Local minimax pointwise estimation of a multivariate density

We consider the problem of the nonparametric minimax estimation of a multivariate density at a given point. A concept of smoothness classes in nonparametric minimax estimation problems is proposed. The smoothness of a function is characterized by the approximability of the function at a point by an integral of the product of this function with an approximate identity. We propose a singular integral estimator, an integral of this approximate identity with respect to the empirical distribution function. Under some assumptions on the approximate identity, the bias of the estimator is shown to be of smaller order asymptotically than the variance, and the estimator itself is shown to be asymptotically locally minimax with respect to the quadratic risk in a proper topology.