# Detecting the Direction of a Signal on High-dimensional Spheres: Non-null and Le Cam Optimality Results

*Davy Paindaveine* () and
*Thomas Verdebout*

Working Papers ECARES from ULB -- Universite Libre de Bruxelles

**Abstract:**
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 seven 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 obtain a full understanding of the non-null behavior of the Watson test, we use martingale CLTs to derive its local asymptotic powers 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.

**Pages:** 49 p.

**Date:** 2017-11

**New Economics Papers:** this item is included in nep-ecm

**References:** View references in EconPapers View complete reference list from CitEc

**Citations:** Track citations by RSS feed

**Published** by:

**Downloads:** (external link)

https://dipot.ulb.ac.be/dspace/bitstream/2013/2603 ... DEBOUT-detecting.pdf Full text for the whole work, or for a work part (application/pdf)

**Related works:**

This item may be available elsewhere in EconPapers: Search for items with the same title.

**Export reference:** BibTeX
RIS (EndNote, ProCite, RefMan)
HTML/Text

**Persistent link:** https://EconPapers.repec.org/RePEc:eca:wpaper:2013/260378

**Ordering information:** This working paper can be ordered from

http://hdl.handle.ne ... lb.ac.be:2013/260378

Access Statistics for this paper

More papers in Working Papers ECARES from ULB -- Universite Libre de Bruxelles Contact information at EDIRC.

Bibliographic data for series maintained by Benoit Pauwels ().