Multivariate convex regression with adaptive partitioning

Abstract

We propose a new, nonparametric method for multivariate regression subject to convexity or con-cavity constraints on the response function. Convexity constraints are common in economics, statistics, operations research, financial engineering and optimization, but there is currently no multivariate method that is stable and computationally feasible for more than a few thousand ob-servations. We introduce convex adaptive partitioning (CAP), which creates a globally convex regression model from locally linear estimates fit on adaptively selected covariate partitions. CAP is a computationally efficient, consistent method for convex regression. We demonstrate empir-ical performance by comparing the performance of CAP to other shape-constrained and uncon-strained regression methods for predicting weekly wages and value function approximation for pricing American basket options.