Some algebraic properties of Archimedean Copula functions and their applications in the statistical estimation of the survival function

Year 2016, Volume: 4 Issue: 3, 88 - 98, 30.09.2016

Abdushukurov Akhmedovich Muradov Rustamjon

Abstract

In this paper we study some algebraic properties of
Archimedean copulas and simple smoothed estimator of distribution function
under random right censored observations in the presence of covariate. Where
the dependence between a life time and a censoring variable may expressed by a
given Archimedean copula. We prove an almost sure asymptotic representation
which provides a key tool for obtaining weak convergence result for estimator.

Keywords

Archimedean copulas, smoothed estimator, asymptotic representation

References

• Abdushukurov A.A., Muradov R.S., Estimation of survival and mean residual life functions from dependent random censored data, New Trends in Mathematical Sciences, (2014).
• Breakers R., Veraverbeke N., A copula-graphic estimator for the conditional survival function under dependent censoring, The Canadian Journal of Statistics, (2005).
• Fisher N.I., Copulas, Encyclopedia of Statistical Sciences, John Wiley & Sons, (1997).
• Fleming T.R.,Harrington D.P., Counting Processes and Survival Analysis, Wiley. New York, (1991).
• Kaplan E.L.,Meier P., Nonparametric estimation from incompelet observations, Journal of American Statistical Association, (1958).
• Muradov R.S.,Abdushukurov A.A., Estimation of multivariate distributions and its mixtures, LAMBERT Academic Publishing. LAP, (In Russian)(2011).
• Nelsen R.B., An Introduction to Copulas, Springer, New York, (2006).
• Rivest L.P., Wells M.T., A martingall aproach to the copula-graphic estimator for the survival function under dependent censoring, Journal of Multivariate Analysis, (2001).
• Sklar A., Fonctions de repartition `a n dimensions et leurs marges, Publications de l’Institut de Statistique de l’Université de Paris, (1959).
• Schweizer B., Advances in Probability Distributions With Given Marginals, Vol. 67 of Mathematics and its applications, Thirty years of copula, Kluwer Academic Publishers, (1991).
• Van.I.Keilegom,Veraverbeke N., Estimation and the bootstrap with censored data in fixed design nonparametric regression, Ann. Inst. Statist. Math., (1997).
• Zeng M.,Klein J.P., Estimates of marginal survival for dependent competing risks based on an assumed copula, Biometrika, (1995).