IMPROVEMENTS IN THE BIAS AND PRECISION OF SAMPLE QUARTILES

The sample quartiles, which are common in robust inference and nonpara metric statistics, have many prevailing definitions, all with the same asymptotic distribution. In this note we examine the higher order ter ms in the asymptotic expansions for the bias, variance and M.S.E. of t he commonly used quartiles, defined as the linear interpolant of two a djacent order statistics. The expansions are used to develop simple im provements 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 o rder O(n(-2)). It is noted that the variances of the traditional quart iles, instead of decreasing monotonically as the sample size increases , exhibit a periodic behavior. This is analogous to a property of samp le medians observed by Hedges (1967), and discussed by Hedges and Lehm ann (1967).