Testing Outliers in Multivariate Data

Summary

Given n random observations on a p-dimensional random vector x, the problem is to test whether a specified number (usually^~small) of suspected observations are outliers (too discordant as compared to the bulk of observations). As a generalization of Tiku’s (1975, 1977) univariate statistic, we propose a statistic g for testing a specified number of outliers in multivariate data; g is the ratio of the product of robust estimators (Tiku, 1980) to the product of ordinary estimators of the scale parameters. For the multivariate normal, g is shown to be considerably more powerful than the prominent statistic R (restricted to the multivariate normal) due to Wilks (1963) under location shifts (model A; Barnett and Lewis, 1978) although slightly less powerful under scale changes (model B; Barnett and Lewis). Like R, g is not sensitive to changes in correlations (orientation). The statistic g can be used (under models A or B) for testing outliers in samples from any multivariate distribution whose marginal distributions are of the type (l/σ)f((x-μ)/σ).