  We analyze four different approaches to estimate a multivariate probability
density (or the log-density) and its first and second order derivatives. Two
methods, local log-likelihood and local Hyv\"arinen score estimation, are in
terms of weighted scoring rules with local quadratic models. The other two
approaches are matching of local moments and kernel density estimation. All
estimators depend on a general kernel, and we use the Gaussian kernel to
provide explicit examples. Asymptotic properties of the estimators are derived
and compared. In terms of rates of convergence, a refined local moment matching
estimator is the best.
