Electronic Journal of Statistics

Log-location-scale-log-concave distributions for survival and reliability analysis

M. C. Jones and Angela Noufaily

Full-text: Open access

Abstract

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.

Article information

Source
Electron. J. Statist., Volume 9, Number 2 (2015), 2732-2750.

Dates
Received: January 2015
First available in Project Euclid: 18 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1450456320

Digital Object Identifier
doi:10.1214/15-EJS1089

Mathematical Reviews number (MathSciNet)
MR3435809

Zentralblatt MATH identifier
1329.62409

Subjects
Primary: 62N99: None of the above, but in this section
Secondary: 60E05: Distributions: general theory 62N05: Reliability and life testing [See also 90B25]

Keywords
Bathtub exponentiated Weibull generalised $F$ generalised gamma hazard shape log-concave log-convex mean residual life

Citation

Jones, M. C.; Noufaily, Angela. Log-location-scale-log-concave distributions for survival and reliability analysis. Electron. J. Statist. 9 (2015), no. 2, 2732--2750. doi:10.1214/15-EJS1089. https://projecteuclid.org/euclid.ejs/1450456320


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References

  • [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.
  • [3] Arnold, B. C. (2014). Univariate and multivariate Pareto models., J. Statist. Distributions Applic. 1 article 11.
  • [4] Bagdonaviçius, V. and Nikulin, M. (2002)., Accelerated Life Model; Modeling and Statistical Analysis. Chapman & Hall/CRC, Boca Raton.
  • [5] Bagnoli, M. and Bergstrom, T. (2005). Log-concave probability and its applications., Economet. Theor. 26 445–469.
  • [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.
  • [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.
  • [12] Gupta, R.D. and Kundu, D. (2003). Closeness of gamma and generalized exponential distribution., Commun. Statist. Theory Meth. 32 705–721.
  • [13] Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994)., Continuous Univariate Distributions, Vol. 2, 2nd ed. Wiley, New York.
  • [14] Jones, M. C. (2008). On a class of distributions with simple exponential tails., Statist. Sinica 18 1101–1110.
  • [15] Lawless, J. F. (2003)., Statistical Models and Methods for Lifetime Data, 2nd ed. Wiley, Hoboken, NJ.
  • [16] Marshall, A. W. and Olkin, I. (2007)., Life Distributions; Structure of Nonparametric, Semiparametric, and Parametric Families. Springer, New York.
  • [17] Mudholkar, G. S. and Srivastava, D. K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data., IEEE Trans. Reliab. 42 299–302.
  • [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.
  • [19] Nadarajah, S., Cordeiro, G. M. and Ortega, E. M. M. (2013). The exponentiated Weibull distribution: a survey., Statist. Pap. 54 839–877.
  • [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.
  • [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.
  • [23] Stacy, E. W. (1962). A generalization of the gamma distribution., Ann. Math. Statist. 33 1187–1192.
  • [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.
  • [25] van Zwet, W. R. (1964)., Convex Transformations of Random Variables. Mathematisch Centrum, Amsterdam.
  • [26] Walther, G. (2009). Inference and modeling with log-concave distributions., Statist. Sci. 24 319–327.