'Smooth' inference for survival functions with arbitrarily censored data.

We propose a procedure for estimating the survival function of a time-to-event random variable under arbitrary patterns of censoring. The method is predicated on the mild assumption that the distribution of the random variable, and hence the survival function, has a density that lies in a class of 'smooth' densities whose elements can be represented by an infinite Hermite series. Truncation of the series yields a 'parametric' expression that can well approximate any plausible survival density, and hence survival function, provided the degree of truncation is suitably chosen. The representation admits a convenient expression for the likelihood for the 'parameters' in the approximation under arbitrary censoring/truncation that is straightforward to compute and maximize. A test statistic for comparing two survival functions, which is based on an integrated weighted difference of estimates of each under this representation, is proposed. Via simulation studies and application to a number of data sets, we demonstrate that the approach yields reliable inferences and can result in gains in efficiency over traditional nonparametric methods.