Abstract
In this paper, we discuss the statistical inference of the lifetime distribution of components based on observing the system lifetimes when the system structure is known. A general proportional hazard rate model for the lifetime of the components is considered, which includes some commonly used lifetime distributions. Different estimation methods—method of moments, maximum likelihood method and least squares method—for the proportionality parameter are discussed. The conditions for existence and uniqueness of method of moments and maximum likelihood estimators are presented. Then, we focus on a special case when the lifetime distributions of the components are exponential. Computational formulas for point and interval estimations of the unknown mean lifetime of the components are provided. A Monte Carlo simulation study is used to compare the performance of these estimation methods and recommendations are made based on these results. Finally, an example is provided to illustrate the methods proposed in this paper.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arcones MA, Kvam PH, Samaniego FJ (2002) Nonparametric estimation of a distribution subject to a stochastic precedence constraint. J Am Stat Assoc 97: 170–182
Arnold BC, Balakrishnan N, Nagaraja HN (1992) A first course in order statistics. Wiley, New York
Balakrishnan N, Basu AP (1995) The exponential distribution: theory, methods and applications. Gordon and Breach, Langhorne
Boland PJ, Samaniego FJ (2004) The signature of a coherent system and its applications in reliability. In: Soyer R, Mazzuchi T, Singpurwalla ND (eds) Mathematical reliability: an expository perspective. Kluwer Publishers, Boston, pp 1–29
Boland PJ, Samaniego FJ, Vestrup EM (2003) Linking dominations and ignatures in network reliability theory. In: Lindqvist BH, Doksum KA (eds) Mathematical and statistical methods in reliability. World Scientific Publishing Co. Pte. Ltd., River Edge, pp 89–104
Dugas MR, Samaniego FJ (2007) On optimal system designs in reliability-economics frameworks. Nav Res Logist 54: 568–582
Efron B (1982) The Jackknife, the Bootstrap, and other resampling plans. Society of Industrial and Applied Mathematics, Philadelphia
Efron B, Tibshirani R (1993) An introduction to the bootstrap. Chapman & Hall, New York
Gåsemyr J, Natvig B (1998) The posterior distribution of the parameters of component lifetimes based on autopsy data in a shock model. Scandinavian J Stat 25: 271–292
Gåsemyr J, Natvig B (2001) Bayesian inference based on partial monitoring of components with applications to preventive system maintenance. Nav Res Logist 48: 551–577
Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, (vol 1, 2nd edn.). Wiley, New York
Kochar S, Mukerjee H, Samaniego FJ (1999) The “Signature” of a coherent system and its application to comparisons among systems. Nav Res Logist 46: 507–523
Li JA, Wu Y, Lai KK, Liu K (2005) Reliability estimation and prediction of multi-state components and coherent systems. Reliab Eng Syst Safety 88: 93–98
Meilijson I (1991) Estimation of the lifetime distribution of the parts from the autopsy statistics of the machine. J Appl Probab 18: 829–836
Navarro J (2007) Tail hazard rate ordering properties of order statistics and coherent systems. Nav Res Logist 58: 820–828
Navarro J (2008) Likelihood ratio ordering of order statistics, mixtures and systems. J Stat Plan Infer 138: 1242–1257
Navarro J, Rubio R (2010) Computations of coherent systems with five components. Commun Stat Simul Comput 39: 68–84
Navarro J, Ruiz JM, Sandoval CJ (2005) A note on comparisons among coherent systems with dependent components using signatures. Stat Probab Lett 72: 179–185
Navarro J, Ruiz JM, Sandoval CJ (2007) Properties of coherent systems with dependent components. Commun Stat Theory Methods 36: 175–191
Navarro J, Ruiz JM, Sandoval CJ (2008) Properties of systems with two exchangeable Pareto components. Stat Papers 49: 177–190
Navarro J, Rychlik T (2007) Reliability and expectation bounds for coherent systems with exchangeable components. J Multivar Anal 98: 102–113
Navarro J, Rychlik T (2010) Comparisons and bounds for expected lifetimes of reliability systems. Eur J Oper Res 207: 309–317
Navarro J, Samaniego FJ, Balakrishnan N (2010) Joint signature of coherent systems with shared components. J Appl Probab 47: 235–253
Navarro J, Samaniego FJ, Balakrishnan N, Bhattacharya D (2008) On the application and extension of system signatures to problems in engineering reliability. Nav Res Logist 55: 313–327
Navarro J, Shaked M (2010) Some properties of the minimum and the maximum of random variables with joint logconcave distributions. Metrika 71: 313–317
Samaniego FJ (1985) On closure of the IFR class under formation of coherent systems. IEEE Trans Reliab 34: 69–72
Samaniego FJ (2007) System signatures and their applications in engineering reliability. Springer, New York
Shaked M, Suarez-Llorens A (1993) On the comparison of reliability experiments based on the convolution order. J Am Stat Assoc 98: 693–702
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ng, H.K.T., Navarro, J. & Balakrishnan, N. Parametric inference from system lifetime data under a proportional hazard rate model. Metrika 75, 367–388 (2012). https://doi.org/10.1007/s00184-010-0331-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-010-0331-7