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Adaptive recursive kernel conditional density estimators under censoring data

Let (T,X) be independent identically distributed pairs of random variables and denote f (t|x) the conditional density of T given X = x, we consider that the random variable T is subject to random censoring by another random variable C. In this paper, we propose and investigate an adaptive recursive kernel conditional density estimation under censored data, which allows us to circumvent the weak performances of Kaplan-Meier estimator (Kaplan and Meier, 1958) in the right-tail of the distribution. The first aim of this paper is to study the properties of the proposed adaptive recursive estimators and compare it with the non-recursive estimator of f (t|x). It turns out that, with an adequate selected bandwidth and a special stepsize, the proposed recursive estimators often provides better results compared to the non-recursive one in terms of estimation error and much better in terms of computational costs. We corroborated these theoretical results through some simulation study.