Abstract
In this paper, progressive stress accelerated life tests are considered when the lifetime of a product under use condition follows a finite mixture of distributions. The experiment is performed when each of the components in the mixture follows a general class of distributions which includes, among others, the Weibull, compound Weibull, power function, Gompertz and compound Gompertz distributions. It is assumed that the scale parameter of each component satisfies the inverse power low, the progressive stress is directly proportional to time and the cumulative exposure model for the effect of changing stress holds. Based on type-I censoring, the maximum likelihood estimates (MLEs) of the parameters under consideration are obtained. A special attention is paid to a mixture of two Rayleigh components. Simulation results are carried out to study the precision of the MLEs and to obtain confidence intervals for the parameters involved.
Similar content being viewed by others
References
AL-Hussaini EK, Abdel-Hamid AH (2004) Bayesian estimation of the parameters, reliability and hazard rate functions of mixtures under accelerated life tests. Commun Statist Simul Compute 33(4):963–982
AL-Hussaini EK, Abdel-Hamid AH (2006) Accelerated life tests under finite mixture models. J Statist Comput Simul 76(8):673–690
AL-Hussaini EK, Ahmad KE (1981) On the identifiability of finite mixtures of distributions. IEEE Trans Inform Theor 27:664–668
AL-Hussaini EK, Fakhry ME (1995) On characterization of finite mixtures of distributions. J Appl Statist Sci 2:249–261
AL-Hussaini EK, Osman MI (1997) On the median of a finite mixture. J Statist Comput Simul 58:121–144
AL-Hussaini EK, Sultan KS (2001) Reliability and Hazard Based on Finite Mixture Models. In: Balakrishnan N, Rao CR (eds) Chapter 5 in Handbook of Statistics: Advances in Reliability, vol 20. Elsevier Science, North Holland, Amsterdam, pp. 139–187
Allen WR (1959) Inference from tests with continuously increasing stress. Oper Res 17:303–312
Apostol FM (1960) Mathematical analysis. Addison Wesley, USA
Bai DS, Cha MS, Chung SW (1992) Optimum simple ramp-tests for the Weibull distribution and type-I censoring. IEEE Trans Rel 41:407–413
Chan CK (1990) A proportional hazard approach to accelerate SiO 2 - breakdown voltage & time distributions. IEEE Trans Rel 39:147–150
Chung SW, Bai DS (1998) Optimal designs of simple step-stress accelerated life tests for lognormal lifetime distributions. Int J Reliab Qual Saf Eng 5(4):315–336
Lindsay BG (1995) Mixture models: theory, geometry and applications. The Institute of Mathematical Statistics, Hayward
Maritz JS, Lwin T (1989) Empirical bayes methods. 2nd Ed Chapman and Hall, London
McLachlan GJ, Basford KE (1988) Mixture models: inferences and applications to clustering. Marcel Dekker, New York
McLachlan GJ, Peel D (2000) Finite mixture models. Wiley, New York
Nelson W (1990) Accelerated testing: statistical models, test plans and data analysis. Wiley, New York
Polovko AM (1968) Fundamentals of reliability theory. Academic Press, New York
Prot EM (1948) Fatigue testing under progressive loading; a new technique for testing materials. Revue de Metallurgie, XIV, 481–489 (in French). Translation in WADC TR-52-148, 1952 Sep
Ronghua W, Heliang F (2004) Statistical inference of Weibull distribution for tampered failure rate model in progressive stress accelerated life testing. J Syst Sci Complex 17:237–243
Solomon P, Klein N, Albert M (1976) A statistical model for step and ramp voltage breakdown tests in thin insulators. Thin Solid Films 35:321–326
Starr WT, Endicott HS (1961) Progressive stress - a new accelerated approach to voltage endurance. Transactions AIEE (Power Apparatus and Systems) 80:515–522
Titterington DM, Smith AFM, Makov UE (1985) Statistical analysis of finite mixture distributions. Wiley, New York
Yakowitz SJ, Spragins JD (1968) On the identifiability of finite mixtures. Ann Math Statist 39:209–214
Yin XK, Sheng BZ (1987) Some aspects of accelerated life testing by progressive stress. IEEE Trans Rel 36:150–155
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Abdel-Hamid, A.H., AL-Hussaini, E.K. Progressive stress accelerated life tests under finite mixture models. Metrika 66, 213–231 (2007). https://doi.org/10.1007/s00184-006-0106-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-006-0106-3