Estimating nonparametric convex functions using gradient data
European Journal of Operational Research, Volume 41, Issue 1, 5 July 1989, Pages 73-85
Malachy Carey
School of Urban and Public Affairs, Carnegie-Mellon University, Pittsburgh, PA 15213, USA
Abstract
The estimation and use of nonparametric convex functions has been discussed in the recent literature, where
it is assumed that the data used in the estimation consists of (a sample of) observations of the convex
function. However, an important additional or alternative source of data consists of observations on the (sub)
gradient of the function. In the present paper we discuss how to use such (sub)gradient data, to estimate
nonparametric functions. This: (a) allows us to estimate nonparametric convex functions when other (point
observation) data is unavailable, or unobservable, or unreliable or costly to obtain, (b) allows us an
alternative even when other (point observation) data is available, and (c) allows us to improve our estimates
by using both types of data together. The resultant function estimators may not be unique, hence we compute
bounds on the (convex) set of estimators. We also consider introducing other data in addition to the gradient
data. The discussion is illustrated by reference to estimating production functions, benefit functions and
demand functions.
Author Keywords: Inference; convex programming; nonparametric