Asymptotic bounds for the expected L error of a multivariate kernel density estimator

The kernel estimator of a multivariate probability density function is studied. An asymptotic upper bound for the expected L¹ error of the estimator is derived. An asymptotic lower bound result and a formula for the exact asymptotic error are also given. The goodness of the smoothing parameter value derived by minimizing an explicit upper bound is examined in numerical simulations that consist of two different experiments. First, the L¹ error is estimated using numerical integration and, second, the effect of the choice of the smoothing parameter in discrimination tasks is studied.