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Block–Savits type characterizations using the Laplace transform and some related issues

Part of: Survival analysis and censored data Special processes

Published online by Cambridge University Press:  22 April 2025

Lisa Parveen Open the ORCID record for Lisa Parveen [Opens in a new window]  and
Murari Mitra
Lisa Parveen*
Affiliation:
Indian Institute of Engineering Science and Technology, Shibpur
Murari Mitra*
Affiliation:
Indian Institute of Engineering Science and Technology, Shibpur
*
*Postal address: Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, P.O. –Botanic Garden, Howrah – 711103, West Bengal, India
*Postal address: Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, P.O. –Botanic Garden, Howrah – 711103, West Bengal, India
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Abstract

We focus on obtaining Block–Savits type characterizations for different ageing classes as well as some important renewal classes by using the Laplace transform. We also introduce a novel approach, based on the equilibrium distribution, to handle situations where the techniques of Block and Savits (1980) either fail or involve tedious calculations. Our approach in conjunction with the theory of total positivity yields Vinogradov’s (1973) result for the increasing failure rate class when the distribution function is continuous. We also present simple but elegant proofs for Block and Savits’ results for the decreasing mean residual life, new better than used in expectation, and harmonic new better than used in expectation classes as applications of our approach. We address several other related issues that are germane to our problem. Finally, we conclude with a short discussion on the issue of convolutions.

Keywords

Laplace transformageingDMTTFIMITequilibrium distributionPFkDMRLHANBUFRNBAFRrenewals

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Secondary: 62N05: Reliability and life testing
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 23
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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