Maximal asymptotic biases and variances of symmetrized interquantile ranges under asymmetric contamination

Abstract

In the location-scale model where random observations come from the distribution we study asymptotic robustness properties of two estimators of the unknown scale parameter ?. The two estimators studied are: (i) the interquantile range, given by the functional where [Ftilde] is a normalizing constant; and (ii) the symmetrized inter-quantile range where F is obtained by symmetrizing F about the median Maximal asymptotic biases and variances of S?(F) and S? ?(F) are found under the e-contamination model where F0 is a fixed strongly unimodal symmetric distribution and G is an unknown and possibly asymmetric distribution. Asymptotic variances are sometimes (depending on e and ?) maximized when G places all its mass in a neighborhood of ? but in some cases the asymptotic variance can be increased by shifting a small amount of mass to a neighborhood of the quantile Numerical comparisons are given for the case when F0 is the standard normal distribution