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 F̂ₙ of F aims at reconstructing F from truncated data. In this paper an almost sure representation of F̂ₙ is derived with improved error bounds on the one hand and under weaker distributional assumptions on the other hand.