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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)
MACSYMA; Maxima
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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
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