Optimal rank-based tests for Common Principal Components

Abstract

This paper provides optimal testing procedures for the $m$-sample null hypothesis of Common Principal Components (CPC) under possibly non-Gaussian and heterogeneous elliptical densities. We first establish, under very mild assumptions that do not require finite moments of order four, the local asymptotic normality (LAN) of the model. Based on that result, we show that the pseudo-Gaussian test proposed in Hallin et al. (J. Nonparametr. Stat. 22 (2010) 879–895) is locally and asymptotically optimal under Gaussian densities, and show how to compute its local powers. A numerical evaluation of those powers, however, reveals that, while remaining valid, this test is poorly efficient away from the Gaussian. Moreover, it still requires finite moments of order four. We therefore propose rank-based procedures that remain valid under any possibly heterogeneous $m$-tuple of elliptical densities, irrespective of the existence of any moments. In elliptical families, indeed, principal components naturally can be based on the scatter matrices characterizing the density contours, hence do not require finite variances. Those rank-based tests, as usual, involve score functions, which may or may not be associated with a reference density at which they achieve optimality. A major advantage of our rank tests is that they are not only validity-robust, in the sense of surviving arbitrary elliptical population densities: unlike their pseudo-Gaussian counterparts, they also are efficiency-robust, in the sense that their local powers do not deteriorate away from the reference density at which they are optimal. We show, in particular, that in the homokurtic case, their normal-score version uniformly dominates, in the Pitman sense, the aforementioned pseudo-Gaussian generalization of Flury’s test. Theoretical results are obtained via a nonstandard application of Le Cam’s methodology in the context of curved LAN experiments. The finite-sample properties of the proposed tests are investigated via simulations.