analysis. The properties of the estimators thereby obtained will be examined via a Monte-Carlo study, and the method will be applied to real data analysis.Part two of the project develops an estimation procedure that combines information from least-squares regularity conditions and quantile regression estimating functions for the purpose of improving estimation efficiency. While this may be cast as an over-determined estimating equation system, common approaches such as Empirical Likelihood (EL) and Generalized Method of Moments (GMM) are not directly applicable in the present context because the estimating functions associated with the quantiles are non-smooth. To circumvent this problem, we propose to apply a kernel-based smoothing procedure on the non-smooth estimating equations. EL and GMM estimation can then be implemented based on the (smoothed) estimating equations; the smoothed EL function will also form the basis for developing EL ratio tests. A Monte-Carlo study will examine the finite-sample properties of the proposed estimators and tests, and the method will be applied to real data. The proposed method will also be extended to regressions with censored and missing observations.The theoretical work of this project is already underway, and the PI and Co-I have succeeded in deriving some preliminary theoretical results (see Section III). The requested amount is mainly for the recruitment of a research assistant to carry out simulation and real data analysis.

## Detail(s)

Project number 7008126 SRG Finished 1/05/11 → 24/03/14