Model Calibration through Statistical Adjustments

Abstract

The problem of shape-constrained regression has been an object of study for many years, dating back to the work of Brunk, Barlow, Wright, and others in the 1960s and 1970s. Such shape-based regression is nonparametric and relieves the statistician of the need to specify the smoothing parameters that arise either implicitly or explicitly in competing methods. Until recently, all such work focused on problems in which the independent variable is one dimensional, in which case there is essentially a closed-form expression for the least-squares estimator, both in the case of isotonic regression and convex regression. Existence of a closed form facilitates the analysis of large-sample behavior of the estimator. In this talk, we will discuss least-squares based convex regression in the higher dimensional setting, in which case no closed-form to the corresponding sample-based optimization problem appears to exist. Such higher dimensional formulations arise naturally in the simulation context, as well as in econometric estimation of utility preference curves and production functions. Our main result concerns large-sample consistency, as well as results pertaining to model mis-specification. Our argument depends critically on the fact that the space of suitably integrable convex functions is itself a closed convex cone of the Hilbert space of square-integrable functions. As such, our argument extends easily to certain other shape-constrained regression problems (e.g. requiring that the estimated function be both convex and monotone). This work is joint with Eunji Lim.?