Sparse ridge Sliced inverse quantile regression without quantile crossing

Quantile regression provides a more complete statistical analysis of the stochastic relationships between random variables. While this technique has become very popular as a comprehensive extension of the classical mean regression it nonetheless suffers the problem of crossing of regression functions estimated at different orders of quantiles. Theoretically, the extension of conditional quantiles to higher dimension of X is straight forward. However, its practical success suffers from the socalled ‘curse of dimensionality’. In this article we propose a method of obtaining quantile regression estimates for high dimension data without the unfavourable quality of quantile crossing. The proposed method is a two step procedure that initially employs sparse ridge sliced inverse regression (SRSIR) to achieve dimension reduction when the predictors are possibly correlated and then followed by the usage of non-parametric method to estimate non-crossing quantile regression. For the second stage of our method we employ double kernel smoothing method (Yu and Jones,1998); monotone-based smoothing method based on the convolution of the distribution (Dette and Volgushev,2008) and joint non-crossing quantile smoothing spline method (Bondell et al., 2010) for estimating the conditional quantile without quantile crossing. Through a simulation and empirical study we compare our estimators with that of Gannon et al. (2004).