Estimating nonparametric convex functions using gradient data

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 of 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.