Abstract
Most of the approaches suggested during the last decades for solving life testing problems are markedly different from those used in the related but wider area of goodness of fit problems. In this paper, it is demonstrate that applying the goodness of fit approach makes sense also for solving life testing problems and that result in simpler procedures that are asymptotically equivalent or better than standard ones. They may have superior finite sample behavior. A U-statistic is obtained for testing exponentiality against the new better than used in the Laplace transform order aging class (NBUL). The pitman asymptotic efficiency of (NBUL) is studied. The power and the critical values of the proposed statistic are calculated. Real examples are given to elucidate the use of the proposed test statistic in the reliability analysis. Finally, the problem with the right censored data is also handled with some applications.
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References
Abouammoh AM, Abdulghani SA and Qamber IS (1994). On partial orderings and testing of new better than renewal used classes. Reliability Eng Syst Safety 43: 37–41
Ahmad IA (1994). A class of statistics useful in testing increasing failure rate average and new better than used life distribution. J Statist Plant Inf 41: 141–149
Ahmad IA (1995). Nonparametric testing of class of life distributions derived from a variability ordering. Parisankhyan Samikkha 2: 13–18
Ahmed H and Kayid M (2004). Preservation properties for the Laplace transform ordering of residual lives. Statist Pap 45: 583–590
Alzaid A, Kim JS and Proschan F (1991). Laplace ordering and its applications. J Appl Probab 28: 116–130
Al-Wasel AH, El-Bassiouny A, Kayid M (2006) Some Results of the NBUL class of life distributions. Under publications
Barlow RE and Proschan F (1981). Statistical theory of reliability and life testing. To Begin with, Silver Spring
Belzunce F, Ortega E and Ruiz JM (1999). The Laplace order and ordering of residual lives. Statist Probab Lett 42: 145–156
Bryson MC and Siddiqui MM (1969). Some criteria for aging. J Amer Stat Assoc 64: 1472–1483
Denuit M (2001). Laplace transform ordering of actuarial quantities. Insur Math Econ 29: 83–102
Ebrahimi M, Habibullah E and Sofi E (1992). Testing exponentiality based on Kullback–Leibler information. J R Stat Soc 54B: 739–748
EL-Arishy SM, Diab LS and Alim A (2006). Testing NRBU class of life distributions using a goodness of fit approach. Int J Reliability Appl 7(2): 77–39
Fisher RA (1966). The design of experiments eight edition. Oliver & Boyd, Edinburgh
Gao X, Belzunce F, Hu T, Pellerey F (2002) Developments on some preservation properties of the Laplace transform order of residual lives. Technical report, Department of Statistics and Finance, University of Science and Technology of China, Hefei, China
Grubbs FE (1971). Fiducial bounds on reliability for the two parameter negative exponential distribution. Technomet 13: 873–876
Hendi MI, Al-Nachawati H and AL-Ruzaiza AS (1999). A test for exponentiality against new better than used average. J King Saud Univ 11(2): 107–121
Hendi MI, Al-Nachawati H and AL-Graian MN (2000). Testing NBUFR and NBAFR classes of life distributions using Kernel methods. Arab J Math Sc 6: 37–53
Kaplan EL and Meier P (1958). Nonparametric estimation from incomplete observation. J Amer Statist Assoc 53: 457–481
Klefsjo B (1983). A useful aging property based on the Laplace transform. J Appl Probab 20: 615–626
Kotz S and Johnson NL (1983). Encyclopedia of statistical sciences, vol 3. Wiley, New York
Lawless JF (1982). Statistical models & methods for lifetime data. Wiley, New York
Lee AJ (1989). U-Statistics. Marcell-Dekker, New York
Mahmoud MAW, EL-arishy SM, Diab LS (2002) A non-parametric test of new renewal better than used class of life distributions. Proceeding of International Conference on Mathematics trends and developments. Cairo (EGYPT) 4:191–203
Mahmoud MAW, EL-arishy SM and Diab LS (2003). Moment inequalities for testing new renewal better than used and renewal new better than used classes of life distributions. Int J Reliability Appl 4(3): 97–123
Mahmoud MAW, EL-arishy SM and Diab LS (2004). Testing renewal new better than used life distributions based on U-test. Appl Math Model 29: 784–796
Mahmoud MAW and Diab LS (2007). On testing exponentiality against HNRBUE based on a goodness of fit. Int J Reliability Appl 8(1): 27–93
Pena AE (2002) Goodness of fit tests with censored data. http://statmanStat.sc.edu pena|ta|kspresented|talk actronel
Stoyan D and Muller A (2002). Comparison methods for queues and other stochastic models. Wiley, New York
Shaked M and Shanthikumar JG (1994). Stochastic orders and their applications. Academic Press, New York
Shapiro SS (1995). Goodness of fit tests. In: Balakrishnan, N and Basu, AP (eds) The exponential distribution theory methods and applications. Gorddon and Breach, Amsterdam
Susarla V and Vanryzin J (1978). Empirical bayes estimations of a survival function right censored observation. Ann Statist 6: 710–755
Yue D and Cao J (2001). The NBUL class of life distribution and replacement policy comparisons. Naval Res Logis 48: 578–591
Wang WY (1996) Life distribution classes and two unit standby redundant system, vol 11. Ph.D. dissertation, Chinese Academy of Science, Beijing
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Diab, L.S. Testing for NBUL using goodness of fit approach with applications. Stat Papers 51, 27–40 (2010). https://doi.org/10.1007/s00362-007-0113-0
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DOI: https://doi.org/10.1007/s00362-007-0113-0