Fine-Tuning Some Resistant Rules for Outlier Labeling

Abstract

Abstract A previous study examined the performance of a standard rule from Exploratory Data Analysis, which uses the sample fourths, FL and FU , and labels as “outside” any observations below FL – k(FU – FL ) or above FU + k(FU – FL ), customarily with k = 1.5. In terms of the order statistics X (1) ≤ X (2) ≤ X (n) the standard definition of the fourths is FL = X(f) and FU = X (n + 1 − f), where f = ½[(n + 3)/2] and [·] denotes the greatest-integer function. The results of that study suggest that finer interpolation for the fourths might yield smoother behavior in the face of varying sample size. In this article we show that using f i = n/4 + (5/12) to define the fourths produces the desired smoothness. Corresponding to a common definition of quartiles, fQ = n/4 + (1/4) leads to similar results. Instead of allowing the some-outside rate per sample (the probability that a sample contains one or more outside observations, analogous to the experimentwise error rate in simultaneous inference) to vary, some users may prefer to maintain it at .10 or .05 for Gaussian data and vary k accordingly. We obtain such values of k at selected sample sizes n ≤ 300.