Quantile Regression Estimation Using Non-Crossing Constraints

Abstract

In this article we are concerned with a collection of multiple linear regressions that enable the researcher to gain an impression of the entire conditional distribution of a response variable given a set of explanatory variables. More specifically, we investigate the advantage of using a new method to estimate a bunch of non-crossing quantile regressions hyperplanes. The main tool is a weighting system of the data elements that aims to reduce the effect of contamination of the sampled population on the estimated parameters by diminishing the effect of outliers. The performances of the new estimators are evaluated on a number of data sets. We had considerable success with avoiding intersections and in the same time improving the global fitting of conditional quantile regressions. We conjecture that in other situations (e.g., data with high level of skewness, non-constant variances, unusual and imputed data) the method of weighted non-crossing quantiles will lead to estimators with good robustness properties.

Solution to the Non-Monotonicity and Crossing Problems in Quantile Regression

“…Two different approaches have been taken: (1) Reduce the crossing problem by applying some constraints or restrictions on the problem, and (2) Apply some smoothing or fitting method for a number of quantiles over a given interval. See [5] for a summary of various methods.…”

“…While the results thus far are interesting, it can be improved further through the use of a smoother check function, as described in this section. It is particularly effective in removing non-monotonicity that has to do with resolution and sensitivity [5]. In particular, if quantiles are spaced 0.001 units apart, then a total of 1000 quantiles are required in the range (0, 1).…”

“…We find that smrq is generally monotonic in small sample data for all quartiles, deciles and percentiles. We now illustrate large-sample data and high-resolution cases using the diabetes data with 442 observations and 10 variables, and the Boston housing data [5] set with 506 observations and 15 variables. The number of non-monotonic events (both spikes and pulses) are fewer for conquer compared to smrq.…”