Applications of the asymmetric eigenvalue problem techniques to robust testing

To study the possibility of constructing a practical alternative to the computation of the p- values of the τ-test statistic, we study the properties of its asymptotic distribution. Using results from perturbation theory, we study the geometry of the matrix that governs the asymptotic distribution of the τ-test statistic. We consider the eigenvalues of this matrix as functions of the leverages, h_i, i = 1,2,...,n, and we obtain power series expansions of the eigenvalues in terms of the factors h_i-p/n, i = 1,2,..., n. From these expansions we see that it is precisely the high leverage cases that cause the eigenvalues to separate. Using Gerschgorin type theorems we try to group the eigenvalues into isolated discs. If the eigenvalues cannot be separated to different isolated discs, but all belong to overlapping discs, then they can be grouped in just one disc and be replaced by their average eigenvalue. In this case the p-value associated with the τ-test statistic can be approximated from the existing chisquare tables.