Kernel estimation of multivariate cumulative distribution function

Abstract

A smooth kernel estimator is proposed for multivariate cumulative distribution functions (cdf), extending the work of Yamato [H. Yamato, Uniform convergence of an estimator of a distribution function, Bull. Math. Statist. 15 (1973), pp. 69–78.] on univariate distribution function estimation. Under assumptions of strict stationarity and geometrically strong mixing, we establish that the proposed estimator follows the same pointwise asymptotically normal distribution of the empirical cdf, while the new estimator is a smooth instead of a step function as the empirical cdf. We also show that under stronger assumptions the smooth kernel estimator has asymptotically smaller mean integrated squared error than the empirical cdf, and converges to the true cdf uniformly almost surely at a rate of (n −1/2log n). Simulated examples are provided to illustrate the theoretical properties. Using the smooth estimator, survival curves for US gross domestic product (GDP) growth are estimated conditional on the unemployment growth rate to examine how GDP growth rate depends on the unemployment policy. Another example of gold and silver price returns is given.