How to Combine M-estimators to Estimate Quantiles and a Score Function

摘要

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 nonparamet-ric 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).