A New Algorithm for Approximating the Least Concave Majorant

Abstract

The least concave majorant, $\hat{F}$, of a continuous function $F$ on a closed interval, $I$, is defined by \[ \hat F (x) = \inf \left\{ G(x): G \geq F, G \mbox{ concave}\right\},\; x \in I. \] We present here an algorithm, in the spirit of the Jarvis March, to approximate the least concave majorant of a differentiable piecewise polynomial function of degree at most three on $I$. Given any function $F\in {\mathcal{C}}^4(I)$, it can be well-approximated on $I$ by a clamped cubic spline $S$. We show that $\hat{S}$ is then a good approximation to $\hat{F}$. We give two examples, one to illustrate, the other to apply our algorithm.

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