On the Effect of Stragglers on the Risk of Some Mean Estimators in Small Samples

Abstract

In a previous paper [2], the risk (essentially, the variance) of certain mean estimators that in some way allow for a possible occurrence of stragglers (random variables with different mean or larger variance) have been numerically computed when the respective variances of stragglers and nonstragglers are given. This restriction is relaxed in the present paper; only the ratio of these variances is assumed to be known. A variety of cases as explained in detail in Sections 2 and 3 is being considered: The underlying distribution may be a normal (Gaussian), a logistic, or a generalized Cauchy distribution; there may be none, one, or two stragglers; the ratio of the standard deviations may be 3 or 6; finally, trimmed and Winsorized means and Bayes estimators are considered. The results are discussed in Section 5. They support the suggestion (J. W. Tukey [3]) to trim the sample by all observations that deviate substantially from the sample mean and to Winsorize those observations that deviate moderately, but trimming off exactly two observations is almost without reservations a strategy superior to always Winsorizing two. A very satisfactory behavior is exhibited by some estimators that formally are Bayes solutions for the Gaussian distribution but that also have been used with the other ones. This suggests to further pursue estimators with similar properties.