On tests for outliers in linear regression through standard linear hypothesis test and a test motivated by Cook's distance

In a linear regression setting, we rigorously explore tests for a possibly multidimensional suspected outlier under a very general framework that can account for a wide range of alternative hypotheses. We consider tests that are (a) based on the standard approach for testing linear hypotheses, and (b) motivated by Cook's distance. These are seen to cover some other intuitively appealing approaches as well. We also indicate how, in special cases, the tests in (a) or (b) relate to common procedures for outlier detection. Even though the test in (a) is dictated by the standard approach for testing linear hypotheses, we find that the test in (b) can have a larger power over an appreciable part of the parameter space under the alternative hypothesis. It is also observed that Cook's distance, as it stands, does not in general entail an F-test, so that a modification thereof is called for.