Empirical likelihood bivariate nonparametric maximum likelihood estimator with right censored data

Jian-Jian Ren () and Tonya Riddlesworth ()

Annals of the Institute of Statistical Mathematics, 2014, vol. 66, issue 5, 913-930

Abstract: This article considers the estimation for bivariate distribution function (d.f.) $$F_0(t, z)$$ F 0 ( t , z ) of survival time $$T$$ T and covariate variable $$Z$$ Z based on bivariate data where $$T$$ T is subject to right censoring. We derive the empirical likelihood-based bivariate nonparametric maximum likelihood estimator $$\hat{F}_n(t,z)$$ F ^ n ( t , z ) for $$F_0(t,z)$$ F 0 ( t , z ) , which has an explicit expression and is unique in the sense of empirical likelihood. Other nice features of $$\hat{F}_n(t,z)$$ F ^ n ( t , z ) include that it has only nonnegative probability masses, thus it is monotone in bivariate sense. We show that under $$\hat{F}_n(t,z)$$ F ^ n ( t , z ) , the conditional d.f. of $$T$$ T given $$Z$$ Z is of the same form as the Kaplan–Meier estimator for the univariate case, and that the marginal d.f. $$\hat{F}_n(\infty ,z)$$ F ^ n ( ∞ , z ) coincides with the empirical d.f. of the covariate sample. We also show that when there is no censoring, $$\hat{F}_n(t,z)$$ F ^ n ( t , z ) coincides with the bivariate empirical d.f. For discrete covariate $$Z$$ Z , the strong consistency and weak convergence of $$\hat{F}_n(t,z)$$ F ^ n ( t , z ) are established. Some simulation results are presented. Copyright The Institute of Statistical Mathematics, Tokyo 2014

Keywords: Bivariate data; Bivariate right censored data; Doubly censored data; Empirical likelihood; Maximum likelihood estimator; Right censored data (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10463-013-0433-x

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