Convergence of D\"umbgen's Algorithm for Estimation of Tail Inflation

Given a density $f$ on the non-negative real line, D\"umbgen's algorithm is a routine for finding the (unique) log-convex, non-decreasing function $\hat\phi$ such that $\int\hat\phi(x)f(x)dx=1$ and such that the likelihood $\prod_{i=1}^{n}f(x_i)\hat\phi(x_i)$ of given data $x_1,\ldots,x_n$ under density $x\mapsto \hat\phi(x)f(x)$ is maximized. We summarize D\"umbgen's algorithm for finding this MLE $\hat\phi$, and we present a novel guarantee of the algorithm's termination and convergence.