Abstract
Here we propose a new definition of memory of distributions in a renewal-theoretic framework and study the corresponding notions of ageing and ordering of distributions. Discrete life times are also considered.
Similar content being viewed by others
References
Alzaid AA (1988) Mean residual life ordering. Statistiche Hefte Statistical Papers 29:35–43
Brown KG, Farlow SJ (1982) Memory of a probability distribution. Int J Math Educ Sci Technol 13:587–592
Chukova S, Dimitrov B (1992) On distributions having the almost-lack-of-memory property. J Appl Probab 29:691–698
Ebrahimi N, Zahedi H (1992) Memory ordering of survival functions. Statistics 23(4):337–348
Feller W (1971) An Introduction to the theory of probability and its applications, \(2^{nd}\) edition. John Wiley and Sons, New York
Galambos J, Kotz S (1978) Characterizations of probability distributions, vol 675. Lecture Notes in Mathematics. Springer, Heidelberg
Gupta RC, Kirmani SNUA (2000) Residual coefficient of variation and some characterization results. J Statist Plan Infer 91:23–31
Hitha N, Nair NU (1989) Characterisation of some discrete models by properties of residual life function. Cal Statist Assoc Bull 38:219–223
Huang JS (1981) On a “lack of memory” property. Ann Inst Statist Math 33:131–134
Marshall AW, Olkin I (1967) A multivariate exponential distribution. J Am Statist Assoc 62:30–44
Muth EJ (1977) Reliability models with positive memory derived from mean residual life function. In: Tsokos CP, Shini IN (eds) Theory and applications of reliability, vol 2. Academic Press, Boston, pp 401–435
Nair NU (1983) A measure of memory for some discrete distributions. J Ind Statist Assoc 21:141–147
Nair NU, Sankaran PG (2010) Properties of a mean residual life arising from renewal theory. Nav Res Log 57:373–379
Nair NU, Sankaran PG, Preeth M (2012) Reliability aspects of discrete equilibrium distributions. Commun Statist - Theor Meth 41:500–515
Nair NU, Sankaran PG, Balakrishnan N (2018) Reliability modelling and analysis in discrete time. Academic Press, UK
Riffi MI (2016) Forgetful lifetime distributions. Pal J Math 5:57–60
Ross SM (1996) Stochastic processes, \(2^{nd}\) edition. John Wiley and Sons, New York
Sandhya E, Rajasekharan KE (2012) On the measure of memory of discrete distributions under change of origin and scale. Int J Math Sci Eng Appl 6:293–301
Stein WE, Dattero R (1999) Bondesson’s functions in reliability theory. Appl Stoch Mo Bus Indus 15:103–109
Acknowledgements
Authors thank the referee for the suggestions that improved the presentation very much.
Funding
The research reported in this article is not funded.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors certify that there is no conflict of interest among them in relation to this article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Ideas in this paper were presented at ISBIS - Kochi - 2020, December 28–30, CUSAT, India.
Rights and permissions
About this article
Cite this article
Nair, N.U., Satheesh, S. & Sandhya, E. Memory of Distributions: A Renewal Theoretic Approach. J Indian Soc Probab Stat 23, 173–185 (2022). https://doi.org/10.1007/s41096-022-00117-6
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41096-022-00117-6