Probability in the Engineering and Informational Sciences



SOME NEW BOUNDS FOR THE RENEWAL FUNCTION


Konstadinos  Politis  a1 and Markos V.  Koutras  a1
a1 Department of Statistics and Insurance Science, University of Piraeus, Piraeus 18534, Greece, E-mail: kpolitis@unipi.gr; mkoutras@unipi.gr

Article author query
politis k   [Google Scholar] 
koutras mv   [Google Scholar] 
 

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 [less-than-or-equal] [infty infinity], and we give both sufficient and necessary conditions for this improvement to hold uniformly for x [greater-than-or-equal] 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.