An empirical saddlepoint approximation based method for smoothing survival functions under right censoring

The Kaplan–Meier (KM) estimator is ubiquitously used for estimating survival functions, but it provides only a discrete approximation at the observation times and does not deliver a proper distribution if the largest observation is censored. Using KM as a starting point, we devise an empirical saddlepoint approximation-based method for producing a smooth survival function that is unencumbered by choice of tuning parameters. The procedure inverts the moment generating function (MGF) defined through a Riemann–Stieltjes integral with respect to an underlying mixed probability measure consisting of the discrete KM mass function weights and an absolutely continuous exponential right-tail completion. Uniform consistency, and weak and strong convergence results are established for the resulting MGF and its derivatives, thus validating their usage as inputs into the saddlepoint routines. Relevant asymptotic results are also derived for the density and distribution function estimates. The performance of the resulting survival approximations is examined in simulation studies, which demonstrate a favourable comparison with the log spline method (Kooperberg & Stone, 1992) in small sample settings. For smoothing survival functions we argue that the methodology has no immediate competitors in its class, and we illustrate its application on several real data sets.