Abstract <p>In many areas such as toxicology, chemistry and environmental sciences, data observed usually include left-censoring, that is, data below some practical limit of detection do not get observed. In this paper, we first provide a review of different non-parametric estimators for the cumulative distribution function under such a left-censoring situation. One of them is based on the chain rule and the other one is based on counting processes. We then propose a new estimator for the cumulative distribution function based on a non-parametric likelihood approach using reversed hazard rate. We then take on a classical non-parametric likelihood approach to derive the same estimator. We also propose another estimator for the cumulative distribution function using the relationship between the cumulative distribution function and the reversed hazard rate function. We then carry out an empirical comparison of all these estimators and make some comparative comments. Finally, we conclude with an application to real data.</p>