CDF and Survival Function Estimation with Infinite-Order Kernels∗

An improved nonparametric estimator of the cumulative distribution function (CDF) and the survival function is proposed using infinite-order kernels. Fourier transform theory on generalized functions is utilized to obtain the improved bias estimates. The new estimators are analyzed in terms of their relative deficiency to the empirical distribution function and Kaplan-Meier estimator, and even improvements in terms of asymptotic relative efficiency (ARE) are present under specified assumptions on the data. The deficiency analysis introduces a deficiency rate which provides a continuum between the classical deficiency analysis and an efficiency analysis. Additionally, an automatic bandwidth selection algorithm, specially tailored to the infinite-order kernels, is incorporated into the estimators. In small sample sizes these estimators can significantly improve the estimation of the CDF and survival function as is illustrated through the deficiency calculations and simulations.