Relative local efficiency of distance and Wald statistics away from independence

Characterize the relative local efficiency of the intrinsic off-log distance statistic and the coordinate Wald statistic away from Gaussian independence, where the Wald statistic whitens GFT coordinate differences while the distance statistic preserves the intrinsic off-log geometry.

Background

The paper compares two quadratic-form tests for equality of correlation structures: the coordinate Wald statistic, which uses the inverse estimated GFT-coordinate covariance to whiten coordinate differences, and the intrinsic off-log distance statistic, which uses the unwhitened Euclidean norm in GFT coordinates.

The statistics are proportional when the GFT-coordinate covariance is a scalar multiple of the identity, including Gaussian independence and the bivariate Gaussian case. Near Gaussian independence, their coefficients differ only at second order in the departure from independence. The unresolved issue is their comparative local efficiency away from independence, where the covariance is generally anisotropic.