Estimation of a bivariate symmetric distribution function

Abstract

We consider the efficient estimation of a bivariate distribution function (DF) under the class of radially symmetric distributions and propose an estimator based on the mean of the empirical distribution and survival functions. We obtain the mean and variance of the estimator and show that it has an asymptotic normal distribution. We also show that the nonparametric maximum likelihood estimator of the bivariate DF coincides with the new estimator under radial symmetry. We study the asymptotic relative efficiency of this estimator and show that it results in a minimum of 50% reduction in sample size over the empirical DF at any point ( x, y) in R 2 . A bootstrap procedure to test whether the data support a radially symmetric model is examined. A simulation study compares the size and power of this test under bivariate normality, against alternatives in the Plackett's family of bivariate distributions, to two other procedures based on Kolmogorov–Smirnov distance.