Local Estimation of a Multivariate Density and its Derivatives.
We present methods for estimating the multivariate probability density (or the $\log$-density) and its first and second order derivatives simultaneously. Two methods, local log-likelihood and Hyv\"arinen score estimation, are in terms of weighted scoring rules with local polynomials. A third approach is matching of local moments. Consistency and asymptotic convergence results are shown and compared with corresponding results for kernel density estimators.