Advertisement

Evaluating the lifetime performance index of products based on generalized order statistics from two-parameter exponential model

  • Mohammad Vali Ahmadi
  • Jafar AhmadiEmail author
  • Mousa Abdi
Original Article
  • 27 Downloads

Abstract

Assessing the lifetime performance of products is one of the most important topics in the manufacturing industries. In this paper, we assume that the lifetimes of products are independent and have a common two-parameter exponential distribution. The lifetime performance index (\(C_L\)) provides a means for evaluating the performance of a process under a known lower lifetime limit L. We consider a sample of generalized order statistics (GOS), introduced by Kamps (A concept of generalized order statistics. Teubner, Stuttgart, 1995), which contains several models of ordered random variables, e.g. ordinary order statistics, progressively censored order statistics and record values. Then, we obtain the maximum likelihood estimator and the uniformly minimum variance unbiased estimator (UMVUE) of \(C_L\) on the basis of a GOS sample. These estimators are compared in terms of mean squared error and Pitman measure of closeness criteria. The UMVUE of \(C_L\) is utilized to develop a novel hypothesis testing procedure in the condition of known L. Finally, in order to illustrate the results, two real data sets due to Lawless (Statistical model and methods for lifetime data, 2nd edn. Wiley, New York, 2003) and Proschan (Technometrics 15:375–383, 1963), and a simulated sample are analyzed.

Keywords

Generalized order statistics Two-parameter exponential distribution Lifetime performance index Hypothesis testing 

Mathematics Subject Classification

62N01 62N0 62N03 62N05 62F25 

Notes

Acknowledgements

The authors would like to thank the Editor and reviewers for their valuable comments and suggestions to improve the presentation of the paper. The research of J. Ahmadi was supported by Ferdowsi University of Mashhad [Grant Number 2/45621].

References

  1. Aboeleneen ZA (2010) Inference for Weibull distribution under generalized order statistics. Math Comput Simul 81:26–36MathSciNetCrossRefzbMATHGoogle Scholar
  2. Ahmadi MV, Doostparast M, Ahmadi J (2013) Estimating the lifetime performance index with Weibull distribution based on progressive first-failure censoring scheme. J Comput Appl Math 239:93–102MathSciNetCrossRefzbMATHGoogle Scholar
  3. Ahmadi MV, Doostparast M, Ahmadi J (2015) Statistical inference for the lifetime performance index based on generalised order statistics from exponential distribution. Int J Syst Sci 46:1094–1107MathSciNetCrossRefzbMATHGoogle Scholar
  4. Ahsanullah M (2000) Generalized order statistics from exponential distribution. J Stat Plan Inference 85:85–91MathSciNetCrossRefzbMATHGoogle Scholar
  5. Arnold BC, Balakrishnan N, Nagaraja HN (1998) Records. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  6. Balakrishnan N, Kundu D, Ng HKT, Kannan N (2007) Point and interval estimation for a simple step-stress model with Type-II censoring. J Qual Technol 39:35–47CrossRefGoogle Scholar
  7. Balakrishnan N, Xie Q (2007) Exact inference for a simple step-stress model with Type-I hybrid censored data from the exponential distribution. J Stat Plan Inference 137:3268–3290MathSciNetCrossRefzbMATHGoogle Scholar
  8. Balakrishnan N, Basu AP (eds) (1995) The exponential distribution: theory, methods and applications. Gordon and Breach, AmsterdamGoogle Scholar
  9. Balasooriya U (1995) Failure censored reliability sampling plans for the exponential distribution. J Stat Comput Simul 52:337–349CrossRefzbMATHGoogle Scholar
  10. Bousquet N, Bertholon H, Celeux G (2006) An alternative competing risk model to the Weibull distribution for modelling aging in lifetime data analysis. Lifetime Data Anal 12:481–504MathSciNetCrossRefzbMATHGoogle Scholar
  11. Casella G, Berger RL (2001) Statistical inference, 2nd edn. Duxbury, CaliforniazbMATHGoogle Scholar
  12. Chen SM, Bhattacharya GK (1988) Exact confidence bound for an exponential parameter under hybrid censoring. Commun Stat Theory Methods 16:1857–1870MathSciNetCrossRefzbMATHGoogle Scholar
  13. Childs A, Chandrasekar B, Balakrishnan N, Kundu D (2003) Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution. Ann Inst Stat Math 55:319–330MathSciNetzbMATHGoogle Scholar
  14. Day S, Sharma VK, Anis MZ, Yadav B (2017) Assessing lifetime performance index of Weibull distributed products using progressive Type-II right censored samples. Int J Syst Assur Eng Manag 8:318–333CrossRefGoogle Scholar
  15. Epstein B, Sobel M (1953) Life-Testing. J Am Stat Assoc 48:486–502MathSciNetCrossRefzbMATHGoogle Scholar
  16. Hong CW, Wu JW, Cheng CH (2007) Computational procedure of performance assessment of lifetime index of businesses for the Pareto lifetime model with the right Type-II censored sample. Appl Math Comput 184:336–350MathSciNetzbMATHGoogle Scholar
  17. Kamps U (1995) A concept of generalized order statistics. Teubner, StuttgartCrossRefzbMATHGoogle Scholar
  18. Kane VE (1986) Process capability indices. J Qual Technol 18:41–52CrossRefGoogle Scholar
  19. Kishan R, Jain D (2014) Classical and Bayesian analysis of reliability characteristics of a two-unit parallel system with Weibull failure and repair laws. Int J Syst Assur Eng Manag 5:252–261CrossRefGoogle Scholar
  20. Laumen B, Cramer E (2015) Likelihood inference for the lifetime performance index under progressive Type-II censoring. Econ Qual Control 30:59–73CrossRefzbMATHGoogle Scholar
  21. Lawless JF (1971) A prediction problem concerning samples from the exponential distribution, with application in life testing. Technometrics 13:725–730CrossRefzbMATHGoogle Scholar
  22. Lawless JF (2003) Statistical model and methods for lifetime data, 2nd edn. Wiley, New YorkzbMATHGoogle Scholar
  23. Lee HM, Lee WC, Lei CL, Wu JW (2011) Computational procedure of assessing lifetime performance index of Weibull lifetime products with the upper record values. Math Comput Simul 81:1177–1189MathSciNetCrossRefzbMATHGoogle Scholar
  24. Lee WC, Wu JW, Hong CW (2009) Assessing the lifetime performance index of products from progressively Type-II right censored data using Burr XII model. Math Comput Simul 79:2167–2179MathSciNetCrossRefzbMATHGoogle Scholar
  25. Lee WC, Wu JW, Hong ML, Lin LS, Chan RL (2011) Assessing the lifetime performance index of Rayleigh products based on the Bayesian estimation under progressive Type-II right censored samples. J Comput Appl Math 235:1676–1688MathSciNetCrossRefzbMATHGoogle Scholar
  26. Lee WC, Wu JW, Lei CL (2010) Evaluating the lifetime performance index for the exponential lifetime products. Appl Math Model 34:1217–1224CrossRefGoogle Scholar
  27. Lee HM, Wu JW, Lei CL, Hung WL (2011) Implementing lifetime performance index of products with two-parameter exponential distribution. Int J Syst Sci 42:1305–1321MathSciNetCrossRefzbMATHGoogle Scholar
  28. Lehmann EL, Casella G (1998) Theory of point estimation, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
  29. Liu MF, Ren HP (2013) Bayesian test procedure of lifetime performance index for exponential distribution under progressive Type-II censoring. Int J Appl Math Stat 32:27–38MathSciNetGoogle Scholar
  30. Montgomery DC (1985) Introduction to statistical quality control. Wiley, New YorkzbMATHGoogle Scholar
  31. Pan JN, Wu SL (1997) Process capability analysis for non-normal relay test data. Microelectron Reliab 37:421–428CrossRefGoogle Scholar
  32. Papadopoulos AS (1978) The Burr distribution as a failure model from a Bayesian approach. IEEE Trans Reliab 27:369–371CrossRefzbMATHGoogle Scholar
  33. Pitman EJG (1937) The “closest” estimate of statistical parameters. Math Proc Camb Philos Soc 33:212–222CrossRefzbMATHGoogle Scholar
  34. Proschan F (1963) Theoretical explanation of observed decreasing failure rate. Technometrics 15:375–383CrossRefGoogle Scholar
  35. Stone GC, Van Heeswijk RG (1977) Parameter estimation for the Weibull distribution. IEEE Trans Electr Insul 12:253–261CrossRefGoogle Scholar
  36. Tong LI, Chen KS, Chen HT (2002) Statistical testing for assessing the performance of lifetime index of electronic components with exponential distribution. Int J Qual Reliab Manag 19:812–824CrossRefGoogle Scholar
  37. Wu JW, Hong CW, Lee WC (2014) Computational procedure of lifetime performance index of products for the Burr XII distribution with upper record values. Appl Math Comput 227:701–716MathSciNetzbMATHGoogle Scholar
  38. Wu JW, Lee HM, Lei CL (2007) Computational testing algorithmic procedure of assessment for lifetime performance index of products with two-parameter exponential distribution. Appl Math Comput 190:116-125MathSciNetzbMATHGoogle Scholar
  39. Wu SF, Lin YP (2016) Computational testing algorithmic procedure of assessment for lifetime performance index of products with one-parameter exponential distribution under progressive type I interval censoring. Math Comput Simul 120:79–90MathSciNetCrossRefGoogle Scholar
  40. Yan A, Liu S, Dong X (2015) Repetitive group sampling plan for Rayleigh products based on the lifetime performance index \(C_L\). J Comput Theor Nanosci 12:3631–3636CrossRefGoogle Scholar
  41. Zehna PW (1966) Invariance of maximum likelihood estimation. Ann Math Stat 37:744MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Society for Reliability Engineering, Quality and Operations Management (SREQOM), India and The Division of Operation and Maintenance, Lulea University of Technology, Sweden 2019

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of BojnordBojnordIran
  2. 2.Department of StatisticsFerdowsi University of MashhadMashhadIran
  3. 3.Department of StatisticsHigher Education Complex of BamBamIran

Personalised recommendations