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SOME NEW BOUNDS FOR THE RENEWAL FUNCTION

Published online by Cambridge University Press:  06 March 2006

Konstadinos Politis  and
Markos V. Koutras
Konstadinos Politis
Affiliation:
Department of Statistics and Insurance Science, University of Piraeus, Piraeus 18534, Greece, E-mail: kpolitis@unipi.gr; mkoutras@unipi.gr
Markos V. Koutras
Affiliation:
Department of Statistics and Insurance Science, University of Piraeus, Piraeus 18534, Greece, E-mail: kpolitis@unipi.gr; mkoutras@unipi.gr
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Abstract

In the literature, most of the bounds for the renewal function U(x) corresponding to a lifetime distribution F are given in terms of the first two moments of F only. The best general upper bound of this type is the one given in Lorden (1970). In the present article, we show that improved bounds can be obtained if one exploits the specific form of the distribution F. We derive a bound that improves upon Lorden's, at least on an interval [0,a) with a ≤ ∞, and we give both sufficient and necessary conditions for this improvement to hold uniformly for x ≥ 0. Refined upper as well as lower bounds are given for the case where F belongs to a class of distributions with monotone aging or when the renewal density is monotone.

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 20 , Issue 2 , April 2006 , pp. 231 - 250
Copyright
© 2006 Cambridge University Press

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