**Sankhya:
The Indian Journal of Statistics**

2005, Volume 67, Pt. 2, 277--294

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

By

Andrzej S. Kozek, Macquarie University, Sydney, Australia

SUMMARY. 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 $J(p)=-\frac{f^{\prime }\left( Q(p)\right) }{f\left(Q(p)\right) },$ where $f=F'$ and $Q(p)$ is the quantile function, $Q(p)=F^{-1}(p)$.

*AMS (1991) subject classification. *Primary 62G99; Secondary 62G35,
62G30.

*Key words and phrases. *Asymptotic properties, kernel estimators,
M-es\-ti\-mators, quantiles, score function, smoothing.