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

TL;DR

This paper analyzes D"umbgen's algorithm for estimating tail inflation, providing a summary and a new proof of its guaranteed termination and convergence for maximizing likelihood under a log-convex density constraint.

Contribution

The paper offers a comprehensive summary of D"umbgen's algorithm and introduces a novel theoretical guarantee of its termination and convergence.

Findings

Algorithm guarantees convergence to the MLE.

Provides a formal proof of termination.

Enhances understanding of tail inflation estimation.

Abstract

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.