Inference on eigenvectors of non-symmetric matrices

• URL: http://arxiv.org/abs/2303.18233v2
• Date: Tue, 4 Apr 2023 12:50:58 GMT
• Title: Inference on eigenvectors of non-symmetric matrices
• Authors: Jerome R. Simons
• Abstract summary: This paper argues that the symmetrisability condition in Tyler (1981) is not necessary to establish inference procedures for eigenvectors. We establish distribution theory for a Wald and t-test for full-vector and individual hypotheses, respectively. As an application, we define confidence sets for Bonacich centralities estimated from adjacency matrices induced by directed graphs.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: This paper argues that the symmetrisability condition in Tyler (1981) is not necessary to establish asymptotic inference procedures for eigenvectors. We establish distribution theory for a Wald and t-test for full-vector and individual coefficient hypotheses, respectively. Our test statistics originate from eigenprojections of non-symmetric matrices. Representing projections as a mapping from the underlying matrix to its spectral data, we find derivatives through analytic perturbation theory. These results demonstrate how the analytic perturbation theory of Sun (1991) is a useful tool in multivariate statistics and are of independent interest. As an application, we define confidence sets for Bonacich centralities estimated from adjacency matrices induced by directed graphs.

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