A binomial model for the kernel density estimator and related inference

We explore the asymptotic normality of the kernel density estimator (KDE) and show, using both theoretical and empirical arguments, that for samples of realistic sizes, the estimator's sampling distribution exhibits a clearly noticeable skewness that becomes more obvious as the population dimension increases. As an alternative, the paper studies the sampling distribution of multivariate KDEs via a binomial model, an exact solution to improve upon the typically used asymptotic normal distribution. The exact distribution of the estimator is used to construct exact confidence bands about the density function along with several approximations that are more accurate than the classical confidence bands. We also use the binomial property of the estimator to derive a list of different tests of hypothesis about the density function which can be used to obtain rejection bands.