Consistency of Multidimensional Convex Regression

Abstract

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 → ∞ 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).

Statistical modeling under partial identification: Distinguishing three types of identification regions in regression analysis with interval data

“…In [3] consistency and further asymptotic results for the sample analogue of SCR are shown. Since the estimation of SR is very similar to the estimation of a convex function via the least squares approach, we think that the results in [52,34,25] can possibly be used to show consistency and maybe also further asymptotic results for the sample analogue estimator of SR under certain assumptions. The situation for the sharp marrow region is more difficult.…”

Fusion of hard and soft information in nonparametric density estimation

“…Specifically, the epi-splines computed from {(P n p,m )} ∞ n=1 tend to a point in the Kullback-Leibler projection, relative to the soft information constraint set, of the true density on the class of epi-splines under consideration. We refer to [24,14,44] for related results on model misspecfication. The second part shows that if the true density is not excluded by the soft information, then {(P n p,m )} ∞ n=1 eventually yields the true density, or possibly a closely related one that deviates at most on m.…”

On Convergence Rates of Convex Regression in Multiple Dimensions

“…In particular, Lim and Glynn (2012) proved the consistency of their least squares estimator and further identified the behavior of the estimator when f * is not convex. In spite of increasing interest in this area, most work in the multidimensional case has focused on almost sure (a.s.) convergence, and little attention has been paid to convergence rates.…”

“…To our knowledge, this is the first paper that computes convergence rates of the estimatorf n for convex functions in multiple dimensions. Our main result can further be used to shed light on the convergence rate of the multidimensional convex regression estimator studied by Lim and Glynn (2012) because our estimator is constructed by imposing a condition of bounded subgradients onto the estimator proposed by Lim and Glynn (2012).…”