Optimal R-estimation of a spherical location

Abstract

In this paper, we provide R-estimators of the location of a rotationally symmetric distribution on the unit sphere of R-k. In order to do so we first prove the local asymptotic normality property of a sequence of rotationally symmetric models; this is a non-standard result due to the curved nature of the unit sphere. We then construct our estimators by adapting the Le Cam one-step methodology to spherical statistics and ranks. We show that they are asymptotically normal under any rotationally symmetric distribution and achieve the efficiency bound under a specific density. Their small sample behavior is studied via a Monte Carlo simulation and our methodology is illustrated on geological data

Efficient ANOVA for directional data

“…Finally the inner-sample independence and the mutual independence between the m samples entail that we can deduce the required ULAN property which is relevant for our purposes (this we state without proof because it follows directly from Proposition 2.2 of Ley et al 2013). …”

“…As a consequence, we choose to base our tests in this section on a rank-based version of the central sequence ∆ ∆ ∆ (n) ϑ ϑ ϑ0;f , namely on The following result, which is a direct corollary (using again the inner-sample independence and the mutual independence between the m samples) of Proposition 3.1 in Ley et al (2013), characterizes the asymptotic behavior of ∆ ∆ ∆ (n) ϑ ϑ ϑ0;K under any m-tuple of densities with respective angular functions g 1 , . .…”

“…The domain of the parameter being the non-linear manifold S k−1 m it is all but easy to establish the ULAN property of a sequence of rotationally symmetric models. A natural way to handle this difficulty consists, as in Ley et al (2013), in resorting to a re-parameterization of the problem in terms of spherical coordinates η η η, say, for which it is possible to prove ULAN, subject to the following technical condition on the angular functions.…”

“…Typical examples of estimators satisfying Assumption D belong to the class of M -estimators (see Chang 2004) or R-estimators (see Ley et al 2013). Put simply, instead of ∆ ∆ ∆ (n) ϑ ϑ ϑ0;φ we have to work with ∆ ∆ ∆ (n) ϑ ϑ ϑ;φ for some estimatorθ ϑ ϑ satisfying Assumption D. The next crucial result quantifies in how far this replacement affects the asymptotic properties established in Proposition 4.1 (a proof is provided in the Appendix).…”

“…The backbone of our approach is the so-called Le Cam methodology (see Le Cam 1986), as adapted to the spherical setup by Ley et al (2013). Of utmost importance for our aims here is the uniform local asymptotic normality (ULAN) of a sequence of rotationally symmetric distributions established therein and which we adapt to our present purpose in Section 3.…”