Optimal Rank-Based Tests for the Location Parameter of a Rotationally Symmetric Distribution on the Hypersphere

Transcription

1 Optimal Rank-Based Tests or the Location Parameter o a Rotationally Symmetric Distribution on the Hypersphere Davy Paindaveine and Thomas Verdebout Université libre de Bruxelles, Brussels, Belgium Laboratoire EQUIPPE, Université Lille Nord de France, France Abstract Rotationally symmetric distributions on the unit hyperpshere are among the most commonly met in directional statistics. These distributions involve a inite-dimensional parameter θ and an ininite-dimensional parameter g, that play the role o location and angular density parameters, respectively. In this paper, we ocus on hypothesis testing on θ, under unspeciied g. We consider (i) the problem o testing that θ is equal to some given θ 0, and (ii) the problem o testing that θ belongs to some given great circle. Using the uniorm local and asymptotic normality result rom Ley et al. (2013), we deine parametric tests that achieve Le Cam optimality at a target angular density. To improve on the poor robustness o these parametric procedures, we then introduce a class o rank tests or these problems. Parallel to parametric tests, the proposed rank tests achieve Le Cam optimality under correctly speciied angular densities. We derive the asymptotic properties o the various tests and investigate their inite-sample behaviour in a Monte Carlo study. 1

2 Keywords and phrases: Group invariance, rank-based tests, rotationally symmetric distributions, spherical statistics, uniorm local and asymptotic normality 1 Introduction Spherical or directional data naturally arise in a plethora o earth sciences such as geology (see, e.g., Fisher (1989)), seismology (Storetvedt and Scheidegger (1992)), astrophysics (Briggs (1993)), oceanography (Bowers and Mould (2000)) or meteorology (Fisher (1987)), as well as in studies o animal behavior (Fisher and Embleton (1987)) or even in neuroscience (Leong and Carlile (1998)). For decades, spherical data were explored through linear approximations trying to circumvent the curved nature o the data. Then the seminal paper Fisher (1953) showed that linearization hampers a correct study o several phenomena (such as, e.g., the remanent magnetism ound in igneous or sedimentary rocks) and that it was thereore crucial to take into account the non-linear, spherical, nature o the data. Since then, a huge literature has been dedicated to a more appropriate study o spherical data; we reer to Mardia (1975), Jupp and Mardia (1989), Mardia and Jupp (2000) or to the second chapter o Merriield (2006) or a detailed overview. Spherical data are commonly viewed as realizations o a random vector X taking values in the unit hypersphere S k 1 := { x R k x := x x = 1 } (k 2). In the last decades, numerous (classes o) distributions on S k 1 have been proposed and investigated. In this paper, we ocus on the class o rotationally symmetric distributions on S k 1, that were introduced in Saw (1978). This class is characterized by the act that the probability mass at x is a monotone nondecreasing unction o the spherical distance x θ between x and a given θ S k 1. This implies that the resulting equiprobability contours are the (k 2)-hyperspheres x θ = c (c [ 1, 1]), and that this north pole θ may be considered as a spherical mode, hence may be interpreted as a location parameter. O course, this assumption o rotational symmetry may seem very restrictive. Yet, the latter is oten used to model real phenomena. Indeed, according to Jupp and 2

3 Mardia (1989), rotationally symmetric spherical data appear inter alia in situations where the observation process imposes such symmetrization (e.g., the rotation o the earth; see Mardia and Edwards (1982)). Another instance where rotational symmetry is appropriate is obtained when the observation scheme does not allow to make a distinction between the measurements x and O θ x or any rotation matrix O θ such that O θ θ = θ. In such a case, indeed, only the projection o x onto the modal axis θ can be observed; see Clark (1983). In the absolutely continuous case (with the dominating measure being the uniorm distribution on S k 1 ), rotationally symmetric distributions have a probability density unction (pd) o the orm x cg(x θ), or some nondecreasing unction g : [ 1, 1] R +. Hence, this model is intrinsically o a semiparametric nature. While inerence about θ has been considered in many papers (see, among others, Chang (2004) and Tsai and Sen (2007)), semiparametrically eicient inerence procedures in the rotationally symmetric case have not been developed in the literature. The only exception is the very recent contribution by Ley et al. (2013), where rankbased estimators o θ that achieve semiparametric eiciency at a target angular density are deined. Their methodology, that builds on Hallin and Werker (2003), relies on invariance arguments and on the uniorm local and asymptotic normality with respect to θ, at a ixed g o the model considered. Ley et al. (2013), however, considers point estimation only, hence does not address situations where one would like to test the null hypothesis that the location parameter θ is equal to a given θ 0. In this paper, we thereore extend the results rom Ley et al. (2013) to hypothesis testing. This leads to a class o rank tests or the aorementioned testing problem, that, when based on correctly speciied scores, are semiparametrically optimal. The proposed tests are invariant both with respect to the group o continuous monotone increasing transormations (o spherical distances) and with respect to the group o orthogonal transormations ixing the null value θ 0. Their main advantage over studentized parametric tests is that they are not only validity-robust but are also eiciency-robust. We also treat a more involved testing problem, in which one needs to test the hypothesis that θ belongs to some given great 3

4 circle more precisely, to the intersection o S k 1 with a given vectorial subspace o R k. The outline o the paper is as ollows. In Section 2, we careully deine the class o rotationally symmetric distributions considered, introduce the main assumptions needed, and state the uniorm local and asymptotic normality result that will be the main technical tool or this work. In Section 3, we ocus on the problem o testing that θ is equal to some given θ 0, derive optimal parametric tests and study their asymptotic behaviour. In Section 4, we discuss the group invariance structure o this testing problem, propose a class o (invariant) rank tests, and study their asymptotic properties. In Section 5, we treat the problem o testing that θ belongs to a given great circle. We conduct in Section 6 a Monte Carlo study to investigate the inite-sample behaviour o the proposed tests. Finally, an Appendix collects technical proos. 2 Rotationally symmetric distributions and ULAN The random vector X, with values in the unit sphere S k 1 o R k, is said to be rotationally symmetric about θ( S k 1 ) i and only i, or all orthogonal k k matrices O satisying Oθ = θ, the random vectors OX and X are equal in distribution. I X is urther absolutely continuous (with respect to the usual surace area measure on S k 1 ), then the corresponding density is o the orm : S k 1 R k (2.1) x c k,g g(x θ), where c k,g (> 0) is a normalization constant and g : [ 1, 1] R is some nonnegative unction called an angular unction in the sequel. Throughout the paper, we then (tacitly) adopt the ollowing assumption on the data generating process. Assumption (A). The observations X 1,..., X n are mutually independent and admit a common density o the orm (2.1), or some θ S k 1 and some angular unction g in the collection F o unctions rom [ 1, 1] to R + that are positive and 4

5 monotone nondecreasing. The notation (instead o g) will be used when considering a ixed angular density. An angular unction that plays a undamental role in directional statistics is then t exp,κ (t) = exp(κt), (2.2) or some concentration parameter κ(> 0). Clearly, exp,κ satisies the conditions in Assumption (A). The resulting rotationally symmetric distribution was introduced in Fisher (1953) and is known as the Fisher-von Mises-Langevin (FvML(κ)) distribution. Other examples are the so-called linear rotationally symmetric distributions (LIN(a)), that are obtained or angular densities deined by (t) = t + a, with a > 1. In the sequel, the joint distribution o X 1,..., X n under Assumption (A) will be denoted as P. Note that, under P, the random variables X 1θ,..., X nθ are mutually independent and admit the common density (with respect to the Lebesgue measure over the real line) t g(t) := ω k c k,g B( 1 2, 1 2 (k 1)) g(t)(1 t2 ) (k 3)/2 I [ 1,1] (t), (2.3) where B(, ) is the beta unction, ω k = 2π k/2 /Γ(k/2) is the surace area o S k 1, and I A ( ) stands or the indicator unction o the set A. The corresponding cd will be denoted by t G(t) = t g(t)dt. Still under P 1, the random vectors S 1(θ),..., S n (θ), where we let S i (θ) := X i (X iθ)θ i = 1,..., n, (2.4) X i (X iθ)θ, are independent o the X iθ s, and are i.i.d., with a common distribution that is uniorm over the unit (k 2)-sphere S k 1 (θ ) := {x S k 1 : x θ = 0}. It is easy to check that the common mean vector and covariance matrix o the S i (θ) s are given by 0 and (I k θθ )/(k 1), respectively. Fix now an angular density and consider the parametric amily o probability measures P := { P θ, θ Sk 1}. For P to be uniormly locally and asymptotically normal (ULAN), the angular density needs to satisy some mild regularity conditions; more precisely, as we will state in Proposition 2.1 below, ULAN holds 5

6 i belongs to the collection F ULAN o angular densities in F (see Assumption (A) above) that (i) are absolutely continuous (with a.e. derivative, say) and such that (ii), letting ϕ := /, the quantity is inite. J k () := 1 1 ϕ 2 (t)(1 t 2 ) (t) dt = 1 0 ϕ 2 ( F 1 (u))(1 ( F 1 (u)) 2 ) du As usual, uniorm local and asymptotic normality describes the asymptotic behaviour o local likelihood ratios o the orm P θ+n 1/2 τ,, P θ, where the sequence (τ ) is bounded. In the present curved setup, (τ ) should be such that θ+n 1/2 τ S k 1 or all n, which imposes that θ τ = O(n 1/2 ). For the sake o simplicity, we will assume throughout that τ = τ + O(n 1/2 ), with θ τ = 0. We then have the ollowing result (see Ley et al. (2013) or the proo). Proposition 2.1 Fix F ULAN. ULAN, with central sequence θ, := 1 n and Fisher inormation matrix Then the amily P = { P θ, θ Sk 1} is n ϕ (X iθ) 1 (X i θ)2 S i (θ) (2.5) i=1 More precisely, (i) or any sequence (τ ) as above, log as n under P θ, Γ θ, := J k() k 1 (I k θθ ). (2.6) ( ) P θ+n 1/2 τ, = (τ ) P θ, 1 2 (τ ) Γ θ, (τ ) + o P (1) θ,, and (ii) θ, mean zero and covariance matrix Γ θ,., still under P θ,, is asymptotically normal with As we show in the next section, this ULAN result allows to deine Le Cam optimal tests or H 0 : θ = θ 0 under speciied angular density. 6

7 3 Optimal parametric tests or H 0 : θ = θ 0 For some ixed θ 0 S k 1 and F ULAN, consider the problem o testing H 0 : θ = θ 0 versus H 1 : θ θ 0 in P, that is, consider the testing problem { H0 : { P } θ 0, H 1 : θ θ 0 { P θ, }. (3.7) For this problem, we deine the test φ null o (3.7) whenever that, at asymptotic level α, rejects the Q := ( θ 0, ) Γ θ 0, θ 0, (3.8) = k 1 n ϕ (X nj k () iθ 0 )ϕ (X jθ 0 ) 1 (X i θ 0 ) 2 (3.9) i,j=1 1 (X j θ 0 ) 2 (S i (θ 0 )) S j (θ 0 ) > χ 2 k 1,1 α, where A denotes the Moore-Penrose inverse o A and χ 2 k 1,1 α stands or the α- upper quantile o a chi-square distribution with k 1 degrees o reedom. Applying in the present context the general results in Hallin et al. (2010) about hypothesis testing in curved ULAN amilies, yields that φ is Le Cam optimal more precisely, locally and asymptotically maximin at asymptotic level α or the problem (3.7). The asymptotic properties o this test are stated in the ollowing result. Theorem 3.1 Fix θ 0 S k 1 and F ULAN. Then, (i) under P θ 0,, Q is asymptotically chi-square with k 1 degrees o reedom; (ii) under P θ 0, where the +n 1/2 τ, sequence (τ ) in R k satisies τ = τ + O(n 1/2 ), with θ 0τ = 0, Q is asymptotically non-central chi-square, still with k 1 degrees o reedom, and non-centrality parameter (iii) the sequence o tests φ τ Γ θ0,τ = J k() k 1 τ 2 ; (3.10) has asymptotic size α under P θ 0,; (iv) φ is locally asymptotically maximin, at asymptotic level α, when testing {P θ 0,} against alternatives o the orm θ θ 0 {P θ, }. 7

8 For the particular case o the ixed-κ Fisher-von Mises-Langevin (FvML(κ)) distribution (obtained or exp,κ ; see (2.2)), we obtain Q exp,κ = κ2 (k 1) n (X i (X nj k ( exp,κ ) iθ 0 )θ 0 ) (X j (X jθ 0 )θ 0 ) = κ2 (k 1) nj k ( exp,κ ) =: i,j=1 n X i(i k θ 0 θ 0)X j i,j=1 κ 2 (k 1)n J k ( exp,κ ) X (I k θ 0 θ 0) X. (3.11) The main drawback o the parametric tests φ is their lack o (validity-)robustness : under angular density g, there is no guarantee that φ the nominal level constraint. Indeed Q asymptotically meets is, in general, not asymptotically χ 2 k 1 under P θ 0,g. In practice, however, the underlying angular density may hardly be assumed to be known, and it is thereore needed to deine robustiied versions o φ that will combine (a) Le Cam optimality at (Theorem 3.1(iv)) and (b) validity under a broad collection o angular densities {g}. A irst way to perorm such a robustiication is to rely on studentization. This simply consists in considering test statistics o the orm Q ;Stud := ( θ 0, ) (ˆΓ g θ 0,) θ 0,, where ˆΓ g θ 0, is an arbitrary consistent estimator o the covariance matrix in the asymptotic multinormal distribution o θ 0, under P θ 0,g. The resulting tests, that reject the null H 0 : θ = θ 0 (with unspeciied angular density) whenever Q ;Stud > χ2 k 1,1 α, are validity-robust that is, they asymptotically meet the level constraint under a broad range o angular densities and remain Le Cam optimal at. O special interest is the FvML studentized test φ exp;stud, say that rejects the null hypothesis whenever Q exp;stud = k 1 n n ˆL X i(i k θ 0 θ 0)X j, (3.12) k i,j=1 where ˆL k := 1 1 n n i=1 (X iθ 0 ) 2 is a consistent estimator o L k (g) := 1 E θ 0,g [(X iθ 0 ) 2 ] (this quantity does not depend on θ 0, which justiies the notation); this test was studied in Watson (1983). From studentization, this test is valid under any rotationally 8

9 symmetric distribution; moreover, since Q exp;stud = Q exp,κ + o P (1) as n under P θ 0, exp,κ or any κ, this test is also optimal in the Le Cam sense under any FvML distribution. Studentization, however, typically leads to tests that ail to be eiciency-robust, in the sense that the resulting type 2 risk may dramatically increase when the underlying angular density g much deviates rom the target density or target densities, in the case o the FvML studentized test φ exp;stud at which they are optimal. That is why studentization will not be considered in this paper. Instead, we will take advantage o the group invariance structure o the testing problem considered, in order to introduce invariant tests that are both validity- and eiciency-robust. As we will see, invariant tests in the present context are rank tests. 4 Optimal rank tests or H 0 : θ = θ 0 We start by describing the group invariance structure o the testing problem considered above (Section 4.1). Then we introduce (and study the properties o) rank-based versions o the central sequences rom Proposition 2.1 (Section 4.2). This will allow us to develop the resulting (optimal) rank tests and to derive their asymptotic properties (Section 4.3). 4.1 Group invariance structure Still or some given θ 0 S k 1, consider the problem o testing H 0 : θ = θ 0 against H 1 : θ θ 0 under unspeciied angular density g, that is, consider the testing problem { H0 : { } g F P θ 0,g H 1 : { } (4.13) θ θ 0 g F P. This testing problem is invariant under two groups o transormations, which we now quickly describe. (i) To deine the irst group, we introduce the tangent-normal decomposition X i = (X iθ 0 )θ 0 + X i (X iθ 0 )θ 0 S i (θ 0 ), i = 1,..., n 9

10 o the observations X i, i = 1,..., n. The irst group o transormations we consider is then G = {g h : h H},, with g h : (S k 1 ) n (S k 1 ) n (X 1,..., X n ) (h(x 1θ 0 )θ 0 + X 1 h(x 1θ 0 )θ 0 S 1 (θ 0 ),..., h(x nθ 0 )θ 0 + X n h(x nθ 0 )θ 0 S n (θ 0 )), where H is the collection o mappings h : [ 1, 1] [ 1, 1] that are continuous, monotone increasing, and satisy h(±1) = ±1. The null hypothesis o (4.13) is clearly invariant under the group G,. The invariance principle thereore suggests restricting to tests that are invariant with respect to this group. As it was shown in Ley et al. (2013), the maximal invariant I (θ 0 ) associated with G, is the sign-and-rank statistic (S 1 (θ 0 ),..., S n (θ 0 ), R 1 (θ 0 ),..., R n (θ 0 )), where R i (θ 0 ) denotes the rank o X iθ 0 among X 1θ 0,..., X nθ 0. Consequently, the class o invariant tests coincides with the collection o tests that are measurable with respect to I (θ 0 ), in short, with the class o (sign-and-)rank or, simply, rank tests. It is easy to check that G, is actually a generating group or the null hypothesis g F {P θ 0,g} in (4.13). As a direct corollary, rank tests are distribution-ree under the whole null hypothesis. This explains why rank tests will be validity-robust. (ii) O course, the null hypothesis in (4.13) is also invariant under orthogonal transormations ixing the null location value θ 0. More precisely, it is invariant under the group G rot = {g O : O O θ0 },, with where O θ0 g O : (S k 1 ) n (S k 1 ) n (X 1,..., X n ) (OX 1,..., OX n ), is the collection o all k k matrices O satisying Oθ 0 = θ 0. Clearly, the vectors o signs and ranks above is not invariant under G rot,, but the statistic ( (S1 (θ 0 ) ) S2 (θ 0 ), ( S 1 (θ 0 ) ) S3 (θ 0 ),..., ( S n 1 (θ 0 ) ) Sn (θ 0 ), R 1 (θ 0 ),..., R n (θ 0 )) (4.14) is. Tests that are measurable with respect to the statistic in (4.14) will thereore be invariant with respect to both groups considered above. 10

11 4.2 Rank-based central sequences To combine validity-robustness/invariance with Le Cam optimality at a target angular density, we introduce rank-based versions o the central sequences that appear in the ULAN property above (Proposition 2.1). More precisely, we consider rank statistics o the orm θ,k = 1 n n ( ) Ri (θ) K S i (θ), n + 1 i=1 where the score unction K : [0, 1] R is throughout assumed to be continuous (which implies that it is bounded and square-integrable over [0, 1]). In order to state the asymptotic properties o the rank-based random vector we introduce the ollowing notation. For any g F, write θ,k,g := 1 n n K ( G(X i θ) ) S i (θ), i=1 where G denotes the cd o X iθ under P. For any g F ULAN, deine urther Γ θ,k := J k(k) k 1 (I k θθ ) and Γ θ,k,g := J k(k, g) k 1 (I k θθ ), with J k (K) := 1 0 K2 (u)du and J k (K, g) := 1 K(u)K 0 g(u)du, where we wrote K g (u) := ϕ g ( G 1 (u)) 1 ( G 1 (u)) 2 or any u [0, 1]. We then have the ollowing result (see Ley et al. (2013)). Proposition 4.1 Fix θ S k 1 and let (τ ) be a sequence in R k that satisies τ = θ,k, τ + O(n 1/2 ), with θ τ = 0. Then (i) under P, with g F, θ,k = θ,k,g + o L 2(1) as n ; (ii) under P, with g F, θ,k is asymptotically multinormal with mean zero and covariance matrix Γ θ,k ; (iii) under P, with g F θ+n 1/2 τ,g ULAN, θ,k is asymptotically multinormal with mean Γ θ,k,gτ and covariance matrix Γ θ,k ; (iv) under P, with g F ULAN, = θ+n 1/2 τ,k θ,k Γ θ,k,gτ +o P (1) as n. For any FULAN C1 C1, where FULAN denotes the collection o angular densities in F ULAN that are continuously dierentiable over [ 1, 1], the unction u K (u) := ϕ ( F 1 (u)) 1 ( F 1 (u)) 2 is a valid score unction to be used in rank-based central sequences. Proposition 4.1(i) then entails that, under P θ,, 11 ϑ,k is asymptotically

12 equivalent in L 2, hence also in probability to the parametric -central sequence θ, = θ,k,. This provides the key to develop rank tests that are Le Cam optimal at any given FULAN C1. As or Proposition 4.1(ii)-(iii), they allow to derive the asymptotic properties o the resulting optimal rank tests. 4.3 Rank tests or H 0 : θ = θ 0 Fix a score unction K as above. The previous sections then make it natural to consider the rank test φk, say that, at asymptotic level α, rejects the null hypothesis H 0 : θ = θ 0 (with unspeciied angular density g) whenever QK = ( = k 1 nj k (K) > χ 2 k 1,1 α. θ 0,K ) Γ θ 0,K n K i,j=1 θ 0,K ( Ri (θ 0 ) n + 1 ) K ( ) Rj (θ 0 ) (S i (θ 0 )) S j (θ 0 ) n + 1 Clearly, this test is invariant with respect to both groups introduced in Section 4.1, since it is measurable with respect to the statistic in (4.14). The ollowing result, that summarizes the asymptotic properties o φ K, easily ollows rom Proposition 4.1. Theorem 1 Let (τ ) be a sequence in R k that satisies τ = τ + O(n 1/2 ), with K θ 0τ = 0. Then, (i) under g F {P θ 0,g}, is asymptotically chi-square with k 1 Q degrees o reedom; (ii) under P, with g F θ 0 +n 1/2 τ,g ULAN, is asymptotically noncentral chi-square, still with k 1 degrees o reedom, and non-centrality parameter τ Γ θ,k,g Γ θ 0,K Γ θ,k,gτ = J k 2 (K, g) (k 1)J k (K) τ 2 ; (4.15) (iii) the sequence o tests φk has asymptotic size α under g F {P θ 0,g}; (iv) K, φ with FULAN C1, is locally and asymptotically maximin, at asymptotic level α, when testing g F {P θ 0,g } against alternatives o the orm θ θ 0 {P θ, }. Note that, or K = K and g = (with FULAN C1 ), the non-centrality parameter in (4.15) above rewrites Jk 2(K, ) (k 1)J k (K ) τ 2 = J k() k 1 τ 2, 12

13 hence coincides with the non-centrality parameter in (3.10). optimality statement in Theorem 1(iv). This establishes the 5 Testing great circle hypotheses In this section, we turn to another classical testing problem involving rotationally symmetric distributions, namely to the problem o testing that θ belongs to some given great circle o S k 1, where the term great circle here reers to the intersection o S k 1 with a vectorial subspace o R k. In other words, letting Υ be some given ullrank k s (s < k) matrix, we consider the problem o testing H Υ 0 against H Υ 1 : θ S k 1 M(Υ) : θ / S k 1 M(Υ), where M(Υ) denotes the s-dimensional subspace o R k that is spanned by the columns o Υ. More precisely, the testing problem is { H Υ 0 : { } θ S k 1 M(Υ) g P H Υ 1 : { } (5.16) θ / S k 1 M(Υ) g P. This problem has been studied by Watson (1983), that provided a FvML score test, and in Fujikoshi and Watamori (1992) and Watamori (1992), that investigated the properties o the FvML likelihood ratio test. For any F ULAN, the construction o -optimal parametric tests or this problem proceeds as ollows. Fix θ S k 1 M(Υ) (the unspeciication o θ under the null will be taken care o later on) and consider a local perturbation o the orm θ + n 1/2 τ where τ is such that τ = τ + O(n 1/2 ), with θ τ = 0 (see the comment just beore Proposition 2.1). It directly ollows rom Hallin et al. (2010) that a locally (in the vicinity o θ) and asymptotically most stringent test can be obtained by considering the most stringent test or the linear constraint τ M(I k θθ ) M(Υ). Letting Υ θ be such that M(Υ θ ) = M(I k θθ ) M(Υ), the resulting -optimal test thereore rejects the null hypothesis H Υ 0, : { } θ S k 1 M(Υ) P θ, or large values o Q θ, = ( ( θ, ) Γ θ, Υ ) θ(υ θγ θ, Υ θ ) Υ θ θ,. Using the identity (I k θθ )Υ θ = Υ θ (which ollows rom the act that I k θθ is the projection matrix onto M(I k θθ ), that, by deinition, contains every column vector 13

14 o Υ θ ), then the act that Υ θ (Υ θυ θ ) Υ θ(i k θθ ) = Υ(Υ Υ) 1 Υ (I k θθ ), we obtain Q θ, = k 1 ( ) J k () ( θ, ) I k Υ θ (Υ θυ θ ) Υ θ θ, We will show below that, under P θ,, Q θ, = k 1 J k () ( θ, ) (I k Υ(Υ Υ) Υ ) θ,. (5.17) is asymptotically chi-square with k s degrees o reedom, so that the resulting test rejects the null, at asymptotic level α, whenever Q θ, exceeds the corresponding upper α-quantile χ2 k s,1 α. Since θ M(Υ) implies that Υ(Υ Υ) 1 Υ θ = θ (or equivalently, that (I k Υ(Υ Υ) 1 Υ )θ = 0), the FvML(κ) version o Q θ, is given by Q θ, exp,κ = nκ2 (k 1) J k ( exp,κ ) X (I k Υ(Υ Υ) 1 Υ ) X. (5.18) As in Section 3, this leads to deining the FvML studentized test φ exp;stud, say that rejects H Υ 0 whenever Q exp;stud = n(k 1) ˆL k (ˆθ) X (I k Υ(Υ Υ) 1 Υ ) X > χ 2 k s,1 α, where ˆL k (θ) = 1 1 n n i=1 (X iθ) 2 is evaluated at an arbitrary consistent estimator ˆθ o θ. When based on ˆθ = X, this test is actually the Watson (1983) score test. Consequently, the ollowing result, that states the asymptotic and optimality properties o φ exp;stud, clariies the exact optimality properties o this classical test. Theorem 5.1 Fix θ S k 1 M(Υ) and let (τ ) be a sequence in R k that satisies τ = τ + O(n 1/2 ), with θ τ = 0. Then, (i) under P, with g F, Q is exp;stud asymptotically chi-square with k s degrees o reedom; (ii) under P θ+n 1/2 τ,g, with g F ULAN, Q exp;stud is asymptotically non-central chi-square, still with k s degrees o reedom, and non-centrality parameter k 1 L k (g) τ (I k Υ(Υ Υ) 1 Υ )τ; (5.19) (iii) the sequence o tests φ = exp;stud I[Q > exp;stud χ2 k s,1 α ] has asymptotic size α under θ S M(Υ) k 1 g F {P }; (iv) φ exp;stud is locally asymptotically most stringent, 14

15 at asymptotic level α, when testing θ S k 1 M(Υ) g F {P } against alternatives o the orm θ S k 1 \M(Υ) κ>0 {P θ, exp,κ }. This test is thereore valid under any rotationally symmetric distribution, hence is validity-robust. It is optimal under any FvML distribution, but is not eiciencyrobust outside the class o FvML distributions. As or the irst testing problem we considered in the previous sections, this motivates building rank-based tests that combine validity- and eiciency-robustness. The appropriate rank test statistics are obtained by replacing in (5.17) the parametric central sequence θ, with its rank-based counterpart generally, we will consider general-score rank statistics o the orm θ, = θ,k. More Qθ,K = k 1 ( ( J k (K) θ,k) Ik Υ(Υ Υ) 1 Υ ) θ,k. As already mentioned, θ is not speciied under the null hypothesis. We will thereore rather consider the test φk, say that rejects the null at asymptotic level α whenever QK := ˆθ,K Q > χ2 k s,1 α. The estimator ˆθ to be used needs to be part o a sequence o estimators that satisies the ollowing assumption. Assumption (B). The sequence o estimators ˆθ = ˆθ is (i) root-n consistent: ˆθ θ = O P (n 1/2 ) under θ S k 1 M(Υ) g F P θ;g ; (ii) locally and asymptotically discrete: or all θ and or all C > 0, there exists a positive integer M = M(C) such that the number o possible values o ˆθ in balls o the orm {θ R k : n θ θ C} is bounded by M, uniormly as n ; (iii) constrained: M(Υ)( S k 1 ). ˆθ takes its values in The ollowing result then states the asymptotic properties o the rank tests φ Theorem 5.2 Let Assumption (B) hold, ix g F ULAN, θ S k 1 M(Υ), and let (τ ) be a sequence in R k that satisies τ = τ + O(n 1/2 ), with θ τ = 0. Then, (i) under P, Q K under P, Q θ+n 1/2 τ,g K. is asymptotically chi-square with k s degrees o reedom; (ii) K is asymptotically non-central chi-square, still with k s 15

16 degrees o reedom, and non-centrality parameter Jk 2 (K, g) (k 1)J k (K) τ (I k Υ(Υ Υ) 1 Υ )τ; (5.20) (iii) the sequence o tests φk = I[Q K > χ2 k s,1 α ] has asymptotic size α under θ S M(Υ) k 1 g F {P }; (iv) K, with FULAN φ C1, is locally asymptotically most stringent, at asymptotic level α, when testing θ S k 1 M(Υ) g FULAN {P } against alternatives o the orm θ S k 1 \M(Υ){P θ, }. Theorems allow to compute in a straightorward way (as usual, as the ratios o the non-centrality parameters in the asymptotic non-null distributions o the corresponding statistics) the asymptotic relative eiciencies (AREs) o the proposed rank tests with respect to their FvML-score competitors rom Watson (1983). These AREs are given by ARE g [ φ K /φ exp;stud ] L 2 = k (g)j k 2 (K, g) (k 1) 2 J k (K), and do not depend on θ nor on τ. It is easy to check that these AREs, that coincides with the ones obtained in Ley et al. (2013) or point estimation, also hold or the testing problem considered in the previous sections. 6 Simulations In this inal section, we conduct a Monte Carlo study to investigate the inite-sample behaviour o the rank tests proposed in Sections 4.3 and 5. Letting θ 0 = ( ) R k = R 3 and Υ = ( we considered the problems o testing H 0 : θ = θ 0 and H 0 : θ S k 1 M(Υ), respectively. We generated M = 10, 000 (mutually independent) random samples o rotationally symmetric random vectors ε (l) i, i = 1,..., n = 250, l = 1, 2, 3, 4, 16 ),

17 with location θ 0 and with angular densities that are FvML(1), FvML(3), LIN(1.1), and LIN(2), or l = 1, 2, 3, and 4, respectively; see Section 2. Each random vector ε (l) i was then transormed into X (l) i;ω := O ω ε (l) i, i = 1,..., n, l = 1, 2, 3, 4, ω = 0, 1, 2, 3, with O ω = cos(πω/50) sin(πω/50) 0 sin(πω/50) cos(πω/50) For both testing problems considered, the value ω = 0 corresponds to the null hypothesis, whereas ω = 1, 2, 3 correspond to increasingly severe alternatives. On each replication o the samples (X (l) 1;ω,..., X (l) n;ω), l = 1, 2, 3, 4, ω = 0, 1, 2, 3, we perormed the ollowing tests or H 0 : θ = θ 0 and or H 0 : θ S k 1 M(Υ), all at asymptotic level α = 5% : (i) the tests φ exp,3, that is, the parametric tests φ using a FvML(3) angular target density, (ii) the tests φ exp;stud that are based on the FvML studentized statistics Q exp;stud, and (iii) the rank-based tests K FvML(1), φ φk FvML(3), φk LIN(2), and φk LIN(2) that are Le Cam optimal at FvML(1), FvML(3), Lin(1.1), and Lin(2) distributions, respectively. The resulting rejection requencies are provided in Tables 1 and 2, or H 0 : θ = θ 0 and or H 0 : θ S k 1 M(Υ), respectively. These empirical results perectly agree with the asymptotic theory : the parametric tests φ exp,3 are the most powerul ones at the FvML(3) distribution, but their null size deviates quite much rom the target size α = 5% away rom the FvML(3). The studentized parametric tests φ exp;stud, on the contrary, show a null behaviour that is satisactory under all distributions considered. They also dominate their rank-based competitors under FvML densities, which is in line with the act that they are optimal in the class o FvML densities. Outside this class, however, the proposed rank tests are more powerul than the studentized tests, which translates their better eiciencyrobustness. The null behaviour o the proposed rank tests is very satisactory, and their optimality under correctly speciied angular densities is conirmed. 17

18 ω Underlying Test density FvML(1) FvML(3) LIN(1.1) LIN(2) φ exp, φ exp;stud φ K FvML(1) φ K FvML(3) φ K LIN(1.1) φ K LIN(2) φ exp,3 φ exp;stud φ K FvML(1) φ K FvML(3) φ K LIN(1.1) φ K LIN(2) φ exp,3 φ exp;stud φ K FvML(1) φ K FvML(3) φ K LIN(1.1) φ K LIN(2) φ exp,3 φ exp;stud φ K FvML(1) φ K FvML(3) φ K LIN(1.1) φ K LIN(2) Table 1: Rejection requencies (out o M = 10, 000 replications), under the null H 0 : θ = θ 0 (ω = 0) and increasingly severe alternatives (ω = 1, 2, 3), o the parametric FvML(3)-test (φ exp,3 ), the studentized FvML-test (φ exp;stud ), and o the rank tests achieving optimality at FvML(1), FvML(3), LIN(1.1), and LIN(2) densities (φ K FvML(1), φk FvML(3), φk LIN(1.1), and φk LIN(2), respectively). The sample size is n = 250 and the nominal level is 5%; see Section 6 or urther details. 18

19 ω Underlying Test density FvML(1) FvML(3) LIN(1.1) LIN(2) φ exp, φ exp;stud φ K FvML(1) φ K FvML(3) φ K LIN(1.1) φ K LIN(2) φ exp,3 φ exp;stud φ K FvML(1) φ K FvML(3) φ K LIN(1.1) φ K LIN(2) φ exp,3 φ exp;stud φ K FvML(1) φ K FvML(3) φ K LIN(1.1) φ K LIN(2) φ exp,3 φ exp;stud φ K FvML(1) φ K FvML(3) φ K LIN(1.1) φ K LIN(2) Table 2: Rejection requencies (out o M = 10, 000 replications), under the null H 0 : θ S k 1 M(Υ) (ω = 0) and increasingly severe alternatives (ω = 1, 2, 3), o the parametric FvML(3)-test (φ exp,3 ), the studentized FvML-test (φ exp;stud ), and o the rank tests achieving optimality at FvML(1), FvML(3), LIN(1.1), and LIN(2) densities (φ K FvML(1), φk FvML(3), φk LIN(1.1), and φk LIN(2), respectively). The sample size is n = 250 and the nominal level is 5%; see Section 6 or urther details. 19

20 Acknowledgments Davy Paindaveine s research is supported by an A.R.C. contract rom the Communauté Française de Belgique and by the IAP research network grant nr. P7/06 o the Belgian government (Belgian Science Policy). The authors are grateul to two anonymous reerees and the Editor or their careul reading and insightul comments that allowed to improve the original manuscript. A Appendix In this Appendix, we prove Theorem 5.1 and Theorem 5.2. Proo o Theorem 5.1. (i) Recalling that ˆθ is an arbitrary consistent estimator o θ, we have that, under P, ˆL k (ˆθ) ˆL k (θ) = 1 n = 1 n n i=1 n { (X i θ) 2 (X iˆθ) 2} i=1 { X i (θ ˆθ)X i(θ + ˆθ) } = (θ ˆθ) { 1 n n } X i X i (θ + ˆθ) = o P (1) as n, so that ˆL k (ˆθ) is a consistent estimator o L k (g) = 1 E [(X iθ) 2 ]. Consequently, Q exp;stud Q is o P(1) as n under P, with i=1 Q := n(k 1) L k (g) = n(k 1) L k (g) X (I k Υ(Υ Υ) 1 Υ ) X X (I k Υ(Υ Υ) 1 Υ )(I k θθ ) X = Y (I k Υ(Υ Υ) 1 Υ )Y, where we used the act that ( I k Υ(Υ Υ) 1 Υ ) (I θθ ) = I k Υ(Υ Υ) 1 Υ and where we let Y := n(k 1)/L k (g)(i k θθ ) X. Under P, n(i θθ ) X = 1 n n { 1 (X i θ) 2 S i (θ) } (A.21) i=1 is asymptotically normal with mean zero and covariance matrix L k (g)(i k θθ )/(k 1), so that Y, under the same, is asymptotically normal with mean zero and covariance 20

21 matrix I k θθ. By using again the act that ( I k Υ(Υ Υ) 1 Υ ) (I θθ ) = I k Υ(Υ Υ) 1 Υ and by noting that tr[i k Υ(Υ Υ) 1 Υ ] = k s, it is easy to check that Theorem in Rao and Mitra (1971) provides the result. (ii) Le Cam s third lemma implies that, under P, the random vector θ+n 1/2 τ,g in (A.21) is asymptotically normal with mean [ n(i lim Cov θθ ) X, ] n τ (A.22) and covariance matrix L k (g)(i k θθ )/(k 1). Using (2.3) and integrating by parts yields E [ (1 (X 1 θ) 2 )ϕ g (X 1θ) ] = 1 1 (1 t 2 )ϕ g (t)g(t) dt = k 1, so that (A.22) can be rewritten as E [ (1 (X 1 θ) 2 )ϕ g (X 1θ) ] E [ S1 (θ)(s 1 (θ)) ] τ = (I k θθ )τ = τ. Thereore, Y, under P, is asymptotically normal with θ+n 1/2 τ,g mean µ := (k 1)/L k (g) τ and covariance matrix I k θθ. From contiguity, we still have that Q exp;stud Y (I k Υ(Υ Υ) 1 Υ )Y is o P (1) under P θ+n 1/2 τ,g. Theorem in Rao and Mitra (1971) then shows that, under this sequence o probability measures, Q exp;stud is asymptotically χ2 k s with non-centrality parameter µ (I k Υ(Υ Υ) 1 Υ )µ, which establishes the result. (iii) This directly ollows rom the asymptotic null distribution given in (i) and the classical Helly-Bray theorem. (iv) Fix κ > 0. Then, it ollows rom Part (i) o the proo that, under P θ, exp,κ, with θ S k 1 M(Υ), Q exp;stud is asymptotically equivalent in probability to Q θ; exp,κ = n(k 1) L k ( exp,κ ) X (I k Υ(Υ Υ) 1 Υ ) X = nκ2 (k 1) J k ( exp,κ ) X (I k Υ(Υ Υ) 1 Υ ) X, which is the FvML(κ)-most stringent statistic we derived in (5.18). In Theorem 5.1(ii), we assumed that g F ULAN to show, through Le Cam s third lemma, that n(i θθ ) X, under P, is asymptotically normal with θ+n 1/2 τ,g mean τ and covariance matrix L k (g)(i k θθ )/(k 1). Actually, the result still holds or g F, as it can be shown that, as n under P θ+n 1/2 τ,g, n(i θθ ) X = M + (I k + θθ )τ + o P (1) = M + τ + o P (1), 21

22 where M := n(i (θ + n 1/2 τ )(θ + n 1/2 τ ) ) X, under the same, is clearly asymptotically normal with mean zero and covariance matrix L k (g)(i k θθ )/(k 1). Proo o Theorem 5.2. (i)-(ii) First note that, since θ τ = O(n 1/2 ), Proposition 4.1(iv) rewrites = J (K, g) θ+n 1/2 τ,k θ,k k 1 τ + o P (1) (A.23) as n under P. Since Assumption (B) holds, Lemma 4.4 in Kreiss (1987) allows to replace in (A.23) the deterministic quantity τ with the random one n(ˆθ θ), which yields ˆθ,K = J (K, g) θ,k n(ˆθ θ) + o P (1), k 1 as n, under P. This, jointly with Assumption (B)(iii) (which implies that (I k Υ(Υ Υ) 1 Υ )ˆθ = 0 almost surely), entails that, under P, with θ M(Υ), ( Ik Υ(Υ Υ) 1 Υ ) ˆθ,K = ( I k Υ(Υ Υ) 1 Υ ) θ,k + o P(1), as n. It ollows that QK = Q θ,k +o P(1) as n under P, with θ M(Υ), hence also under sequences o local alternatives. The results in (i)-(ii) then ollow, as in the proo o Theorem 5.1(i)-(ii), rom Theorem in Rao and Mitra (1971) and Proposition 4.1(ii)-(iii) (recall that (I k Υ(Υ Υ) 1 Υ )(I θθ ) = I k Υ(Υ Υ) 1 Υ ). (iii) As in the proo o Theorem 5.1(iii), this is a direct consequence o Part (i) o the result and the classical Helly-Bray theorem. (iv) Then, under P θ,, with θ Sk 1 M(Υ), QK = Qθ,K + o P (1) as n. Now, Proposition 4.1(i) entails that, under the same sequence o hypotheses, Qθ,K is asymptotically equivalent in probability to Q θ,k = k 1 ( ) ( θ,k Ik J k (K ) Υ(Υ Υ) 1 Υ ) θ,k, which coincides with the -most stringent statistic in (5.17). The result ollows. Reerences Bowers, J. A., M. I. D. and Mould, G. I. (2000), Directional statistics o the wind and waves, Applied Ocean Research, 22,

23 Briggs, M. S. (1993), Dipole and quadrupole tests o the isotropy o gamma-ray burst locations, Astrophys. J., 407, Chang, T. (2004), Spatial statistics, Statist. Sci., 19, Clark, R. M. (1983), Estimation o parameters in the marginal Fisher distribution, Austr. J. Statist., 25, Fisher, N. I., L. T. and Embleton, B. J. J. (1987), Statistical Analysis o Spherical Data, Cambridge University Press, UK. Fisher, N. I. (1953), Dispersion on a sphere, Proceedings o the Royal Society o London A, 217, (1987), Problems with the current deinitions o the standard deviation o wind direction, J. Clim. Appl. Meteorol., 26, (1989), Smoothing a sample o circular data, J. Struct. Geol., 11, Fujikoshi, Y. and Watamori, Y. (1992), Tests or the mean direction o the Langevin distribution with large concentration parameter, J. Multivariate Anal., 42, Hallin, M., Paindaveine, D., and Verdebout, T. (2010), Optimal rank-based testing or principal components, Ann. Statist., 38, Hallin, M. and Werker, B. J. (2003), Semi-parametric eiciency, distribution-reeness and invariance, Bernoulli, 9, Jupp, P. and Mardia, K. (1989), A uniied view o the theory o directional statistics, Internat. Statist. Rev., 57, Kreiss, J. (1987), On adaptive estimation in stationary ARMA processes, Ann. Statist., 15, Leong, P. and Carlile, S. (1998), Methods or spherical data analysis and visualization, J. Neurosci. Meth., 80, Ley, C., Swan, Y., Thiam, B., and Verdebout, T. (2013), Optimal R-estimation o 23

24 a spherical location, Statist. Sinica, 23, Mardia, K. (1975), Statistics o directional data (with discussion), J. Roy. Statist. Soc. Ser. B, 37, Mardia, K. V. and Edwards, R. (1982), Weighted distributions and rotating caps, Biometrika, 69, Mardia, K. V. and Jupp, P. (2000), Directional Statistics, Wiley, New York. Merriield, A. (2006), An investigation o mathematical models or animal group movement, using classical and statistical approaches, Ph.D. thesis, The University o Sydney, Sydney, Australia. Rao, C. and Mitra, S. (1971), Generalized Inverses o Matrices and its Apllications, New York: J. Wiley. Saw, J. G. (1978), A amily o distributions on the m-sphere and some hypothesis tests, Biometrika, 65, Storetvedt, K. M. and Scheidegger, A. E. (1992), Orthogonal joint structures in the Bergen area, southwest Norway, and their regional signiicance, Phys. Earth Planet. In., 73, Tsai, M. and Sen, P. (2007), Locally best rotation-invariant rank tests or modal location, J. Multivariate Anal., 98, Watamori, Y. (1992), Tests or a given linear structure o the mean direction o the Langevin distribution, Ann. Inst. Statist. Math., 44. Watson, G. (1983), Statistics on Spheres, Wiley, New York. 24