Log-location-scale-log-concave distributions for survival and reliability analysis. (English) Zbl 1329.62409

Summary: We consider a novel sub-class of log-location-scale models for survival and reliability data formed by restricting the density of the underlying location-scale distribution to be log-concave. These models display a number of attractive properties. We particularly explore the shapes of the hazard functions of these, LLSLC, models. A relatively elegant, if partial, theory of hazard shape arises under a further minor constraint on the hazard function of the underlying log-concave distribution. Perhaps the most useful LLSLC models are contained in a class of three-parameter distributions which allow constant, increasing, decreasing, bathtub and upside-down bathtub shapes for their hazard functions.


62N99 Survival analysis and censored data
60E05 Probability distributions: general theory
62N05 Reliability and life testing
Full Text: DOI Euclid


[1] An, M. Y. (1995). Log-concave probability distributions: theory and statistical testing. Technical Report, Department of Economics, Duke, University.
[2] An, M. Y. (1998). Logconcavity versus logconvexity: a complete characterization., J. Econ. Theor. 80 350-369. · Zbl 0911.90071
[3] Arnold, B. C. (2014). Univariate and multivariate Pareto models., J. Statist. Distributions Applic. 1 article 11. · Zbl 1329.62061
[4] Bagdonaviçius, V. and Nikulin, M. (2002)., Accelerated Life Model; Modeling and Statistical Analysis . Chapman & Hall/CRC, Boca Raton. · Zbl 1001.62035
[5] Bagnoli, M. and Bergstrom, T. (2005). Log-concave probability and its applications., Economet. Theor. 26 445-469. · Zbl 1077.60012
[6] Cooray, K. and Ananda, M. M. A. (2008). A generalization of the half-normal distribution with applications to lifetime data., Commun. Statist. Theor. Meth. 37 1323-1337. · Zbl 1163.62006
[7] Cox, C. (2008). The generalized \(F\) distribution: an umbrella for parametric survival analysis., Statist. Med. 27 4301-4312.
[8] Cox, C., Chu, H., Schneider, M. F. and Muñoz, A. (2007). Parametric survival analysis and taxonomy of hazard functions for the generalized gamma distribution., Statist. Med. 26 4352-4374.
[9] Cox, C. and Matheson, M. (2014). A comparison of the generalized gamma and exponentiated Weibull distributions., Statist. Med. 33 3772-3780.
[10] Dimitrakopoulou, T., Adamidis, K. and Loukas, S. (2007). A lifetime distribution with an upside-down bathtub-shaped hazard function., IEEE Trans. Reliab. 56 308-311.
[11] Glaser, R.E. (1980). Bathtub and related failure rate characterizations., J. Amer. Statist. Assoc. 75 667-672. · Zbl 0497.62017
[12] Gupta, R.D. and Kundu, D. (2003). Closeness of gamma and generalized exponential distribution., Commun. Statist. Theory Meth. 32 705-721. · Zbl 1048.62013
[13] Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994)., Continuous Univariate Distributions , Vol. 2, 2nd ed. Wiley, New York. · Zbl 0821.62001
[14] Jones, M. C. (2008). On a class of distributions with simple exponential tails., Statist. Sinica 18 1101-1110. · Zbl 1149.62007
[15] Lawless, J. F. (2003)., Statistical Models and Methods for Lifetime Data , 2nd ed. Wiley, Hoboken, NJ. · Zbl 1015.62093
[16] Marshall, A. W. and Olkin, I. (2007)., Life Distributions; Structure of Nonparametric, Semiparametric, and Parametric Families . Springer, New York. · Zbl 1304.62019
[17] Mudholkar, G. S. and Srivastava, D. K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data., IEEE Trans. Reliab. 42 299-302. · Zbl 0800.62609
[18] Mudholkar, G. S., Srivastava, D. K. and Freimer, M. (1995). The exponentiated Weibull family: a reanalysis of the bus-motor-failure data., Technometrics 37 436-445. · Zbl 0900.62531
[19] Nadarajah, S., Cordeiro, G. M. and Ortega, E. M. M. (2013). The exponentiated Weibull distribution: a survey., Statist. Pap. 54 839-877. · Zbl 1307.62033
[20] Nikulin, M. and Haghighi, F. (2006). A chi-squared test for the generalized power Weibull family for the head-and-neck cancer censored data., J. Math. Sci. 133 1333-1341. · Zbl 1085.62125
[21] Nikulin, M. and Haghighi, F. (2009). On the power generalized Weibull family: model for cancer censored data., Metron 67 75-86.
[22] Saumard, A. and Wellner, J. A. (2014). Log-concavity and strong log-concavity: a review., Statist. Surveys 8 45-114. · Zbl 1360.62055
[23] Stacy, E. W. (1962). A generalization of the gamma distribution., Ann. Math. Statist. 33 1187-1192. · Zbl 0121.36802
[24] Vallejos, C. A. and Steel, M. F. J. (2015). Objective Bayesian survival analysis using shape mixtures of log-normal distributions., J. Amer. Statist. Assoc. 110 697-710. · Zbl 1373.62506
[25] van Zwet, W. R. (1964)., Convex Transformations of Random Variables. Mathematisch Centrum, Amsterdam. · Zbl 0142.15002
[26] Walther, G. (2009). Inference and modeling with log-concave distributions., Statist. Sci. 24 319-327. · Zbl 1329.62192
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.