Almost Sure Representations of the Product-Limit Estimator for Truncated Data

In the left-truncation model, one observes data (Xᵢ, Yᵢ) only when Yᵢ Xᵢ. Let F denote the marginal d. f. of Xᵢ, the variable of interest. The nonparametric MLE Fₙ of F aims at reconstructing F from truncated data. In this paper an almost sure representation of Fₙ is derived with improved error bounds on the one hand and under weaker distributional assumptions on the other hand.