U-statistics based on the Green's function of the Laplacian on the circle and the sphere

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Abstract

We show that the Watson and Cramér–von Mises statistics are related to Green's function of the Laplacian on a circle. A generalization leads to a new U-statistic whose kernel is the Green function of the Laplacian on the sphere.

Introduction

In the field of directional statistics the problem of testing uniformity remains widely open, especially when the sample space is the unit sphere. The present paper deals with the case where the sample space is the unit circle or the unit sphere. For general surveys about tests for uniformity on the circle and the sphere the reader is referred to Fisher (1993, pp. 64–71) and Mardia and Jupp (2000, Sections 6.3 and 10.4.1). Many tests for uniformity on the circle and the sphere fit into the general framework established by Giné (1975), whose most important features are expounded in Mardia and Jupp (2000, Section 10.8). In the present paper we propose a new approach based upon the following remarks. The problem of testing uniformity on the unit circle is closely related to that of testing uniformity on [0,1]. A classical way to test the hypothesis

H0: a random sample x1,,xn (with order statistics x(1)<<x(n)) has been drawn from a population uniformly distributed on [0,1], is based on the rejection of H0 for large values taken by one of the celebrated statisticsWn2=i=1nx(i)-i-12n2+112n(Cramérvon Mises)Un2=Wn2-n(x¯-12)2(Watson)An2=-n-1ni=1n(2i-1){logx(i)+log[1-x(n-i+1)]}(AndersonDarling)where x¯=n-1i=1nxi (see Durbin, 1973, formulas (4.17)–(4.18) p. 27 and (5.4.2) p. 36). Elementary computations enable to express these Cramér–von Mises type statistics in the alternative form of the von Mises functionals (or V-statistics)Un2=1ni=1nj=1n(|xi-xj|-1/2)22-124with the kernelKU(x1,x2)(|x1-x2|-1/2)22-124=k=12sin(2kπx1)sin(2kπx2)+2cos(2kπx1)cos(2kπx2)4k2π2,Wn2=1ni=1nj=1nxi2-xi+xj2-xj-|xi-xj|2+13withKW(x1,x2)x12-x1+x22-x2-|x1-x2|2+13=k=12cos(kπx1)cos(kπx2)k2π2,An2=1ni=1nj=1n{-log[max(xi,xj)-xixj]-1}.For basic definitions and results about U- and V-statistics, see Koroljuk and Borovskich (1994, Chapter 1). The explicit Kac-Siegert or Karhunen–Loève expansions (5) and (7) of the Watson and Cramér–von Mises kernels (KU and KW respectively) are given in Durbin (1973, formula (5.6.7) p. 38) and Dym and McKean (1972, p. 60). The definition and usefulness of such an expansion of the kernel, in the study of the corresponding Cramér–von Mises, U- or V-statistic is well known. Some basic facts will be shortly recalled in Section 3. For details, the reader is referred to Shorack and Wellner (1986, Chapter 5) (for Cramér–von Mises type statistics) and Koroljuk and Borovskich (1994, Section 4.3). (for U- and V-statistics). Unfortunately, such explicit expansions can rarely be derived. However, recent advances (see Deheuvels and Martynov, 2003; Henze and Nikitin, 2002; Lachal, 2001; Pycke, 2003) show a renewal of the interest in this field, in which a central problem is to find a general method for deriving explicit expansions having a statistical interest. The aim of this paper is to show how the three classical examples mentioned above can be generalized. It leads to a new family of statistics since to our knowledge, the asymptotic distribution given by (39) in Proposition 4.2 is not that of an already known test for uniformity on the sphere.

To start with, relations (14) of Proposition 3.2 and (22) of Proposition 3.3 show that KU and KW are simply related to the zero-mean Green function of the Laplacian on the circle, say G1. In Section 2, we recall the definition of the Green's function of the Laplacian on a compact manifold M, and give in Proposition 2.1 a Karhunen–Loève expansion of this function, in the particular case where dimM3. In Propositions 3.2 and 3.3, we show that the general way to cover the circle by an interval leads to new V-statistics whose kernel, defined by (15) and (24), are as well simply related to G1. In Section 4 we generalize these ideas to the sphere. This interpretation enables us to introduce a new U-statistic, defined in (39), arising as a generalization of the Watson statistic in order to test uniformity on the sphere. Its kernel is the centered Green function of the Laplacian on the sphere, for which we obtain different explicit orthogonal decompositions in Proposition 4.1. The asymptotic distribution of a subset of principal components of the new statistic (in the line of Durbin and Knott, 1972) is given in Propositions 4.3 and 4.4. In particular, Proposition 4.3 shows that the Anderson–Darling statistic can be viewed as the radial principal component, with respect to a pole, of our new statistic. Generalizations of these ideas to a wider class of compact manifolds will be studied in a forthcoming paper.

Before stating our results, we recall some basic facts about Green's function and Karhunen–Loève expansions.

Section snippets

Green's function and Karhunen–Loève expansions

For details concerning the Green's function of the Laplacian, the reader is referred to Aubin (1982, Chapter 4), particularly Section 2.3, p. 108.

Assume M is a compact and connected, C, Riemannian manifold without boundary having volume V. The density of the Riemannian measure on M (see Chavel, 1984, Chapter 1, Section 2) is denoted by dQ. We let L2(M) be the space of real measurable functions f on M for which Mf2(Q)dQ<, equipped with the usual inner product (f|g)=Mf(Q)g(Q)dQ,f,gL2(M).For

The Watson, Cramér–von Mises statistics and the circle

Un2 was introduced by Watson (1961) for use with observations P1,,Pn recorded on the circumference of the circle. For the circle of radius R S1(R){Reiθ:θR},endowed with the measure dP=Rdθ, a point PS1(R) corresponding to the argument θ will be denoted by P(θ) and we write θ=argP. The principal value of the argument of P, denoted by ArgP, is the argument satisfying -π<θπ. The north pole corresponds to θ=0, the south pole to θ=π. We denote by d(P1,P2) the distance on the circle between the

Tests of uniformity on the sphere

Suppose now that wish to test the hypothesis

H0: a random sample P1,,Pn has been drawn from a population uniformly distributed on the unit sphere S2{(x,y,z)R3,x2+y2+z2=1}.

On S2 the generic point Q(x,y,z) has spherical coordinates (θ,φ)[0,π]×[0,2π] withx=sinθsinφ,y=sinθcosφ,z=cosθ.The Riemannian measure is dQ=sinθdθdφ.For the following basic facts, see e.g. Magnus et al. (1966, Section 4.9). The Laplacian of f:Pf(θ,φ) is given by Δf=(sinθ)-1θ(sinθθf)+(sinθ)-2φ2f.For each N, -(+1) is

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