Large sample nonparametric rejection of outlying observations

Abstract

Letx 1≦, ..., ≦x n be independent observations from continuous populations. The null hypothesis,H 0, is that these observations are a sample. The alternative hypothesis is that thei smallest observations are too small (or that thei largest observations are too large) to be consistent withH 0. Herei is a small number and should be specified without knowledge of the observation values. The common population hypothesized for the null case is assumed to be well-behaved but no specific assumptions are made about its shape. The alternative that thei smallest observations are too small is accepted if a statistic of the formx i−(1+A)x i+1+Ax k is negative, whereA>0,k is the largest integer contained ini+√2n, andn is sufficiently large. Similarly, the alternative that thei largest observations are too large is accepted ifx n+1−i −(1+A)x n−i+Ax n+1−k is positive. Two-sided tests are obtained as combinations of these one-sided tests. ForA suitably chosen, an approximate upper bound for the significance level of a test is evaluated from Techebycheff’s inequality. Using this relation, the value ofA is expressed as a function ofi, n, and the significance level upper bound. For fixed population shapes, these tests have powers that tend to unity for the case wheren−i of the populations are the same and the minimum difference between the median of this common population and the medians of the other populations increases in the appropriate direction The results of this paper may be useful in population statistics, operations research, and other applied fields.