Estimates for the score function under random censoring

Assuming that X1, ..., Xn is a random sample of lifetimes from a distribution with density [latin small letter f with hook] subject to random right-censoring, an estimator of the score function. [latin small letter f with hook]'(x)/[latin small letter f with hook](x), is then proposed and proved to be strong consistent. This estimator is based on random spacings of order statistics. It extends to the censored data case the idea of the histogram density estimator in Van Ryzin (1973) as well as the estimation of the derivative of a density in Kim and Van Ryzin (1980).