Estimating Smooth and Convex Functions

We propose a new method for estimating an unknown regression function $f_*-[\alpha, \beta] \rightarrow \mathbb{R}$ from a dataset $(X_1, Y_1), \dots, (X_n,$ $Y_n)$ when the only information available on $f_*$ is the fact that $f_*$ is convex and twice differentiable. In the proposed method, we fit a convex function to the dataset that minimizes the sum of the roughness of the fitted function and the average squared differences between the fitted function and $f_*$. We prove that the proposed estimator can be computed by solving a convex quadratic programming problem with linear constraints. Numerical results illustrate the superior performance of the proposed estimator compared to existing methods when i) $f_*$ is the price of a stock option as a function of the strike price and ii) $f_*$ is the steady-state mean waiting time of a customer in a single server queue.