Tests for detection of outliers based on robust estimators of nuisance parameters

The problem of detection of anomalous observations (outliers) in multivariate data sets in the presence of nuisance parameters is considered and an asymptotical approach is used [1]. A counting process is constructed on the tails of the normalized empirical distribution and conditions are formulated for weak convergence to a Poisson process. Crossing some level by counting process trajectories indicates the presence of anomalous observations and crossing points determine observations subjected to gross errors. For elliptic families of multivariate distributions, robust estimators of unknown parameters with a high breakdown point (the smallest portion of outliers with inadmissible large estimator values) and a bounded influence function, defining estimator sensitivity to gross errors, in Hampel's terminology [5], [6], are considered. The estimators, which have these properties, retain high efficiency in the presence of outliers and reduce the “masking effect” when outliers are masked and look like “proper” observations, whereas the nearest “proper” observations are deleted. An example is given.