Estimation of reliability in a series system with random sample size.

*(English)*Zbl 1284.62620Summary: We are interested in the estimation of the reliability coefficient \(R=P(X>Y)\), when the data on the minimum of two exponential samples, with random sample size, are available. The confidence intervals of R, based on maximum likelihood and bootstrap methods, are developed. The performance of these confidence intervals is studied through extensive simulation. A numerical example, based on a real data, is presented to illustrate the implementation of the proposed procedure.

##### MSC:

62N05 | Reliability and life testing |

62-04 | Software, source code, etc. for problems pertaining to statistics |

##### Keywords:

truncated Poisson distribution; series system; random sample size; asymptotic and bootstrap confidence intervals; simulation study
PDF
BibTeX
Cite

\textit{D. K. Al-Mutairi} et al., Comput. Stat. Data Anal. 55, No. 2, 964--972 (2011; Zbl 1284.62620)

Full Text:
DOI

##### References:

[1] | Bartoszewicz, J., Stochastic comparisons of random minima and maxima from life distributions, Statistics and probability letters, 55, 107-112, (2001) · Zbl 0989.60020 |

[2] | Canty, A., Ripley, B., (2009). boot: Bootstrap R (S-Plus) Functions. R package version 1.2-37. |

[3] | Church, J.D.; Harris, B., The estimation of reliability from stress – strength relationship, Technometrics, 12, 49-54, (1970) · Zbl 0195.20001 |

[4] | Consul, P.C., On the distribution of order statistics for a random sample size, Statistica neerlandica, 38, 249-256, (1984) · Zbl 0552.62010 |

[5] | Davison, A.C.; Hinkley, D.V., Bootstrap methods and their application, (2005), Cambridge University Press UK · Zbl 0886.62001 |

[6] | Downton, F., The estimation of \(\Pr(Y < X)\) in the normal case, Technometrics, 15, 551-558, (1973) · Zbl 0262.62016 |

[7] | Gupta, D.; Gupta, R.C., On the distribution of order statistics for a random sample size, Statistica neerlandica, 38, 13-19, (1984) · Zbl 0531.62011 |

[8] | Gupta, R.C., Modified power series distribution and some of its applications, Sankhya, series B, 36, 288-298, (1974) · Zbl 0318.62009 |

[9] | Gupta, R.C.; Ramakrishnan, S.; Zhou, X., Point and interval estimation of P\((X < Y)\): the normal case with common coefficient of variation, Annals of the institute of statistical mathematics, 51, 571-584, (1999) · Zbl 0938.62014 |

[10] | Gupta, R.C; Brown, N., Reliability studies of the skew-normal distribution and its application to strength-stress models, Communications in statistics, theory and methods, 30, 2427-2445, (2001) · Zbl 1009.62513 |

[11] | Gupta, R.C.; Li, X., Statistical inference for the common Mean of two log-normal distributions and some applications in reliability, Computational statistics and data analysis, 50, 3141-3164, (2006) · Zbl 1445.62268 |

[12] | Gupta, R.C.; Peng, C., Estimating reliability in proportional odds ratio models, Computational statistics and data analysis, 53, 1495-1510, (2009) · Zbl 1452.62726 |

[13] | Gupta, R.D.; Gupta, R.C., Estimation of \(\Pr(a^\prime x > b^\prime y)\) in the multivariate normal case, Statistics, 21, 91-97, (1990) · Zbl 0699.62053 |

[14] | Johnson, N.L.; Kemp, A.W.; Kotz, S., Univariate discrete distributions, (2005), Wiley New York · Zbl 1092.62010 |

[15] | Kotz, S.; Lumelskii, Y.; Pensky, M., The strength-stress model and its generalization, (2003), World Scientific Press Singapore |

[16] | Kundu, D.; Gupta, R.D., Estimation of \(P [Y < X]\) for generalized exponential distribution, Metrika, 61, 291-308, (2005) · Zbl 1079.62032 |

[17] | Kundu, D.; Raqab, M.Z., Estimation of \(R = P(Y < X)\) for three-parameter Weibull distribution, Statistics and probability letters, 79, 1839-1846, (2009) · Zbl 1169.62012 |

[18] | Kus, C., A new lifetime distribution, Computational statistics and data analysis, 51, 4497-4509, (2007) · Zbl 1162.62309 |

[19] | Lehmann, L.E.; Casella, G., Theory of point estimation, (1998), Springer New York · Zbl 0916.62017 |

[20] | Li, X.; Zuo, M.J., Preservation of stochastic orders for random minima and maxima, with applications, Naval research logistics, 51, 332-344, (2004) · Zbl 1055.60008 |

[21] | Mazumdar, M., Some estimates of reliability using interference theory, Naval research. logistics quarterly, 17, 159-165, (1970) · Zbl 0205.23001 |

[22] | Nanda, A.K.; Shaked, M., Partial ordering and aging properties of order statistics when the sample size is random: A brief review, Communications in statistics, theory and methods, 37, 1710-1720, (2008) · Zbl 1145.60305 |

[23] | Proschan, F., Theoretical explanation of observed decreasing failure rate, Technometrics, 5, 375-383, (1963) |

[24] | Raqab, M.Z.; Madi, M.D.; Kundu, D., Estimation of \(P(Y < X)\) for the 3-parameter generalized exponential distribution, Communications in statistics, theory and methods, 37, 2854-2864, (2008) · Zbl 1292.62041 |

[25] | Reiser, B.; Guttman, I., Statistical inference for \(\Pr(Y < X)\): the normal case, Technometrics, 28, 253-257, (1986) · Zbl 0631.62033 |

[26] | Tahmasbi, R.; Rezaei, S., A two-parameter lifetime distribution with decreasing failure rate, Computational statistics and data analysis, 52, 3889-3901, (2008) · Zbl 1245.62128 |

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.