# Nonparametric Quantile Estimation

Ichiro Takeuchi, Quoc V. Le, Timothy D. Sears, Alexander J. Smola.

Year: 2006, Volume: 7, Issue: 45, Pages: 1231−1264

#### Abstract

In regression, the desired estimate of *y*|*x* is not always given by a
conditional mean, although this is most common. Sometimes one wants to
obtain a good estimate that satisfies the property that a proportion,
τ, of *y*|*x*, will be below the estimate. For τ = 0.5 this is
an estimate of the *median*. 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 a simple quadratic programming problem and provide
uniform convergence statements and bounds on the quantile property of
our estimator. Experimental results show the feasibility of the
approach and competitiveness of our method with existing ones. We
discuss several types of extensions including an approach to solve the
*quantile crossing* problems, as well as a method to incorporate
prior qualitative knowledge such as monotonicity constraints.