This paper addresses the nonparametric estimation of a distribution function F when data are incomplete due to grouping, censoring, and truncation. The problem involves estimating F based on observations that are not fully known, with each observation falling within a specific interval or set. The approach uses the concept of self-consistency to develop an algorithm that converges monotonically to provide a maximum likelihood estimate of F. This method is compared favorably to the Newton-Raphson method. The paper also proposes a test for comparing two distributions when data may be incomplete and discusses various applications of the empirical distribution function. The paper outlines the problem of estimating F when data are incomplete, considering subsets of the real line and observations that are truncated or censored. It introduces the idea of self-consistency to construct a simple algorithm that converges to the maximum likelihood estimate of F. The algorithm is shown to converge monotonically and is compared with the Newton-Raphson method. The paper also discusses the equivalence of self-consistency and maximum likelihood, and the convergence properties of the algorithm. The paper presents a reduction of the problem to estimating the parameters of a multinomial distribution with censoring and truncation. It describes the self-consistency algorithm and shows that it converges to the maximum likelihood estimate of F. The algorithm is compared with the Newton-Raphson method, and the paper discusses the properties of the algorithm and its application to hypothesis testing. The paper also discusses the large sample properties of the maximum likelihood estimate and the handling of concomitant variables. The paper concludes that the self-consistency algorithm provides a valid and efficient method for estimating F in the presence of incomplete data.This paper addresses the nonparametric estimation of a distribution function F when data are incomplete due to grouping, censoring, and truncation. The problem involves estimating F based on observations that are not fully known, with each observation falling within a specific interval or set. The approach uses the concept of self-consistency to develop an algorithm that converges monotonically to provide a maximum likelihood estimate of F. This method is compared favorably to the Newton-Raphson method. The paper also proposes a test for comparing two distributions when data may be incomplete and discusses various applications of the empirical distribution function. The paper outlines the problem of estimating F when data are incomplete, considering subsets of the real line and observations that are truncated or censored. It introduces the idea of self-consistency to construct a simple algorithm that converges to the maximum likelihood estimate of F. The algorithm is shown to converge monotonically and is compared with the Newton-Raphson method. The paper also discusses the equivalence of self-consistency and maximum likelihood, and the convergence properties of the algorithm. The paper presents a reduction of the problem to estimating the parameters of a multinomial distribution with censoring and truncation. It describes the self-consistency algorithm and shows that it converges to the maximum likelihood estimate of F. The algorithm is compared with the Newton-Raphson method, and the paper discusses the properties of the algorithm and its application to hypothesis testing. The paper also discusses the large sample properties of the maximum likelihood estimate and the handling of concomitant variables. The paper concludes that the self-consistency algorithm provides a valid and efficient method for estimating F in the presence of incomplete data.