This paper addresses the nonparametric estimation of a distribution function \( F \) when the data are incomplete due to grouping, censoring, and/or truncation. The author, Bruce W. Turnbull, proposes a simple iterative algorithm based on the concept of self-consistency to estimate \( F \). The algorithm is shown to converge monotonically to the maximum likelihood estimate (MLE) of \( F \). The paper also includes a proof of the equivalence between self-consistency and maximum likelihood, and discusses the properties of the algorithm, comparing it favorably with the more complex Newton-Raphson method. Additionally, the paper introduces a test for comparing two distributions when data on one or both is incomplete, and explores further applications of the empirical distribution function. The work is motivated by practical situations such as survivorship analysis, reliability analysis, and recidivism analysis, where data may be incomplete due to various forms of censoring and truncation.This paper addresses the nonparametric estimation of a distribution function \( F \) when the data are incomplete due to grouping, censoring, and/or truncation. The author, Bruce W. Turnbull, proposes a simple iterative algorithm based on the concept of self-consistency to estimate \( F \). The algorithm is shown to converge monotonically to the maximum likelihood estimate (MLE) of \( F \). The paper also includes a proof of the equivalence between self-consistency and maximum likelihood, and discusses the properties of the algorithm, comparing it favorably with the more complex Newton-Raphson method. Additionally, the paper introduces a test for comparing two distributions when data on one or both is incomplete, and explores further applications of the empirical distribution function. The work is motivated by practical situations such as survivorship analysis, reliability analysis, and recidivism analysis, where data may be incomplete due to various forms of censoring and truncation.