A Differential Equation for a Class of Discrete Lifetime Distributions with an Application in Reliability

Methodology and Computing in Applied Probability - Tập 17 - Trang 647-660 - 2013
Attila Csenki1
1School of Computing, Informatics and Media, University of Bradford, Bradford, UK

Tóm tắt

It is shown that the probability generating function of a lifetime random variable T on a finite lattice with polynomial failure rate satisfies a certain differential equation. The interrelationship with Markov chain theory is highlighted. The differential equation gives rise to a system of differential equations which, when inverted, can be used in the limit to express the polynomial coefficients in terms of the factorial moments of T. This then can be used to estimate the polynomial coefficients. Some special cases are worked through symbolically using Computer Algebra. A simulation study is used to validate the approach and to explore its potential in the reliability context.

Tài liệu tham khảo

Abramowitz M, Stegun A (1972) Handbook of mathematical functions. Dover Publications, New York Berg MP (1996) Towards rational age-based failure modelling. In: Özekici S (ed) Reliability and maintenance of complex systems. Proceedings of the NATO advanced study institute on current issues and challenges in the reliability and maintenance of complex systems, Kemer-Antalya, Turkey, 12–22 Jun 1995. NATO ASI Series F, vol 154. Springer, pp 107–113 Biggs NL (1989) Discrete mathematics. Clarendon Press, Oxford Charalambides CA (2005) Moments of a class of discrete q-distributions. J Stat Plan Inference 135:64–76 Crippa D, Simon K (1997) q-distributions and Markov processes. Discret Math 170:81–98 Csenki A (1994) Dependability for systems with a partitioned state space—Markov and Semi-Markov theory and computational implementation. In: Lecture Notes in Statistics, vol 90. Springer, New York Csenki A (2011) On continuous lifetime distributions with polynomial failure rate with an application in reliability. Reliab Eng Syst Saf 96:1587–1590 Csenki A (2012) Asymptotics for continuous lifetime distributions with polynomial failure rate with an application in reliability. Reliab Eng Syst Saf 102:1–4 Grimmett G, Welsh D (1986) Probability—an introduction. Clarendon Press, Oxford Heller B (1991) MACSYMA for statisticians. Wiley–Interscience, New York Jazi MA, Lai CD, Alamatsaz MH (2010) A discrete inverse Weibull distribution and estimation of its parameters. Stat Methodol 7:121–132 Khan MSA, Kalique A, Abouammoh AM (1989) On estimating parameters in a discrete Weibull distribution. IEEE Trans Reliab 38:348–348 Lawless JF (1982) Statistical models and methods for lifetime data. Wiley, New York Lehmann EL (1999) Elements of large-sample theory. Springer, New York Limnios N (2011) Reliability measures of semi-Markov systems with general state space. Methodol Comput Appl Probab. doi:10.1007/s11009-011-9211-5 Ma Y, Genton MG, Parzen E (2011) Asymptotic properties of sample quantiles of discrete distributions. Ann Inst Stat Math 63:227–243 Nakagawa T A, Osaki S (1975) The discrete Weibull distribution. IEEE Trans Reliab 24:300–301 (1975) Neuts M (1981) Matrix-geometric solutions in stochastic models: an algorithmic approach. The Johns Hopkins University Press, Baltimore Rand RH (2010) Introduction to maxima. Dept. of Theoretical and Applied Mechanics, Cornell University. http://maxima.sourceforge.net/docs/intromax/intromax.html. Accessed 15 Aug 2012 Rényi A (1970) Probability theory. North-Holland, Amsterdam, London (1970) Sarhan AM, Hamilton DC, Smith B, Kundu D (2011) The bivariate generalized linear failure rate distribution and its multivariate extension. Comput Stat Data An 55:644–654 Schelter WF (2006) Maxima Manual, version 5.9.3. http://maxima.sourceforge.net/docs/manual/en/maxima.html. Accessed 15 Aug 2012 Shaked M, Shantikumar GJ, Valdez-Torres JB (1995) Discrete hazard rate functions. Comput Oper Res 22:391–402 Stein WE, Dattero R (1984) A new discrete Weibull distribution. IEEE Trans Reliab 33:196–197 Wang Z (2011) One mixed negative binomial distribution with application. J Stat Plan Inference 141:1153–1160 Withers C, Nadarajah S (2010) Stabilizing the asymptotic covariance of an estimate. Electron J Stat 4:161–171