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.