Optimal Rank-based Tests for Common Principal Components

This paper provides optimal testing procedures for the m-sample null hypothesis of Common Principal Components (CPC) under possibly non Gaussian and heterogenous 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. (2010a) is locally and asymptotically optimal under Gaussian densities. We also show how to compute its local powers and asymptotic relative efficiencies (AREs). A numerical evaluation of those AREs, ho wever, reveals that, while remain- ing valid, this test is poorly efficient away from the Gaussian. Moreover, it still requires finite moments of order four. We therefore propose rank-based proc edures that remain valid under any possibly heterogenous m-tuple of elliptical densities, irrespective of any moment assumptions--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 are not only validity-robust in the sense that they survive arbitrary elliptical populationdensities: we show that they also are efficiency-robust, in the sense that their local powers do not deteriorate under non-Gaussian alternatives. In the homogeneous case, the normal-score version of our tests uniformly dominates, in the Pitman sens e, the optimal pseudo-Gaussian 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 through simulations