Asymptotic Properties of Self-Consistent Estimators with Doubly-Censored Data

The asymptotic properties of the self-consistent estimator (SCE) of a distribution function F of a random variable X with doubly-censored data are examined by several authors under the assumption that X is observable everywhere in the interval [a,b], where a = inf{x : F(x) > 0} and b = sup{x : F(x) < 1}. Such an assumption does not allow the situation that X is discrete and the situation that X is only observable in a nontrivial subinterval of [a,b]. However, often in practice this assumption is not satisfied. In this manuscript we establish strong uniform consistency, asymptotic normality and asymptotic efficiency of the SCE under a set of assumptions that allow the situation that X is discrete and the situation that X is only observable in a nontrivial subinterval of [a,b]. Finally, we point out a gap in the proofs of the existing results in the literature due to the definition of the SCE.