## Journal of Applied Probability

### Concave renewal functions do not imply DFR interrenewal times

Yaming Yu

#### Abstract

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

#### Article information

Source
J. Appl. Probab. Volume 48, Number 2 (2011), 583-588.

Dates
First available in Project Euclid: 21 June 2011

Permanent link to this document
https://projecteuclid.org/euclid.jap/1308662647

Digital Object Identifier
doi:10.1239/jap/1308662647

Mathematical Reviews number (MathSciNet)
MR2840319

Zentralblatt MATH identifier
1219.60074

Subjects
Primary: 60K05: Renewal theory

Keywords
Renewal theory log-convexity

#### Citation

Yu, Yaming. Concave renewal functions do not imply DFR interrenewal times. J. Appl. Probab. 48 (2011), no. 2, 583--588. doi:10.1239/jap/1308662647. https://projecteuclid.org/euclid.jap/1308662647

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