Robust Tests for Spherical Symmetry and Their Application to Least Squares Regression

Invariance is used to show that Kariya and Eaton's test for multivariate spherical symmetry is UMP invariant against elliptically symmetric distributions. Also both the null and alternative distributions of the test statistic are found to be the same as those which occur when the sample is normally distributed. UMP and UMPU tests for serial correlation derived assuming normality are found to be even more robust against departure from this assumption than was recently demonstrated by Kariya. When applied to the linear regression model, these results give useful robustness properties for Kadiyala's $T1$ test and the Durbin-Watson test.