Empirical Likelihood for Right Censored Lifetime Data

Abstract

When the empirical likelihood (EL) of a parameter θ is constructed with right censored data, literature shows that − 2log (empirical likelihood ratio) typically has an asymptotic scaled chi-squared distribution, where the scale parameter is a function of some unknown asymptotic variances. Therefore, the EL construction of confidence intervals for θ requires an additional estimation of the scale parameter. Additional estimation would reduce the coverage accuracy for θ. By using a special influence function as an estimating function, we prove that under very general conditions, − 2log (empirical likelihood ratio) has an asymptotic standard chi-squared distribution with one degree of freedom. This eliminates the need for estimating the scale parameter as well as eases some of the often demanding computations of the EL method. Our estimating function yields a smaller asymptotic variance than those of Wang and Jing (2001 Wang, Q. H., and Jing, B. Y. (2001), “Empirical Likelihood for a Class of Functions of Survival Distribution With Censored Data,” Annals of the Institute of Statistical Mathematics, 53, 517–527.[Crossref], [Web of Science ®] , [Google Scholar]) and Qin and Zhao (2007 Qin, G. S., and Zhao, Y. C. (2007), “Empirical Likelihood Inference for the Mean Residual Life Under Random Censorship,” Statistics and Probability Letters, 77, 549–557.[Crossref], [Web of Science ®] , [Google Scholar]). Thus, it is not surprising that confidence intervals using the special influence functions give a better coverage accuracy as demonstrated by simulations.