Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-30T19:00:27.520Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized Measures of Divergence in Survival Analysis and Reliability

Part of: Survival analysis and censored data Operations research and management science Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Filia Vonta  and
Alex Karagrigoriou
Filia Vonta*
Affiliation:
National Technical University of Athens
Alex Karagrigoriou*
Affiliation:
University of Cyprus
*
∗Postal address: Department of Mathematics, National Technical University of Athens, 9 Iroon Polytechneiou, Zografou Campus, 15780 Athens, Greece. Email address: [email protected]
∗∗Postal address: Department of Mathematics and Statistics, University of Cyprus, University Campus, CY-1678 Nicosia, Cyprus.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Measures of divergence or discrepancy are used either to measure mutual information concerning two variables or to construct model selection criteria. In this paper we focus on divergence measures that are based on a class of measures known as Csiszár's divergence measures. In particular, we propose a measure of divergence between residual lives of two items that have both survived up to some time t as well as a measure of divergence between past lives, both based on Csiszár's class of measures. Furthermore, we derive properties of these measures and provide examples based on the Cox model and frailty or transformation model.

Keywords

Divergence measureCsiszár's family of measuresCox modelproportional reverse hazards modelfrailty model

MSC classification

Secondary: 62N05: Reliability and life testing 60E15: Inequalities; stochastic orderings 90B25: Reliability, availability, maintenance, inspection
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 1 , March 2010 , pp. 216 - 234
Copyright
Copyright © Applied Probability Trust 2010 

References

Ali, S. M. and Silvey, S. D. (1966). A general class of coefficients of divergence of one distribution from another. J. R. Statist. Soc. B 28, 131–142.Google Scholar
Balakrishnan, V. and Sanghvi, L. D. (1968). Distance between populations on the basis of attribute. Biometrics 24, 859–865.Google Scholar
Basu, A., Harris, I. R., Hjort, N. L. and Jones, M. C. (1998). Robust and efficient estimation by minimising a density power divergence. Biometrika 85, 549–559.Google Scholar
Cox, D. R. (1972). Regression models and life tables. J. R. Statist. Soc. B 34, 187–202.Google Scholar
Cressie, N. and Read, T. R. C. (1984). Multinomial goodness-of-fit tests. J. R. Statist. Soc. B 46, 440–464.Google Scholar
Csiszár, I. (1963). Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizitat on Markhoffschen Ketten. Magyar Tud. Akad. Mat. Kutató Int. Közl. 8, 84–108.Google Scholar
Di Crescenzo, A. (2000). Some results on the proportional reversed hazards model. Statist. Prob. Lett. 50, 313–321.Google Scholar
Di Crescenzo, A. and Longobardi, M. (2004). A measure of discrimination between past lifetime distributions. Statist. Prob. Lett., 67, 173–182.Google Scholar
Ebrahimi, N. and Kirmani, S. N. U. A. (1996a). A characterization of the proportional hazards model through a measure of discrimination between two residual life distributions. Biometrika 83, 233–235.CrossRefGoogle Scholar
Ebrahimi, N. and Kirmani, S. N. U. A. (1996b). A measure of discrimination between two residual life-time distributions and its applications. Ann. Inst. Statist. Math. 48, 257–265.Google Scholar
Gupta, R. C., Gupta, P. L. and Gupta, R. D. (1998). Modeling failure time data by Lehmann alternatives. Commun. Statist. Theory Meth. 27, 887–904.Google Scholar
Kagan, A. M. (1963). On the theory of Fisher's amount of information. Soviet Math. Dokl. 4, 991–993.Google Scholar
Kirmani, S. N. U. A. and Gupta, R. C. (2001). On the proportional odds model in survival analysis. Ann. Inst. Statist. Math. 53, 203–216.CrossRefGoogle Scholar
Kolmogorov, A. N. (1933). Sulla determinazione empirica di una legge di distribuzione. Giornale dell'Instituto Italiano degli Attuari 4, 83–91.Google Scholar
Kullback, S. and Leibler, R. A. (1951). On information and sufficiency. Ann. Math. Statist. 22, 79–86.Google Scholar
Lévy, P. (1925). Calcul des Probabilites. Gauthiers-Villars, Paris.Google Scholar
Mahalanobis, P. C. (1936). On the generalized distance in statistics. Proc. Nat. Acad. Sci. India 2, 49–55.Google Scholar
Mattheou, K., Lee, S. and Karagrigoriou, A. (2009). A model selection criterion based on the BHHJ measure of divergence. J. Statist. Planning Infer. 139, 228–235.CrossRefGoogle Scholar
Matusita, K. (1967). On the notion of affinity of several distributions and some of its applications. Ann. Inst. Statist. Math. 19, 181–192.Google Scholar
Pardo, L. (2006). Statistical Inference Based on Divergence Measures. Chapman and Hall/CRC, Boca Raton, FL.Google Scholar
Pearson, K. (1900). On the criterion that a given system of deviation from the probable in the case of a correlated system of variables in such that it can be reasonably supposed to have arisen from random sampling. Phil. Mag. 50, 157–172.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, Boston, MA.Google Scholar
Vonta, F. (1996). Efficient estimation in a non-proportional hazards model in survival analysis. Scand. J. Statist. 23, 49–61.Google Scholar