A simple hermitian estimator of the quantile mjnction

Abstract

The most commonly used quantile estimator, based upon the linear interpolation between two order statistics, lacks smoothness, and because of its nondifferentiability is not always usable to estimate the density-quantile function. In this note a monotone cubic version of the Hemite interpolation formula is used to construct and study a smooth quadratic estimator of the entire quantile function

presented at (2.8), and the associated estimator for the density quantile function. These are simple enough to replace even where the latter does-not exist. The asymptotic expansions up to O(n⁻³) for the expectation and variance for are provided. As special cases the Hermitian medians and Hermitian quartiles are examined. It is shown that an apparent anomaly of the variance of considered by Hodges [I] and Hodgcs and Lehmann [2] is alleviated for the Hermitian median .

Keywords:

• Hennite interpolation
• Median
• Quartiles; Mad
• IQR and Q-Q curves