Estimating a Distribution Function with Truncated Data

Let P be a finite population with N ≥ 1 elements; for each e ∈ P, let Xₑ and Yₑ be independent, positive random variables with unknown distribution functions F and G; and suppose that the pairs (Xₑ, Yₑ) are i.i.d. We consider the problem of estimating $F, G$, and N when the data consist of those pairs (Xₑ, Yₑ) for which e ∈ P and Yₑ ≤ Xₑ. The nonparametric maximum likelihood estimators (MLEs) of F and G are described; and their asymptotic properties as N → ∞ are derived. It is shown that the MLEs are consistent against pairs $(F, G)$ for which F and G are continuous, G⁻¹(0) ≤ F⁻¹(0), and G⁻¹(1) ≤ F⁻¹(1). √ N × estimation error for F converges in distribution to a Gaussian process if ∫^∞₀ (1/G) dF < ∞, but may fail to converge if this integral is infinite.