Sensitivity Curves for Asymmetric Trimming Hinge Estimators

Introduction: Our objective is to supplement previous simulation studies of robust estimators, hinge estimators, by examining the behavior of these adaptive location estimators using sensitivity curves first developed by the Princeton Robust Study. Methods: Skewness, tail length, and peakedness all describe distribution characteristics. These selector statistics are used to classify symmetric distributions as light-tailed (uniform, medium-tailed, or heavy-tailed). We defined a set of asymmetric 'hinge estimators” that set a total trimming proportion to be trimmed from the sample á from either the upper or lower tail depending on the data distribution characteristics and examined the effect of a single outlier using a simple sensitivity curve. Results: The sensitivity curve for a sample mean is a straight line, suggesting that the mean changes linearly with the value of the added point. The sensitivity curves for our set of adaptive estimators HQ, HQ1, HH3 suggest that the adaptive trimming causes the value of the estimator to decrease only. However, HH1, HH3, HSK2 and HSK5 are at least symmetric in their reaction to the value of the added point. The estimator HH1 has a somewhat unique property that when the value of the added point gets a bit outside the symmetric part of the sample, its influence is zero.

Conclusions: Real-world data sets may be described as messy with everything but a normal distribution presenting to the data analyst. In the asymmetric data distributions faced on a daily basis, estimators that adapt themselves to the data may be formulated and used.