Nonexistence of Rigorous Tests for Multiple Outlier Detection in Least-Squares Adjustment

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The present paper focuses on the theory of outlier detection in least-squares adjustment. Although the case of a single outlier can be efficiently handled, extensions of the testing theory to the multiple outlier case seem questionable in rigor or applicability. This contribution is a demonstration that unambiguous determination of the vector of outliers from least-squares residuals is impossible without additional hypotheses. One such hypothesis, the single outlier hypothesis, is also proven to be sufficient (with just one exception) for the residual analysis to be conclusive in the process of outlier identification.