Weighted Empirical Likelihood in Some Two-Sample Semiparametric Models with Various Types of Censored Data

In this article, the weighted empirical likelihood is applied to a general setting of two-sample semiparametric models, which includes biased sampling models and case-control logistic regression models as special cases. For various types of censored data, such as right censored data, doubly censored data, interval censored data and partly interval-censored data, the weighted empirical likelihood-based semiparametric maximum likelihood estimator $(\tilde{\theta}_{n},\tilde{F}_{n})$ for the underlying parameter .. and distribution F. is derived, and the strong consistency of $(\tilde{\theta}_{n},\tilde{F}_{n})$ and the asymptotic normality of $\tilde{\theta}_{n}$ are established. Under biased sampling models, the weighted empirical log-likelihood ratio is shown to have an asymptotic scaled chi-squared distribution for censored data aforementioned. For right censored data, doubly censored data and partly interval-censored data, it is shown that $\sqrt{n}(\tilde{F}_{n}-F_{0})$ weakly converges to a centered Gaussian process, which leads to a consistent goodness-of-fit test for the case-control logistic regression models.