One of the objectives of quantile-based reliability analysis is to make use of quantile functions as models in lifetime data analysis. Accordingly, in this chapter, we discuss the characteristics of certain quantile functions known in the literature. The models considered are the generalized lambda distribution of Ramberg and Schmeiser, the generalized Tukey lambda family of Freimer, Kollia, Mudholkar and Lin, the four-parameter distribution of van Staden and Loots, the five-parameter lambda family and the power-Pareto model of Gilchrist, the Govindarajulu distribution and the generalized Weibull family of Mudholkar and Kollia.
The shapes of the different systems and their descriptive measures of location, dispersion, skewness and kurtosis in terms of conventional moments, L-moments and percentiles are provided. Various methods of estimation based on moments, percentiles, L-moments, least squares and maximum likelihood are reviewed. Also included are the starship method, the discretized approach specifically introduced for the estimation of parameters in the quantile functions and details of the packages and tables that facilitate the estimation process.
In analysing the reliability aspects, one also needs various functions that describe the ageing phenomenon. The expressions for the hazard quantile function, mean residual quantile function, variance residual quantile function, percentile residual life function and their counter parts in reversed time given in the preceding chapters provide the necessary tools in this direction. Some characterization theorems show the relationships between reliability functions unique to various distributions. Applications of selected models and the estimation procedures are also demonstrated by fitting them to some data on failure times.
Failure Time Quantile Function Lifetime Data Reliability Function Life Distribution
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access
Arnold, B.C., Balakrishnan, N., Nagaraja, H.N.: A First Course in Order Statistics. Wiley, New York (1992)MATHGoogle Scholar
Balakrishnan, N., Balasubramanian, K.: Equivalence of Hartley-David-Gumbel and Papathanasiou bounds and some further remarks. Stat. Probab. Lett. 16, 39–41 (1993)MathSciNetCrossRefGoogle Scholar
Bigerelle, M., Najjar, D., Fournier, B., Rupin, N., Iost, A.: Application of lambda distributions and bootstrap analysis to the prediction of fatigue lifetime and confidence intervals. Int. J. Fatig. 28, 233–236 (2005)Google Scholar
Cox, D.R., Oakes, D.: Analysis of Survival Data. Chapman and Hall, London (1984)Google Scholar
Dudewicz, E.J., Karian, A.: The extended generalized lambda distribution (EGLD) system for fitting distribution to data with moments, II: Tables. Am. J. Math. Manag. Sci. 19, 1–73 (1996)Google Scholar
Filliben, J.J.: Simple and robust linear estimation of the location parameter of a symmetric distribution. Ph.D. thesis, Princeton University, Princeton (1969)Google Scholar
Fournier, B., Rupin, N., Bigerelle, M., Najjar, D., Iost, A.: Application of the generalized lambda distributions in a statistical process control methodology. J. Process Contr. 16, 1087–1098 (2006)CrossRefGoogle Scholar
Fournier, B., Rupin, N., Bigerelle, M., Najjar, D., Iost, A., Wilcox, R.: Estimating the parameters of a generalized lambda distribution. Comput. Stat. Data Anal. 51, 2813–2835 (2007)MathSciNetMATHCrossRefGoogle Scholar
Freimer, M., Mudholkar, G.S., Kollia, G., Lin, C.T.: A study of the generalised Tukey lambda family. Comm. Stat. Theor. Meth. 17, 3547–3567 (1988)MathSciNetMATHCrossRefGoogle Scholar
Gilchrist, W.G.: Statistical Modelling with Quantile Functions. Chapman and Hall/CRC Press, Boca Raton (2000)CrossRefGoogle Scholar
Govindarajulu, Z.: A class of distributions useful in life testing and reliability with applications to nonparametric testing. In: Tsokos, C.P., Shimi, I.N. (eds.) Theory and Applications of Reliability, vol. 1, pp. 109–130. Academic, New York (1977)Google Scholar
Karian, A., Dudewicz, E.J.: Fitting Statistical Distributions, the Generalized Lambda Distribution and Generalized Bootstrap Methods. Chapman and Hall/CRC Press, Boca Raton (2000)MATHCrossRefGoogle Scholar
Karian, A., Dudewicz, E.J.: Comparison of GLD fitting methods, superiority of percentile fits to moments in L2 norm. J. Iran. Stat. Soc. 2, 171–187 (2003)Google Scholar
Karian, A., Dudewicz, E.J.: Computational issues in fitting statistical distributions to data. Am. J. Math. Manag. Sci. 27, 319–349 (2007)MathSciNetMATHGoogle Scholar
Karian, A., Dudewicz, E.J.: Handbook of Fitting Statistical Distributions with R. CRC Press, Boca Raton (2011)MATHGoogle Scholar
Mercy, J., Kumaran, M.: Estimation of the generalized lambda distribution from censored data. Braz. J. Probab. Stat. 24, 42–56 (2010)MathSciNetCrossRefGoogle Scholar
Mudholkar, G.S., Kollia, G.D.: The isotones of the test of exponentiality. In: ASA Proceedings, Statistical Graphics, Alexandria (1990)Google Scholar
Mudholkar, G.S., Srivastava, D.K., Kollia, G.D.: A generalization of the Weibull distribution with applications to the analysis of survival data. J. Am. Stat. Assoc. 91, 1575–1583 (1996)MathSciNetMATHCrossRefGoogle Scholar
Ramberg, J.S., Dudewicz, E., Tadikamalla, P., Mykytka, E.: A probability distribution and its uses in fitting data. Technometrics 21, 210–214 (1979)CrossRefGoogle Scholar
Ramberg, J.S., Schmeiser, B.W.: An approximate method for generating symmetric random variables. Comm. Assoc. Comput. Mach. 15, 987–990 (1972)MATHGoogle Scholar
Ramberg, J.S., Schmeiser, B.W.: An approximate method for generating asymmetric random variables. Comm. Assoc. Comput. Mach. 17, 78–82 (1974)MathSciNetMATHGoogle Scholar
Ramos-Fernandez, A., Paradela, A., Narajas, R., Albar, J.P.: Generalized method for probability based peptitude and protein identification from tandem mass spectrometry data and sequence data base searching. Mol. Cell. Proteomics 7, 1745–1754 (2008)Google Scholar
Robinson, L.W., Chan, R.R.: Scheduling doctor’s appointment, optimal and empirically based heuristic policies. IIE Trans. 35, 295–307 (2003)CrossRefGoogle Scholar