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Likelihood Testing With Censored and Missing Duration Data

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Abstract

Duration data of a component, product, or system are often incomplete. We give guidelines for likelihood ratio testing for small samples with some data imperfections, that is, when some information is missing and some information is censored. We introduce the model of the missing time-to-failure mechanism, which still enables the exact likelihood ratio testing of the scale and homogeneity. Such a model encompasses well both the generalized gamma and Pareto duration time models with missing individual times to failure. The exact distribution of the likelihood ratio test for the censored sample from Weibull distribution with known shape parameter is derived and discussed. We discuss separately Type I, Type II, and progressively censored samples. We show that the Type I censoring and Type II censoring differ substantially. The construction of the pivotal quantity is possible for Type II and progressively Type II censoring; however, it is not available for the case of Type I censoring. Thus, for the latter case the usage of the exact likelihood ratio test is a natural option. We also discuss the case of the generalized gamma distribution. For the case with nuisance shape parameters we use an integrated likelihood approach. Convenient examples illustrate the methods developed in the article.

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References

  • Amstadter, B. L. 1971. Reliability mathematics, chap. 9. New York, NY: McGraw-Hill.

    Google Scholar 

  • Alvarez-Andradea, S., N. Balakrishnan, and L. Bordes. 2007. Homogeneity tests based on several progressively Type-II censored samples. J. Multivariate Anal., 98, 1195–1213.

    Article  MathSciNet  MATH  Google Scholar 

  • Bahadur, R. R. 1965. An optimal property of the likelihood ratio statistic. In Proc. 5th Berkeley Symposium on Probability Theory and Mathematical Statistics, vol. 1, ed. L. Le Cam and J. Neyman, 13–26. Berkeley, CA: University of California Press.

    Google Scholar 

  • Bain, L. J., and M. Engelhardt. 1992. Introduction to probability and mathematical statistics (2nd ed.). Boston, MA: PWSKENT Publishing Company.

    Google Scholar 

  • Balakrishnan, N. 2007. Progressive censoring methodology: An appraisal. TEST, 16, 211–259.

    Article  MathSciNet  MATH  Google Scholar 

  • Balakrishnan, N., and R. Aggarwala. 2000. Progressive censoring theory, methods, and applications, Series: Statistics for Industry and Technology, XV. Boston, MA: Birkhäuser.

    Book  Google Scholar 

  • Balakrishnan, N., and R. A. Sandhu. 1996. Best linear unbiased and maximum likelihood estimation for exponential distributions under general progressive Type-II censored samples. Sankhya, Ser. B, 58, 1–9.

    MathSciNet  MATH  Google Scholar 

  • Barndorf-Nielsen, O. E. 1978. Information and exponential families in statistical theory, 111–115. New York, NY: John Wiley Sons.

    Google Scholar 

  • Bartholomew, D. J. 1963. The sampling distribution of an estimate arising in life testing. Technometrics, 5, 361–374.

    Article  MathSciNet  MATH  Google Scholar 

  • Childs, A., B. Chandrasekar, N. Balakrishnan, and D. Kundu. 2003. Exact likelihood inference based on Type I and Type II hybrid censored samples from the exponential distribution. Ann. Inst. Stat. Math., 55(2), 319–330.

    MATH  Google Scholar 

  • Ciuperca, G. 2002. Likelihood ratio statistic for exponential mixtures. Ann. Inst. Stat. Math. 54(3), 585–594.

    Article  MathSciNet  MATH  Google Scholar 

  • Cohen, A. C. 1991. Truncated and censored samples. Statistics, a Series of Textbooks and Monographs. New York, NY: Marcel Dekker.

    MATH  Google Scholar 

  • Coit, D. W., and T. Jin. 2000. Gamma distribution parameter estimation for field reliability data with missing failure times. IIE Trans. 32, 1161–1166.

    Google Scholar 

  • Coit, D. W., and Dey, K. A. 1999. Analysis of grouped data from field-failure reporting systems. Reliability Eng. System Safety, 65, 95–101.

    Article  Google Scholar 

  • Cole, K. N., P. B. Nagarsenker, and B. N. Nagarsenker. 1987. A test for equality of exponential distributions based on Type-II censored samples. IEEE Trans. Reliability, 36(1), 94–97.

    Article  MATH  Google Scholar 

  • Corless, R. M., G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth. 1996. On the Lambert W function. Adv. Comput. Math. 5, 329–359.

    Article  MathSciNet  MATH  Google Scholar 

  • Corless, R. M., G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth. 1993. Lambert’s W function in Maple. Maple Tech. Newslett., 9, 12–22.

    Google Scholar 

  • Dey, K. A. 1982. Statistical analysis of noisy and incomplete failure data. In Proceedings Annual Reliability and Maintainability Symposium (RAMS), IEEE, Piscataway, NJ, 93–97.

  • Economou, P., and M. Stehlík. 2014. On small samples testing for frailty through homogeneity test. Commun. Stat. Simul. Comput. http://dx.doi.org/10.1080/03610918.2013.763982.

  • Epstein, B., and M. Sobel. 1954. Some theorems relevant to life testing from an exponential distribution. Ann. Math. Stat. 25, 373–381.

    Article  MathSciNet  MATH  Google Scholar 

  • Gaudoin, O., and J. L. Soler. 1992. Statistical analysis of the geometric de-eutrophication software reliability model. IEEE Trans. on Reliability, 41(4), 518–524.

    Article  MATH  Google Scholar 

  • Gautam, N. 1999. Erlang distribution: Description and case study. In Industrial engineering applications and practice: Users encyclopedia, eds. A. Mitaland J. Chen.

  • Hamada, M. S., A. G Wilson, C. S. Reese, and H. F. Martz. 2008. Bayesian reliability. Springer Series in Statistics. New York, NY: Springer.

    Book  MATH  Google Scholar 

  • Johansen, S. 1979. Introduction to the theory of regular exponential families. Lecture Notes, vol. 3. Copenhagen, Denmark: Institute of Mathematical Statistics, University of Copenhagen.

    Google Scholar 

  • Kleinrock, L. 1975. Queueing systems, vol. 1, 71–72, 119–134. Toronto, Canada: John Wiley & Sons.

    MATH  Google Scholar 

  • Kundu, D., and A. Joarder. 2006. Analysis of Type-II progressively hybrid censored data. Comput. Stat. Data Anal. 50, 2509–2528.

    Article  MathSciNet  MATH  Google Scholar 

  • Lehmann, E. L., and J. P. Romano. 2005. Testing statistical hypotheses. New York, NY: Springer-Verlag, LLC.

    MATH  Google Scholar 

  • Lin, D. K. J., J. S. Usher, and F. M. Guess. 1996. Bayes estimation of component from masked system-life data. IEEE Trans. Reliability, 45, 233–237.

    Article  Google Scholar 

  • Little, R. J. A., and D. B. Rubin. 1987. Statistical analysis with missing data. New York, NY: Wiley.

    MATH  Google Scholar 

  • Lomax, K. S. 1954. Business failures: Another example of the analysis of failure data. J. Am. Stat. Assoc. 49, 847–852.

    Article  MATH  Google Scholar 

  • Marshall, A. W., and I. Olkin. 2007. Life distributions. New York, NY: Springer.

    MATH  Google Scholar 

  • Mosler, K., and L. Haferkamp. 2007. Size and power of recent tests for homogeneity in exponential mixtures. Commun. Stat. Simulation Comput. 36, 493–504.

    Article  MathSciNet  MATH  Google Scholar 

  • Mosler, K., and C. Scheicher. 2008. Homogeneity testing in a Weibull mixture model. Stat. Papers, 49, 315–332.

    Article  MathSciNet  MATH  Google Scholar 

  • Nagarsenker, P. B. 1980. On a test of equality of several exponential survival distributions. Biometrika, 67(2), 475–478.

    Article  MathSciNet  MATH  Google Scholar 

  • Philbrick, S. W. 1985. A practical guide to the single parameter Pareto distribution. Proc. Casualty Actuarial Society, LXXII, 44–84.

    Google Scholar 

  • Rublík, F. 1989a. On optimality of the LR tests in the sense of exact slopes, Part 1, General case. Kybernetika, 25, 13–25.

    MathSciNet  MATH  Google Scholar 

  • Rublík, F. 1989b. On optimality of the LR tests in the sense of exact slopes, Part 2, Application to individual distributions. Kybernetika, 25, 117–135.

    MathSciNet  MATH  Google Scholar 

  • Soland, R. M. 1966. Use of Weibull distribution in Bayesian decision theory, Report No. RAC-TP-225. McLean, VA: Research Analysis Corporation.

    Google Scholar 

  • Soland, R. M. 1969. Bayesian analysis of the Weibull process with unknown scale and shape parameters. IEEE Trans. Reliability, 18(4), 181–184.

    Article  Google Scholar 

  • Severini, T. A. 1999. On the relationship between Bayesian and non-Bayesian elimination of nuisance parameters. Stat. Sin. 9, 713–724.

    MathSciNet  MATH  Google Scholar 

  • Severini, T. A. 2006. Likelihood methods in statistics. Oxford Statistical Science Series. New York, NY: Oxford University Press.

    MATH  Google Scholar 

  • Severini, T. A. 2010. Likelihood ratio statistics based on an integrated likelihood. Biometrika, 97(2), 481–496.

    Article  MathSciNet  MATH  Google Scholar 

  • Shoukri, M. M. 1987. Simple Bayes test of equality of exponential means. IEEE Trans. Reliability, 36(5), 613–616.

    Article  Google Scholar 

  • Stehlík, M. 2003. Distributions of exact tests in the exponential family. Metrika, 57, 145–164.

    Article  MathSciNet  MATH  Google Scholar 

  • Stehlík, M. 2006. Exact likelihood ratio scale and homogeneity testing of some loss processes. Stat. Probability Lett., 76, 19–26.

    Article  MathSciNet  MATH  Google Scholar 

  • Stehlík, M. 2007. Exact testing of the scale with the missing time-to-failure information. Commun. Dependability Qual. Manage. Eng. (CDQM), 10(2), 124–129.

    Google Scholar 

  • Stehlík, M. 2008. Homogeneity and scale testing of generalized gamma distribution. Reliability Eng. System Safety, 93, 1809–1813.

    Article  Google Scholar 

  • Stehlík, M. 2009. Scale testing in small samples with missing time-to-failure information. Int. J. Reliability Qual. Safety Eng. (IJRQSE), 16(6), 1–13.

    Article  Google Scholar 

  • Stehlík, M., R. Potocký, H. Waldl, and Z. Fabián. 2010. On the favourable estimation of fitting heavy tailed data. Comput. Stat., 25, 485–503.

    Article  MATH  Google Scholar 

  • Stehlík, M., and H. Wagner. 2011. Exact likelihood ratio testing for homogeneity of exponential distribution. Commun. Stat. Simulation Comput., 40, 663–684.

    Article  MathSciNet  MATH  Google Scholar 

  • Stehlík, M., P. Economou, J. Kiseľák, and W.-D. Richter. 2014. Kullback-Leibler life time testing. App. Math. Comput., 240, 122–139.

    Article  MathSciNet  MATH  Google Scholar 

  • Sukhatme, P. V. 1937. Tests of significance for samples of the χ2 population with two degrees of freedom. Ann. Eugenics, 8, 52–56.

    Article  Google Scholar 

  • Thiagarajah, K. R. 1995. Homogeneity tests for scale parameters of 2-parameter exponential populations under time censoring. IEEE Trans. Reliability. 44(2), 297–301.

    Article  Google Scholar 

  • Thomas, D. R., and W. M. Wilson. 1972. Linear order statistic estimation for the two parameter Weibull and extreme value distributions from Type-II progressively censored samples. Technometrics, 14, 679–691.

    Article  MATH  Google Scholar 

  • Usher, J. S. 1996. Weibull component reliability—Prediction in the presence of masked data. IEEE Trans. Reliability, 45, 229–232.

    Article  Google Scholar 

  • Viveros, R., and N. Balakrishnan. 1994. Interval estimation of parameters of life from progressively censored data. Technometrics, 36(1), 84–91.

    Article  MathSciNet  MATH  Google Scholar 

  • Wilks, S. S. 1962. Mathematical statistics. New York, NY: John Wiley & Sons.

    MATH  Google Scholar 

  • Wu, S. J., D. H. Chen, and S. T. Chen. 2006. Bayesian inference for Rayleigh distribution under progressive censored sample. App. Stochastic Models Business Ind., 22, 269–279.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada

    Narayanaswamy Balakrishnan

  2. Department of Applied Statistics, Johannes Kepler University, Linz, Austria

    Milan Stehlík

  3. Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile

    Milan Stehlík

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  1. Narayanaswamy Balakrishnan
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  2. Milan Stehlík
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Correspondence to Milan Stehlík.

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Balakrishnan, N., Stehlík, M. Likelihood Testing With Censored and Missing Duration Data. J Stat Theory Pract 9, 2–22 (2015). https://doi.org/10.1080/15598608.2014.927811

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