Nonparametric Bootstrap Inference for the Eigenvalues of Geophysical Tensors

Symmetric matrices (tensors) are measured in geophysics and other disciplines, including in medical imaging, and typically their eigenvalues have valuable scientific interpretations. We design pivotal bootstrap hypothesis tests of specified eigenvalues or eigenvalue multiplicities in one-sample situations and for equal eigenvalues in k-sample situations. Our tests are more broadly applicable than existing tests by allowing very general distributions, allowing three or more samples, and accounting for common constraints in geophysical measurements. Simulations indicate that our tests generally perform well and have improved power in situations where there are existing formal hypothesis tests (eigenvalue multiplicity tests and 2-sample tests of unconstrained eigenvalues). We show fast O(n−2) convergence of test size for our pivotal k-sample bootstrap tests. We also propose confidence regions for eigenvalues and apply our tests to four geophysical datasets. An R package accompanies this article. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.