Symmetric quantiles and their applications

To develop estimators with stronger efficiencies than the trimmed means which use the empirical quantile, Kim (1992) Kim, S. J. 1992. The metrically trimmed means as a robust estimator of location. Annals of Statistics, 20: 1534–1547. [Crossref], [Web of Science ®] [Google Scholar] and Chen & Chiang (1996) Chen, L. A. and Chiang, Y. C. 1996. Symmetric type quantile and trimmed means for location and linear regression model. Journal of Nonparametric Statistics, 7: 171–185. [Taylor & Francis Online] [Google Scholar], implicitly or explicitly used the symmetric quantile, and thus introduced new trimmed means for location and linear regression models, respectively. This study further investigates the properties of the symmetric quantile and extends its application in several aspects. (a) The symmetric quantile is more efficient than the empirical quantiles in asymptotic variances when quantile percentage α is either small or large. This reveals that for any proposal involving the α th quantile of small or large α s, the symmetric quantile is the right choice; (b) a trimmed mean based on it has asymptotic variance achieving a Cramer-Rao lower bound in one heavy tail distribution; (c) an improvement of the quantiles-based control chart by Grimshaw & Alt (1997) Grimshaw, S. D. and Alt, F. B. 1997. Control charts for quantile function values. Journal of Quality Technology, 29: 1–7. [Taylor & Francis Online] [Google Scholar] is discussed; (d) Monte Carlo simulations of two new scale estimators based on symmetric quantiles also support this new quantile.