Small Sample Behavior of Some Robust Linear Estimators of Location

Originally, Tukey justified the use of the trimmed mean as a robust estimator because of its high efficiency in large samples from contaminated normal data. The means, variances and covariances of the order statistics in small samples (of size ≤ 20) from a standard normal distribution contaminated by various fractions (.01, .05 and .10) of a normal distribution with mean 0 and scale parameter 3 are derived. The small sample behavior of various robust linear estimators is then discussed. At sample size 20, the variances of the usual robust linear estimators are seen to be well approximated by asymptotic theory. If one assumes that the observations come from one of the following distributions: Cauchy, double-exponential, normal, contaminated normal or logistic, then the asymptotically maximin efficient estimator, the 27 1/2 percent trimmed mean, is the maximin efficient estimator in our family at sample size 16. However, its minimum relative efficiency is less at sample size 16 than asymptotic theory suggests. With the Cauchy distribution deleted from the above family, the maximin efficient linear estimator is the 20 percent trimmed mean for all sample sizes studied.