How to combine M-estimators to estimate quantiles and a score function

In Kozek (2003) it has been shown that proper linear combinations of some M-estimators provide efficient and robust estimators of quantiles of near normal probability distributions. In the present paper we show that this approach can be extended in a natural way to a general case, not restricted to a vicinity of a specified probability distribution. The new class of nonparametric quantile estimators obtained this way can also be viewed as a special class of linear combinations of kernel-smoothed quantile estimators with a varying window width. The new estimators are consistent and can be made more efficient than the popular quantile estimators based on kernel smoothing with a single bandwidth choice, like those considered in Nadaraya (1964), Azzalini (1981), Falk (1984) and Falk (1985). The present approach also yields simple and efficient nonparametric estimators of a, score function J(p) = -f′(Q(p))/f(Q(p)), where f = F′ and Q(p) is the quantile function, Q(p) = F^-1(p).