The rate of strong uniform consistency for the product-limit estimator

Abstract

The maximal deviation of the product-limit estimate from the estimated distribution function is investigated. As a consequence of a functional law of the iterated logarithm, the log log law is proved for this deviation on appropriate half lines, with the precise constants. This result implies that the log log law need not hold in general for the maximal deviation on the whole line. Then a general asymptotic order of magnitude is determined for the latter deviation. This order depends on the joint behaviour of the censoring and censored distributions in a well-defined way. Corresponding to specific joint behaviours, several limsup results are deduced as consequence including all the previously known log log-type laws in improved form. The results are also extended to the variable censoring model. © 1983 Springer-Verlag.