arXiv 31 Mar 2023 · Mathematics — Statistics Theory
arXiv:2303.18233 · PDF · DOI · OpenAlex · Extracted main text
We generalise the inference procedure for eigenvectors of symmetrizable matrices of Tyler (1981) to that of invariant and singular subspaces of non-diagonalizable matrices. Wald tests for invariant vectors and $t$-tests for their individual coefficients perform well in simulations, despite the matrix being not symmetric. Using these results, it is now possible to perform inference on network statistics that depend on eigenvectors of non-symmetric adjacency matrices as they arise in empirical applications from directed networks. Further, we find that statisticians only need control over the first-order Davis-Kahan bound to control convergence rates of invariant subspace estimators to higher-orders. For general invariant subspaces, the minimal eigenvalue separation dominates the first-order bound potentially slowing convergence rates considerably. In an example, we find that accounting for uncertainty in network estimates changes empirical conclusions about the ranking of nodes' popularity.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Tyler, D. E (1981) Asymptotic Inference for Eigenvectors | 0.874 | 9 | 6 | 67% |
| 2 | Davis, C. and W. M. Kahan (1969) Some new bounds on perturbation of subspaces | 0.843 | 3 | 3 | 100% |
| 3 | Yu, Y., T. Wang, and R. J. Samworth (2015, 04) (2015) A useful variant of the davis–kahan theorem for statisticians | 0.794 | 6 | 5 | 50% |
| 4 | Lunde, R. and P. Sarkar (2023, 06) (2023) Subsampling sparse graphons under minimal assumptions | 0.644 | 2 | 2 | 100% |
| 5 | Benaych-Georges, F. and A. Knowles (2018) Lectures on the local semicircle law for wigner matrices | 0.644 | 2 | 2 | 100% |
| 6 | De Paula, A., I. Rasul, and P. C. Souza (2024) Identifying network ties from panel data: Theory and an application to tax competition | 0.644 | 2 | 2 | 100% |
| 7 | Manresa, E (2013) Estimating the structure of social interactions using panel data | 0.644 | 2 | 2 | 100% |
| 8 | Rothenhäusler, D., C. Heinze, J. Peters, and N. Meinshausen (2015) Backshift: Learning causal cyclic graphs from unknown shift interventions | 0.644 | 2 | 2 | 100% |
| 9 | Magnus, J. R. and H. Neudecker (2019) Matrix differential calculus with applications in statistics and econometrics | 0.405 | 10 | 2 | 10% |
| 10 | Young, S. J. and E. R. Scheinerman (2007) Random dot product graph models for social networks | 0.405 | 1 | 1 | 100% |
Showing the top 10 of 44 scored citations.