Statistica Sinica

Yuly Koshevnik and William R. Schucany

Abstract:Nonparametric estimation of a cumulative distribution function,F, is accomplished from data containing independent observations of two types. The first type of observation is simply a recorded value of a random variableXdistributed according toF. The second type is incomplete (or censored) and contains partial information aboutX, namely only the indicator of the event [X≤d] is available. The valuedbelongs to a grid {d_{1}≤...≤d_{r}} , so the second type can be thought of as a stratified sample of dichotomous observations, each of them being represented as a pair containing a nonrandomd_{j}and realizations of the indicatorY_{j}=I[X≤d_{j}].

Asymptotically efficient estimates are derived for a cumulative distribution function (CDF) and therefore, for a wide class of functionals that can be expressed via the CDF. Their limit distribution turns out to be normal, while this asymptotic normality can be established uniformly with respect to any precompact set of CDF's. This uniformity implies asymptotic efficiency of the proposed estimates.

Key words and phrases:Combining information, contingent evaluation, estimation under constraints, geometric approach, incomplete observations.