Consistency of Multidimensional Convex Regression

Convex regression is concerned with computing the best fit of a convex function to a data set of n observations in which the independent variable is (possibly) multi–dimensional. Such regression problems arise in operations research, economics, and other disciplines in which imposing a convexity constraint on the regression function is natural. This paper studies a least squares estimator that is computable as the solution of a quadratic program and establishes that it converges almost surely to the “true” function as n→∞ under modest technical assumptions. In addition to this multi–dimensional consistency result, we identify the behavior of the estimator when the model is mis–specified (so that the “true” function is non–convex) and extend the consistency result to settings in which the function must be both convex and non–decreasing (as is needed for consumer preference utility functions).