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Log-concavity and other concepts of bivariate increasing failure rate distributions

Part of: Operations research and management science Stochastic processes Survival analysis and censored data

Published online by Cambridge University Press:  08 February 2022

Ramesh C. Gupta Open the ORCID record for Ramesh C. Gupta [Opens in a new window]  and
S. N. U. A. Kirmani
Ramesh C. Gupta*
Affiliation:
University of Maine
S. N. U. A. Kirmani*
Affiliation:
University of Northern Iowa
*
*Postal address: University of Maine, Orono, Maine, ME 04469, USA. Email address: rcgupta@maine.edu
**Postal address: University of Northern Iowa, Cedar Falls, Iowa, IA 50614, USA. Email address: kirmani@math.uni.edu
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Abstract

Log-concavity of a joint survival function is proposed as a model for bivariate increasing failure rate (BIFR) distributions. Its connections with or distinctness from other notions of BIFR are discussed. A necessary and sufficient condition for a bivariate survival function to be log-concave (BIFR-LCC) is given that elucidates the impact of dependence between lifetimes on ageing. Illustrative examples are provided to explain BIFR-LCC for both positive and negative dependence.

Keywords

Clayton’s measure of associationhazard gradientHessian matrixlocal dependence functionlog-concave densitySchur-concavity

MSC classification

Primary: 62N05: Reliability and life testing
Secondary: 60G09: Exchangeability 90B25: Reliability, availability, maintenance, inspection

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 2 , June 2022 , pp. 325 - 337
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Arias-Nicolas, J. P., Belzunce, F., Nunez-Barrera, O. and Suárez-Llorens, A. (2009). A multivariate IFR notion based on the multivariate dispersive ordering. Appl. Stoch. Models Business Industry 25, 339–358.CrossRefGoogle Scholar
Arias-Nicolas, J. P., Mulero, J., Nunez-Barrera, O. and Suárez-Llorens, A. (2010). New aging properties of the Clayton–Oakes model based on multivariate dispersion. Statist. Operat. Res. Trans. 34, 79–94.Google Scholar
Arnold, B. C. and Gupta, R. C. (2016). Alternative approaches for conditional specification of bivariate distributions. Metron 74, 21–36.CrossRefGoogle Scholar
Arriaza, A., Belzunce, F., Mulero, J. and Suárez-Llorens, A. (2016). On a new multivariate IFR ageing notion based on the standard construction. Appl. Stoch. Models Business Industry 32, 292–306.CrossRefGoogle Scholar
Barlow, R. E. and Spizzichino, F. (1993). Schur-Concave survival functions and survival analysis. J. Comput. Appl. Math. 46, 437–447.CrossRefGoogle Scholar
Bassan, B. and Spizzichino, F. (1999). Stochastic comparisons for residual lifetimes and Bayesian notions of multivariate ageing. Adv. Appl. Prob. 31, 1078–1094.CrossRefGoogle Scholar
Bassan, B. and Spizzichino, F. (2005). Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes. J. Multivariate Anal. 93, 313–339.CrossRefGoogle Scholar
Bassan. B., Kochar, S. and Spizzichino, F. (2002). Some bivariate notions of IFR and DMRL and related properties. J. Appl. Prob. 39, 533–544.CrossRefGoogle Scholar
Basu, A. P. (1971). Bivariate failure rate. J. Amer. Statist. Assoc. 66, 103–104.CrossRefGoogle Scholar
Brindley, E. C. and Thompson, W. A. (1972). Dependence and aging aspects of multivariate survival. J. Amer. Statist. Assoc. 67, 822–830.CrossRefGoogle Scholar
Clayton, D. G. (1978). A model for association in bivariate life tables and its applications in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65, 141–151.CrossRefGoogle Scholar
Durante, F. (2009). Construction of non-exchangeable bivariate distribution functions. Statist. Papers 50, 383–391.CrossRefGoogle Scholar
Finkelstein, M. (2003). On one class of bivariate distributions. Statist. Prob. Lett. 65, 1–6.CrossRefGoogle Scholar
Finkelstein, M. and Esaulova, V. (2005). On the weak IFR aging of bivariate lifetime distributions. Appl. Stoch. Models Business Industry 21, 265–272.CrossRefGoogle Scholar
Freund, J. E. (1961). A bivariate extension of the Exponential Distribution. J. Amer. Statist. Assoc. 56, 971–977.CrossRefGoogle Scholar
Gupta, R. C. (2001). Reliability studies of bivariate distributions with Pareto conditionals. J. Multivariate Anal. 76, 214–225.CrossRefGoogle Scholar
Gupta, R. C. (2003). On some association measures in bivariate distributions and their relationships. J. Statist. Planning Infer. 117, 83–98.CrossRefGoogle Scholar
Gupta, R. C. and Balakrishnan, N. (2012). Log-concavity and monotonicity of hazard and reversed hazard functions of univariate and multivariate skew normal distributions. Metrika 75, 181–191.CrossRefGoogle Scholar
Harris, R. (1970). A multivariate definition for increasing hazard rate distribution functions. Ann. Math. Statist. 41, 713–717.CrossRefGoogle Scholar
Holland, P. W. and Wang, Y. J. (1987). Dependence function for continuous bivariate densities. Commun. Statist. Theory Methods 16, 663–687.Google Scholar
Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall, London.Google Scholar
Johnson, N. L. and Kotz, S. (1975). A vector multivariate hazard rate. J. Multivariate Anal. 5, 53–66.CrossRefGoogle Scholar
Jones, M. C. (1998). Constant local dependence. J. Multivariate Anal. 64, 148–155.CrossRefGoogle Scholar
Kallberg, J. G. and Ziemba, W. T. (1981). Generalized concave functions in stochastic programming and portfolio theory. In Generalized Concavity in Optimization and Economics, eds S. Schaible and W. T. Ziemba, pp. 719–767. Academic Press, New York.Google Scholar
Klinger, A. and Mangasarian, G. L. (1968). Logarithmic convexity and geometric programming.J. Math. Anal. Appl. 24, 388–408 CrossRefGoogle Scholar
Lai, C. D. and Xie, M. (2006). Stochastic Ageing and Dependence Function for Reliability. Springer, New York.Google Scholar
Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and its Applications. Academic Press, New York.Google Scholar
Navarro, J. (2008). Characterizations using the bivariate failure rate function. Statist. Prob. Lett. 78, 1349–1354.CrossRefGoogle Scholar
Navarro, J. and Shaked, M. (2010). Some properties of the minimum and maximum of random variables with joint log-concave distributions. Metrika 71, 313–317.CrossRefGoogle Scholar
Oakes, D. (1989). Bivariate survival models induced by frailties. J. Amer. Statist. Assoc. 84, 487–493.CrossRefGoogle Scholar
Prekopa, A. (1971). Logarithmic concave measures with applications to Stochastic Programming. Acta Sci. Math. 32, 301–316.Google Scholar
Prekopa, A. (1980). Logarithmic concave measures and related topics. In Stochastic Programming, ed. M. A. H. Dempster, pp. 63–82. Academic Press, New York.Google Scholar
Roy, D. (2002). Classification of multivariate life distributions based on partial orderings. Prob. Eng. Inf. Sci. 16, 129–137.CrossRefGoogle Scholar
Savits, T. H. (1985). A multivariate IFR distribution. J. Appl. Prob. 22, 197–204.CrossRefGoogle Scholar
Shaked, M. (1977). A family of concepts of dependence for bivariate distributions. J. Amer. Statist. Assoc. 72, 642–650.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (1987). The multivariate hazard construction. Stoch. Process. Appl. 24, 241–258.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.CrossRefGoogle Scholar
Spizzichino, F. (2001). Subjective Probability Models for Lifetimes. Chapman & Hall/CRC, Boca Raton, FL.CrossRefGoogle Scholar
Wang, Y. (1993). Construction of continuous bivariate density functions. Statist. Sinica 3, 173–187.Google Scholar