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) multidimensional. 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 -> infinity under modest technical assumptions. In addition to this multidimensional consistency result, we identify the behavior of the estimator when the model is misspecified (so that the true function is nonconvex), and we extend the consistency result to settings in which the function must be both convex and nondecreasing (as is needed for consumer preference utility functions).