On a certain class of nonparametric density estimators with reduced bias

A class of kernel-based nonparametric density estimators with reduced bias is considered which is constructed from a multiplicative adjustment scheme. Estimators in the class are connected by a real parameter [alpha] and an interesting fact is that the leading term of the bias is linear in [alpha] and that of the variance is free for [alpha]. This shows that the asymptotic mean integrated squared error is quadratic in [alpha]. Consequently, we can find the best estimator in the class. Suggestions for practical choices of [alpha] are given.