Large Sample Asymptotic Theory of Tests for Uniformity on the Grassmann Manifold

AbstractThe Grassmann manifold Gk,m − k consists of k-dimensional linear subspaces V in Rm. To each V in Gk,m − k, corresponds a unique m × m orthogonal projection matrix P idempotent of rank k. Let Pk,m − k denote the set of all such orthogonal projection matrices. We discuss distribution theory on Pk,m − k, presenting the differential form for the invariant measure and properties of the uniform distribution, and suggest a general family F(P) of non-uniform distributions. We are mainly concerned with large sample asymptotic theory of tests for uniformity on Pk,m − k. We investigate the asymptotic distribution of the standardized sample mean matrix U taken from the family F(P) under a sequence of local alternatives for large sample size n. For tests of uniformity versus the matrix Langevin distribution which belongs to the family F(P), we consider three optimal tests-the Rayleigh-style, the likelihood ratio, and the locally best invariant tests. They are discussed in relation to the statistic U, and are shown to be approximately, near uniformity, equivalent to one another. Zonal and invariant polynomials in matrix arguments are utilized in derivations

Noncentral bimatrix variate generalised beta distributions

In this paper, we determine the density functions of nonsymmetrised doubly noncentral matrix variate beta type I and II distributions. The nonsymetrised density functions of doubly noncentral and noncentral bimatrix variate generalised beta type I and II distributions are also obtained.Comment: 14 page

Detecting the direction of a signal on high-dimensional spheres: Non-null and Le Cam optimality results

We consider one of the most important problems in directional statistics, namely the problem of testing the null hypothesis that the spike direction $\theta$ of a Fisher-von Mises-Langevin distribution on the $p$-dimensional unit hypersphere is equal to a given direction $\theta_0$. After a reduction through invariance arguments, we derive local asymptotic normality (LAN) results in a general high-dimensional framework where the dimension $p_n$ goes to infinity at an arbitrary rate with the sample size $n$, and where the concentration $\kappa_n$ behaves in a completely free way with $n$, which offers a spectrum of problems ranging from arbitrarily easy to arbitrarily challenging ones. We identify various asymptotic regimes, depending on the convergence/divergence properties of $(\kappa_n)$, that yield different contiguity rates and different limiting experiments. In each regime, we derive Le Cam optimal tests under specified $\kappa_n$ and we compute, from the Le Cam third lemma, asymptotic powers of the classical Watson test under contiguous alternatives. We further establish LAN results with respect to both spike direction and concentration, which allows us to discuss optimality also under unspecified $\kappa_n$. To investigate the non-null behavior of the Watson test outside the parametric framework above, we derive its local asymptotic powers through martingale CLTs in the broader, semiparametric, model of rotationally symmetric distributions. A Monte Carlo study shows that the finite-sample behaviors of the various tests remarkably agree with our asymptotic results.Comment: 47 pages, 4 figure

Warped Riemannian metrics for location-scale models

The present paper shows that warped Riemannian metrics, a class of Riemannian metrics which play a prominent role in Riemannian geometry, are also of fundamental importance in information geometry. Precisely, the paper features a new theorem, which states that the Rao-Fisher information metric of any location-scale model, defined on a Riemannian manifold, is a warped Riemannian metric, whenever this model is invariant under the action of some Lie group. This theorem is a valuable tool in finding the expression of the Rao-Fisher information metric of location-scale models defined on high-dimensional Riemannian manifolds. Indeed, a warped Riemannian metric is fully determined by only two functions of a single variable, irrespective of the dimension of the underlying Riemannian manifold. Starting from this theorem, several original contributions are made. The expression of the Rao-Fisher information metric of the Riemannian Gaussian model is provided, for the first time in the literature. A generalised definition of the Mahalanobis distance is introduced, which is applicable to any location-scale model defined on a Riemannian manifold. The solution of the geodesic equation is obtained, for any Rao-Fisher information metric defined in terms of warped Riemannian metrics. Finally, using a mixture of analytical and numerical computations, it is shown that the parameter space of the von Mises-Fisher model of $n$-dimensional directional data, when equipped with its Rao-Fisher information metric, becomes a Hadamard manifold, a simply-connected complete Riemannian manifold of negative sectional curvature, for $n = 2,\ldots,8$. Hopefully, in upcoming work, this will be proved for any value of $n$.Comment: first version, before submissio

Re-examining the consumption-wealth relationship : the role of model uncertainty

This paper discusses the consumption-wealth relationship. Following the recent influential workof Lettau and Ludvigson [e.g. Lettau and Ludvigson (2001), (2004)], we use data on consumption, assets andlabor income and a vector error correction framework. Key …ndings of their work are that consumption doesrespond to permanent changes in wealth in the expected manner, but that most changes in wealth are transitoryand have no e¤ect on consumption. We investigate the robustness of these results to model uncertainty andargue for the use of Bayesian model averaging. We …nd that there is model uncertainty with regards to thenumber of cointegrating vectors, the form of deterministic components, lag length and whether the cointegratingresiduals a¤ect consumption and income directly. Whether this uncertainty has important empirical implicationsdepends on the researcher's attitude towards the economic theory used by Lettau and Ludvigson. If we workwith their model, our findings are very similar to theirs. However, if we work with a broader set of models andlet the data speak, we obtain somewhat di¤erent results. In the latter case, we …nd that the exact magnitudeof the role of permanent shocks is hard to estimate precisely. Thus, although some support exists for the viewthat their role is small, we cannot rule out the possibility that they have a substantive role to play

High Dimensional Asymptotic Expansions for the Matrix Langevin Distributions on the Stiefel Manifold

Let Vk,m denote the Stiefel manifold whose elements are m - k (m >= k) matrices X such that X'X = Ik. We may be interested in high dimensional (as m --> [infinity]) asymptotic behaviors of statistics on Vk,m. High dimensional Stiefel manifolds may appear in a geometrical study in other contexts, e.g., for the analysis of compositional data with an arbitrary number m of components. We consider the matrix Langevin L(m, k; F) and L(m, k; m1/2F) distributions, each with the singular value decomposition F = [Gamma] [Delta][Theta]' of an m - k parameter matrix F, where [Gamma] [set membership, variant] Vp,m, [Theta] [set membership, variant] Vp,k, and [Delta] = diag([lambda]1, ..., [lambda]p), [lambda]j > 0. For a random matrix X having each of the two distributions, we derive asymptotic expansions, for large m, for the probability density functions of the matrix variates Y = m1/2[Gamma]'X and W = YY' and of the related functions y = tr MY' /(tr MM')1/2 and w = tr W. Here M is an arbitrary p - k constant matrix. Putting [Delta] = 0 in the asymptotic expansions yields those for the uniform distribution. The asymptotic expansions derived in this paper may be useful for statistical inference on Vk,m.