Multivariate kernel density estimators are known to systematically dev
iate from the true value near critical points of the density surface.
To overcome this difficulty a method based on Rao-Blackwell's theorem
is proposed. Local corrections of kernel density estimators are achiev
ed by conditioning these estimators with respect to locally sufficient
statistics. The asymptotic as well as the small sample size behavior
of the improved estimators are studied. Asymptotic bias and variance a
re investigated and weak and complete consistency are derived under mi
ld hypothesis. (C) 1998 Academic Press.