Concave Renewal Functions Do Not Imply DFR Inter-Renewal Times

TL;DR

This paper disproves Brown's conjecture by providing counter-examples showing that concave renewal functions do not necessarily imply DFR inter-renewal times, and also confirms the closure of DFR under geometric compounding.

Contribution

It provides the first counter-examples to Brown's conjecture and offers a concise proof of DFR closure under geometric compounding.

Findings

Counter-examples to Brown's conjecture

Concave renewal functions do not imply DFR inter-renewal times

DFR property is closed under geometric compounding

Abstract

Brown (1980, 1981) proved that the renewal function is concave if the inter-renewal distribution is DFR (decreasing failure rate), and conjectured the converse. This note settles Brown's conjecture with a class of counter-examples. We also give a short proof of Shanthikumar's (1988) result that the DFR property is closed under geometric compounding.