Asymptotic properties of self-consistent estimators with mixed interval-censored data

Mixed interval-censored (MIC) data consist of n intervals with endpoints L-i and R-i, i = 1,..., n. At least one of them is a singleton set and one is a finite non-singleton interval. The survival time X-i is only known to lie between L-i and R-i, i = 1, 2,..., n. Peto (1973, Applied Statistics, 22, 86-91) and Turnbull (1976, J. Roy. Statist. Soc. Ser. B, 38, 290-295) obtained, respectively, the generalized MLE (GMLE) and the self-consistent estimator (SCE) of the distribution function of X with MIC data. In this paper, we introduce a model for MIC data and establish strong consistency, asymptotic normality and asymptotic efficiency of the SCE and GMLE with MIC data under this model with mild conditions.