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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.

Distribution-free tolerance regions are hyper-rectangles in terms of the number of variables that include at least a pre-specified proportion of normal subjects with a confidence bounded from below by a prescribed probability. This paper has two main goals. The first is to propose an innovative method for constructing multidimensional tolerance regions that work well when the only assumption that can be made about the underlying distribution is that it is continuous. Although our proposal is, in essence, an extension of the Wilks-Wald method to higher dimensions, this research is far less immediate and straightforward than it has appeared to many authors, who, moreover, have not really used it in practical work. In particular, the order statistics dividing the regions are not fixed beforehand, but are determined by an optimization procedure. The second goal is to suggest a way of overcoming a problem of practical importance concerning the order in which the variables are included, which has remained unsolved since the introduction of the argument almost eighty years ago. This device facilitates the search for a good solution while avoiding being drawn away into the black hole of combinatorial computations. Simulation and applications to laboratory medicine data illustrate the advantages of using the method presented in this paper.