Improvements in the Bias and Precision of Sample Quartiles

Abstract

The sample quartiles, which are common in robust inference and nonparametric statistics, have many prevailing definitions, all with the same asymptotic distribution. In this note we examine the higher order terms in the asymptotic expansions for the bias, variance and M.S.E. of the commonly used quartiles, defined as the linear interpolant of two adjacent order statistics. The expansions are used to develop simple improvements of the interpolation based definition by removing the O(n ^-1) term in the bias and by minimizing the variance and M.S.E. up to order O(n ^-2). It is noted that the variances of the traditional quartiles, instead of decreasing monotonically as the sample size increases, exhibit a periodic behavior. This is analogous to a property of sample medians observed by Hodges (1967), and discussed by Hodges and Lehmann (1967).

AMS 1991 Subject Classifications::

• Primary 62G30
• 62G20
• Secondary 62F12

Keywords:

• Asymptotic expansions
• bias-variance tradeoff