Nonparametric quantile estimation

In regression, the desired estimate of y|x is not always given by a conditional mean, althoughthis is most common. Sometimes one wants to obtain a good estimate that satisfies the propertythat a proportion, t, of y|x, will be below the estimate. For t = 0.5 this is an estimate of themedian. What might be called median regression, is subsumed under the term quantile regression.We present a nonparametric version of a quantile estimator, which can be obtained by solving asimple quadratic programming problem and provide uniform convergence statements and boundson the quantile property of our estimator. Experimental results show the feasibility of the approachand competitiveness of our method with existing ones. We discuss several types of extensionsincluding an approach to solve the quantile crossing problems, as well as a method to incorporateprior qualitative knowledge such as monotonicity constraints.