×

zbMATH — the first resource for mathematics

A differential equation for a class of discrete lifetime distributions with an application in reliability. A demonstration of the utility of computer algebra. (English) Zbl 1337.60224
Summary: 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 can then 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.
MSC:
60K10 Applications of renewal theory (reliability, demand theory, etc.)
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J22 Computational methods in Markov chains
90B25 Reliability, availability, maintenance, inspection in operations research
62M05 Markov processes: estimation; hidden Markov models
65C40 Numerical analysis or methods applied to Markov chains
65C50 Other computational problems in probability (MSC2010)
Software:
MACSYMA; Maxima
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abramowitz M, Stegun A (1972) Handbook of mathematical functions. Dover Publications, New York · Zbl 0543.33001
[2] 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 · Zbl 0563.62079
[3] Biggs NL (1989) Discrete mathematics. Clarendon Press, Oxford
[4] Charalambides, CA, Moments of a class of discrete \(q\)-distributions, J Stat Plan Inference, 135, 64-76, (2005) · Zbl 1075.60087
[5] Crippa, D; Simon, K, \(q\)-distributions and Markov processes, Discret Math, 170, 81-98, (1997) · Zbl 0884.60065
[6] Csenki, A, Dependability for systems with a partitioned state space—markov and semi-Markov theory and computational implementation, (1994), New York · Zbl 0805.60090
[7] Csenki, A, On continuous lifetime distributions with polynomial failure rate with an application in reliability, Reliab Eng Syst Saf, 96, 1587-1590, (2011)
[8] Csenki, A, Asymptotics for continuous lifetime distributions with polynomial failure rate with an application in reliability, Reliab Eng Syst Saf, 102, 1-4, (2012)
[9] Grimmett G, Welsh D (1986) Probability—an introduction. Clarendon Press, Oxford · Zbl 0606.60002
[10] Heller B (1991) MACSYMA for statisticians. Wiley-Interscience, New York · Zbl 0778.62002
[11] Jazi, MA; Lai, CD; Alamatsaz, MH, A discrete inverse Weibull distribution and estimation of its parameters, Stat Methodol, 7, 121-132, (2010) · Zbl 1230.62130
[12] Khan, MSA; Kalique, A; Abouammoh, AM, On estimating parameters in a discrete Weibull distribution, IEEE Trans Reliab, 38, 348-348, (1989) · Zbl 0709.62640
[13] Lawless JF (1982) Statistical models and methods for lifetime data. Wiley, New York · Zbl 0541.62081
[14] Lehmann EL (1999) Elements of large-sample theory. Springer, New York · Zbl 0914.62001
[15] Limnios, N, Reliability measures of semi-Markov systems with general state space, Methodol Comput Appl Probab, (2011) · Zbl 1259.60106
[16] Ma, Y; Genton, MG; Parzen, E, Asymptotic properties of sample quantiles of discrete distributions, Ann Inst Stat Math, 63, 227-243, (2011) · Zbl 1432.62035
[17] Nakagawa, TA; Osaki, S, The discrete Weibull distribution, IEEE Trans Reliab, 24, 300-301, (1975)
[18] Neuts M (1981) Matrix-geometric solutions in stochastic models: an algorithmic approach. The Johns Hopkins University Press, Baltimore · Zbl 0469.60002
[19] 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
[20] Rényi A (1970) Probability theory. North-Holland, Amsterdam, London (1970)
[21] Sarhan, AM; Hamilton, DC; Smith, B; Kundu, D, The bivariate generalized linear failure rate distribution and its multivariate extension, Comput Stat Data An, 55, 644-654, (2011) · Zbl 1247.62137
[22] Schelter WF (2006) Maxima Manual, version 5.9.3. http://maxima.sourceforge.net/docs/manual/en/maxima.html. Accessed 15 Aug 2012
[23] Shaked, M; Shantikumar, GJ; Valdez-Torres, JB, Discrete hazard rate functions, Comput Oper Res, 22, 391-402, (1995) · Zbl 0822.90073
[24] Stein, WE; Dattero, R, A new discrete Weibull distribution, IEEE Trans Reliab, 33, 196-197, (1984) · Zbl 0563.62079
[25] Wang, Z, One mixed negative binomial distribution with application, J Stat Plan Inference, 141, 1153-1160, (2011) · Zbl 1206.62014
[26] Withers, C; Nadarajah, S, Stabilizing the asymptotic covariance of an estimate, Electron J Stat, 4, 161-171, (2010) · Zbl 1329.62112
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.