Simple optimal tests for circular reflective symmetry about a ... - ULB

Simple optimal tests for circular reflective symmetry about a specified median direction Christophe Ley∗ and Thomas Verdebout† Université Libre de Bruxelles and Université Lille Nord de France

Abstract In this paper, we propose optimal tests for circular reflective symmetry about a fixed median direction. The distributions against which optimality is achieved are the k sine-skewed distributions of Umbach and Jammalamadaka (2009). We first show that sequences of k -sine-skewed models are locally and asymptotically normal in the vicinity of reflective symmetry. Following the Le Cam methodology, we then construct optimal (in the maximin sense) parametric tests for reflective symmetry, which we render semi-parametric by a studentization argument. These asymptotically distribution-free tests happen to be uniformly optimal (under any reference density) and are moreover of a very simple and intuitive form. They furthermore exhibit nice small sample properties, as we show through a Monte Carlo simulation study. Our new tests also allow us to re-visit the famous red wood ants data set of Jander (1957). The choice of k -sine-skewed alternatives, which are the circular analogues of the Azzalini-type linear skew-symmetric distributions, permits us a Fisher singularity analysis ` a la Hallin and Ley (2012) with the result that only the prominent sine-skewed von Mises distribution suffers from these inferential drawbacks. Finally, we conclude the paper by discussing the unspecified location case.

Keywords: Circular statistics, Fisher information singularity, skewed distributions, tests for symmetry

1

Introduction

Symmetry is a fundamental and ubiquitous structural assumption in statistics, underpinning most classical inferential methods, be it for univariate data on the real line or for circular data. Its acceptance generally simplifies the statistician’s task, both in the elaboration of new theoretical tools and in the analysis of a given set of observations. For instance, the classical models for circular data, such as, e.g., the von Mises, cardioid, wrapped normal or wrapped Cauchy distributions (see Mardia and Jupp 2000, Section 3.5) are all symmetric about their unique mode. This form of symmetry on the circle is called reflective symmetry. However, quoting Mardia (1972, p. 10), “symmetrical distributions on the circle are comparatively rare”, ∗ †

E-mail address: [email protected]; URL: http://homepages.ulb.ac.be/˜chrisley E-mail address: [email protected]

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and recent years have seen an increasing interest in non-symmetric models (see, e.g., Umbach and Jammalamadaka 2009, Kato and Jones 2010, Abe and Pewsey 2011 or Jones and Pewsey 2012); therefore, it is all the more important to be able to test whether the hypothesis of symmetry holds or not, in order to know whether the classical or rather the modern models should be used. Since circular distributions are encountered in several domains of scientific investigation, with particular emphasis on the analysis of (i) phases of periodic phenomena (physics, biology, etc.) and (ii) directions (animal movements as a response to some stimulus, pigeon homing, earth sciences, etc.), practical examples needing tests for circular symmetry are all but rare. In this paper, we are interested in those settings where the experimental setup suggests a specific direction about which to test symmetry, e.g. in animal orientation problems. While testing for symmetry about a fixed center (the median) is a classical issue on the real line and has generated an important number of publications, the situation is very different in the circular case. Indeed, the null hypothesis of circular reflective symmetry is way less explored in the literature. There exist essentially three proposals ∗ 1 for such tests: • Schach (1969) constructs locally optimal linear rank tests against rotation alternatives, the circular analogue of a linear shift alternative. His construction comprises the circular sign and Wilcoxon tests. • Universally consistent tests from the linear setting have been adapted to the circular case (such as the runs tests, see Pewsey 2004). • A “true” test for circular symmetry has been studied in Pewsey (2004) by having recourse to the second sine moment about the fixed median direction, a classical measure of circular skewness first proposed by Batschelet (1965). The scarcity of existing tests for circular symmetry might at first sight seem puzzling as one may be tempted to say that all tests for (linear) univariate symmetry should be adaptable to the circular setup (such as done for the runs tests), replacing the real line by [−π, π). However, this translation from one setup to the other is not so straightforward, due to several facts including that the points at π and −π coincide as periodicity is an essential feature of circular distributions. As stated in Pewsey (2004), when the observations are distributed on a large arc of the circle, it is likely that adapted tests suffer from a loss of power. It seems also very unlikely that optimal tests on the real line will keep their optimality features on the circle, as nothing a priori ensures that they behave well against the (certainly complicated) wrapped versions of the univariate skew distributions they were designed for. Thus, except against rotation alternatives, there exist so far no optimal tests for reflective symmetry. In view of this absence of optimal tests and the growing interest in skew circular distributions, our aim in the present paper is to fill in this gap by proposing tests for circular reflective symmetry about a fixed center that behave extremely well against a certain (general) type of skew alternatives. More precisely, we shall build locally and asymptotically optimal (in the maximin sense) tests for symmetry against k -sine-skewed alternatives (Umbach and 1∗

One should not confuse the problem of testing for reflective symmetry treated here with that of testing for l -fold symmetry on the circle; this issue has been addressed in Jupp and Spurr 1983 (see also Mardia and Jupp 2000, page 146).

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Jammalamadaka 2009, Abe and Pewsey 2011), a broad class of recently proposed skew circular distributions that has received an increasing interest over the past few years (see Section 2 for a description). In a nutshell, these skew distributions are obtained by perturbation of a base symmetric distribution via a factor involving sines and a parameter to regulate skewness. Apart from the general interest in these skew circular distributions, the motivations for this choice are mainly twofold. First, they are the circular analogues of the skew-symmetric distributions on the real line (see Azzalini and Capitano 2003 or Wang, Boyer and Genton 2004) inspired from the skew-normal distribution proposed in the seminal paper Azzalini (1985). Second, as we shall see, the resulting test statistics are based on the (trigonometric) sine moments, hence we provide these classical measures of circular skewness as well as the test of Pewsey (2004) with so far not known optimality properties. As nice by-product, our findings also enable us to discuss Fisher singularity issues exactly as in the linear case. The backbone of our approach is the Le Cam methodology which, although of linear nature, lends itself well for a transcription to circular settings (and even, with much more complications, to data living on unit hyperspheres in higher dimensions, see Ley, Swan, Thiam and Verdebout 2013). In a first stage, we will obtain optimal parametric tests, and then, by means of studentization arguments, we shall turn them into semi-parametric ones, valid under the entire null hypothesis of symmetry and optimal not only, as is usually the case, under the symmetric base distribution their parametric antecedents are based on, but uniformly optimal under any given symmetric base distribution. We will hence derive, as Schach (1969), a family of fully efficient semi-parametric tests which, in our case, are always optimal. For a given density, our tests will thus behave asymptotically like the likelihood ratio tests, but they clearly improve on the latter by their simplicity and the fact that, thanks to the Le Cam approach, one can derive explicit power expressions against sequences of contiguous skew alternatives. The paper is organized as follows. In Section 2, we first describe the family of k -sine-skewed distributions, then establish their ULAN property in the vicinity of symmetry, the crucial step in the Le Cam approach, and discuss some aspects of this property. In Section 3, we construct our optimal tests for reflective symmetry about a known center and investigate their asymptotic properties. The finite-sample performances of all our tests for reflective symmetry are evaluated and compared to existing tests in a large Monte Carlo simulation study, see Section 4. Our tests are then applied to the famous red wood ants data set of Jander (1957) in Section 5. The Fisher information singularity issue is tackled in Section 6. Section 7 concludes the paper with final comments and an outlook on the case where the median direction is not specified, as considered e.g. in Pewsey (2002), while an Appendix collects the technical proofs.

2

k -sine-skewed distributions and the ULAN property

In this section, we shall first describe in details the class of k -sine-skewed circular distributions and then establish their ULAN property. Throughout this paper (and as already used in the Introduction), all angles are measured in radians and we choose without loss of generality the zero direction as initial direction and the anti-clock-wise orientation of the unit circle. Moreover, in order to stress the difference between symmetry and asymmetry, we consider the interval

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[−π, π) instead of [0, 2π) and define quantities such as the cumulative distribution function (cdf) accordingly. This may lead to some differences w.r.t. the commonly adopted notation of, e.g., Mardia and Jupp (2000, Chapter 3), but these of course do not affect the mathematical outcomes.

2.1

k -sine-skewed densities

As briefly depicted in the Introduction, k -sine-skewed distributions are obtained by perturbation of a base symmetric density. Define the collection F := f0 : f0 (θ) > 0 a.e., f0 (θ + 2πk) = f0 (θ)∀k ∈ Z, f0 (−θ) = f0 (θ), Z π f0 (θ)dθ = 1 f0 unimodal at 0, −π

of unimodal reflectively symmetric (about the zero direction) circular densities. We attract the reader’s attention on the fact that the periodicity requirement is both classical and essential when dealing with circular distributions. The most well-known representatives of the collection F are the von Mises, cardioid and wrapped Cauchy distributions, with respective densities fVMκ (θ) := 2πI10 (κ) exp(κ cos(θ)) for κ > 0 (I0 stands for the modified Bessel function of the first 2

1 1 kind and order zero), fCA` (θ) := 2π (1+` cos(θ)) for ` ∈ (0, 1), and fWCρ (θ) := 1−ρ 2π 1+ρ2 −2ρ cos(θ) for ρ ∈ (0, 1). A location parameter µ ∈ [−π, π) is readily introduced as center of symmetry, leading to densities f (θ−µ), θ ∈ [−π, π), with mode µ. Inspired by the classical one-dimensional skewing method of Azzalini and Capitanio (2003), Umbach and Jammalamadaka (2009) have skewed such symmetric densities f0 by transforming them into

2f0 (θ − µ)G(ω(θ − µ)),

θ ∈ [−π, π),

Rθ where G(θ) = −π g(y)dy is the cdf of some circular symmetric density g and ω is a weighting function satisfying for all θ ∈ [−π, π) the three conditions ω(−θ) = −ω(θ), ω(θ + 2πk) = ω(θ)∀k ∈ Z, and |ω(θ)| ≤ π . This construction being too general and for the sake of mathematical tractability, Umbach and Jammalamadaka have particularized their choice to G(θ) = (π + θ)/(2π), the cdf of the uniform circular distribution, and ω(θ) = λπ sin(kθ), k ∈ N0 , with λ ∈ (−1, 1) playing the role of a skewness parameter. This finally yields what we call the k -sine-skewed densities k fµ,λ (θ) := f0 (θ − µ)(1 + λ sin(k(θ − µ))),

θ ∈ [−π, π),

(2.2)

with location parameter µ ∈ [−π, π) and skewness parameter λ ∈ (−1, 1). When λ = 0, no perturbation occurs and we retrieve the base symmetric density, otherwise (2.2) is skewed to the left (λ > 0) or to the right (λ < 0). Further properties of k -sine-skewed distributions are that k (µ − θ) = f k k fµ,λ µ,−λ (µ + θ), fµ,λ (µ) = f0 (0) whatever the value of λ, and the two endpoints, k (µ − π) and f k (µ + π), coincide. However, for k ≥ 2, f k fµ,λ µ,λ µ,λ is multimodal, whereas, for k = 1, multimodality only rarely occurs. This explains why Abe and Pewsey (2011) have

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restricted their attention to the study of the densities fµ,λ (θ) := f0 (θ − µ)(1 + λ sin(θ − µ)),

θ ∈ [−π, π),

(2.3)

which they have called sine-skewed circular densities (hence our terminology k -sine-skewed densities for general k ). Abe and Pewsey have shown the conditions under which the densities (2.3) happen to be multimodal. In the present paper, we establish all our theoretical results and propose tests for general k -sine-skewed distributions. Note that, when f0 is the circular uniform density, then (2.3) is the cardioid density fCAλ with mode at µ+π/2(mod 2π), hence, in passing, we will as well consider an optimal test for uniformity against the cardioid distribution. Sine-skewed (and k -sine-skewed) distributions lend themselves pretty well for modeling real data phenomena. Aside from Abe and Pewsey (2011) where this aspect is thoroughly described, these skew-circular distributions have been used, inter alia, in the analysis of the CO2 daily cycle in the low atmosphere at a rural site (Pérez, Sánchez, Garc´ıa, and Pardo 2012) and of forest disturbance regimes (Abe, Kubota, Shimatani, Aakala, and Kuuluvainen 2012). This, combined with the motivations stated in the Introduction, makes k -sine-skewed distributions an appealing choice as asymmetric alternatives in the construction of tests for circular reflective symmetry.

2.2

The ULAN property for k -sine-skewed densities

As explained in the Introduction, we shall use the Le Cam methodology in order to construct locally and asymptotically optimal tests for reflective symmetry against k -sine-skewed alternatives. For the sake of generality and in view of future research (see Section 7), we here do not assume µ to be fixed. This of course contains the µ-fixed case, which we need in this paper. Let θ1 , . . . , θn be i.i.d. circular observations with common density (2.2). For any symmetric base (n) density f0 ∈ F and any k ∈ N0 , denote by Pϑ ;f0 ,k , where ϑ := (µ, λ)0 ∈ [−π, π) × (−1, 1), the k reduces to f and joint distribution of the n-tuple θ1 , . . . , θn . Since, for λ = 0, the density fµ,λ 0 (n) hence does not depend on k , we drop the index k and simply write Pϑ;f0 at ϑ = ϑ 0 := (µ, 0)0 . Any pair (f0 , k) induces the parametric location-skewness model n o (n) (n) Pf0 ,k := Pϑ ;f0 ,k : ϑ ∈ [−π, π) × (−1, 1) , (n)

(n)

whereas any k ∈ N0 induces the semi-parametric location-skewness model Pk := ∪f0 ∈F Pf0 ,k . The very first step in our construction of tests for symmetry about a fixed center consists in establishing the Uniform Local Asymptotic Normality (ULAN) property, in the vicinity of (n) symmetry (i.e., at λ = 0), of the parametric model Pf0 ,k . This property of k -sine-skewed distributions happens to be interesting per se, as it paves the way to numerous other applications of the Le Cam theory (such as, e.g., the construction of tests for symmetry about an unspecified center or of the one-step optimal estimators, see e.g. van der Vaart 2002). ULAN requires the following mild regularity condition on the base densities f0 . Assumption (A). The function f0 (θ) is a.e.-C 1 over [−π, π) (or equivalently over R by periodicity) with a.e.-derivative f˙0 . 5

Most classical reflectively symmetric densities satisfy this requirement. Note that the condition over a bounded set combined with the fact that f0 > 0 and the periodicity condition entails that, letting ϕf0 = −f˙0 /f0 , the Fisher information quantity for location Rπ (n) If0 := −π ϕ2f0 (θ)f0 (θ)dθ is finite. ULAN of the parametric model Pf0 ,k with respect to ϑ = (µ, λ)0 , in the vicinity of symmetry, then takes the following form. C1

Theorem 2.1. Let f0 ∈ F and k ∈ N0 , and assume that Assumption (A) holds. Then, for (n) any µ ∈ [−π, π), the parametric family of densities Pf0 ,k is ULAN at ϑ 0 = (µ, 0)0 with central sequence (n)

∆(n) f0 ,k (µ)

∆f0 ,k;1 (µ)

:=

!

(n)

∆k;2 (µ) :=

n 1 X √ n i=1

ϕf0 (θi − µ) sin(k(θi − µ))

! ,

and corresponding Fisher information matrix Γf0 ,k;11 Γf0 ,k;12 Γf0 ,k;12 Γf0 ,k;22

Γ f0 ,k :=

! ,

Rπ where Γf0 ,k;11 and := I f0 , Γf0 ,k;12 := − −π sin(kθ)f˙0 (θ)dθ Rπ Γf0 ,k;22 := −π sin2 (kθ)f0 (θ)dθ . More precisely, for any µ(n) = µ + O(n−1/2 ) and for any (n) (n) (n) bounded sequence τ (n) = (τ1 , τ2 )0 ∈ R2 such that n−1/2 τ2 belongs to (−1, 1), we have, (n) (n) letting Λ(n) := log(dP (n) −1/2 (n) −1/2 (n) 0 /dP(µ(n) ,0)0 ;f ,k ), (µ

+n

τ1

,n

τ2

) ;f0 ,k

0

0

0

(n) Λ(n) = τ (n) ∆ f0 ,k (µ(n) ) − (1/2)ττ (n) Γ f0 ,kτ (n) + oP (1)

(2.4) L

(n) (n) and ∆ f0 ,k (µ(n) ) → N2 (00, Γ f0 ,k ), both under P(µ(n) ,0)0 ;f

0 ,k

as n → ∞.

The proof is given in the Appendix. One easily sees that the Fisher information for skewness Γf0 ,k;22 , and hence the cross-information quantity Γf0 ,k;12 , is finite by bounding sin2 by 1 under the integral sign. Note that the constant k has no effect on the validity of Theorem 2.1. Note also (n) that ∆k;2 (µ) does not depend on f0 , a fact that will become of great interest in the sequel. With this ULAN property in hand, we are ready to derive our optimal tests for reflective symmetry about a fixed center θ , as explained below in Section 2.3. Moreover, since we do not fix µ in Theorem 2.1, our result also paves the way for deriving optimal tests for symmetry about an unknown center; see Section 7. We conclude the present section with a brief discussion on the minimal conditions required to ensure the ULAN property. Indeed, in view of the proof of Lemma A.1 in the Appendix which is the main step to demonstrate Theorem 2.1, Assumption (A) can be further weakened to 1/2

Assumption (A min ). The mapping θ 7→ f0 (θ) is differentiable in quadratic mean over 6

1/2

[−π, π) (or equivalently over R by periodicity) with quadratic mean or weak derivative (f0 )0 (θ) 1/2 1/2 and, letting ψf0 (θ) = −2(f0 )0 (θ)/f0 (θ), the Fisher information quantity for location Rπ 2 Jf0 := −π ψf0 (θ)f0 (θ)dθ is finite. 1/2

Quadratic mean differentiability of f0 , a classical requirement in the Le Cam framework, R π 1/2 1/2 means that −π (f0 (θ + h) − f0 (θ) − hψf0 (θ))2 dθ = o(h2 ) as h → 0, which corresponds exactly to the integral in expression (A.8) of the proof of Theorem 2.1 with h = −t and hence (n) is the minimal condition in order to have the ULAN property of the parametric model Pf0 ,k . Note that, under Assumption (A), these two derivatives of course coincide a.e., as well as ψf0 and ϕf0 , and Jf0 = If0 (as already mentioned above, the C 1 condition ensures finiteness of If0 , while in the weaker Assumption (A min ) one requires that Jf0 < ∞).

2.3

Constructing Le Cam optimal tests from the ULAN property

The central idea of the Le Cam theory we are using here is the concept of convergence of statistical models (experiments in the Le Cam vocabulary). More precisely, Le Cam’s idea consists in approximating a family of probability measures by a family of a simpler nature. The ULAN property is an essential ingredient in this approximation, as it allows to deduce (n) that (see Le Cam 1986 for details) our parametric location-skewness model Pf0 ,k is locally (around (µ, 0)0 ) and asymptotically (for large sample sizes) equivalent to a simple Gaussian shift model. Intuitively, this follows from the fact that the likelihood ratio expansion (2.4), up to the remainder terms, strongly resembles the likelihood ratio of a Gaussian shift model Γf0 ,kτ (n) , Γ f0 ,k ) with a single observation denoted by ∆ (n) N2 (Γ f0 ,k . Since the optimal procedures for Gaussian shift experiments are well-known, we can translate them into our circular locationskewness model and hence obtain inferential procedures that are asymptotically optimal, here in the maximin sense. Recall that a test φ∗ is called maximin in the class Cα of level-α tests for the null H0 against the alternative H1 if (i) φ∗ has level α and (ii) the power of φ∗ is such that inf EP [φ∗ ] ≥ sup inf EP [φ]. P∈H1

φ∈Cα P∈H1

Here, we shall employ this working scheme for testing the null hypothesis H0µ of symmetry about a known central direction µ ∈ [−π, π). As explained above, our procedures will be (asymptotically) optimal against a fixed k -sine-skewed alternative (2.2). For the present testing (n) µ problem, we first construct f0 -parametric tests for H0;f = P(µ,0)0 ;f0 : the optimality of these 0 tests under the base density f0 is thwarted by the fact that they are valid (in the sense that they meet the asymptotic level-α constraint) only under f0 . In order to avoid this non-validity beyond f0 , we make use of a classical studentization argument allowing us to turn our parametric (n) tests into tests for the semi-parametric null hypothesis H0µ = ∪f0 ∈F P(µ,0)0 ;f0 . The next section contains the detailed derivations of these tests.

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3

The test statistic and its asymptotic properties (n);µ

Fix µ ∈ [−π, π). The f0 -parametric test φf0 ;k for circular reflective symmetry about a known µ central direction µ we propose rejects H0;f at asymptotic level α whenever the statistic 0 (n)

(n);µ Qf0 ;k

:=

|∆k;2 (µ)| 1/2 Γf0 ,k;22

=

|n−1/2

Pn

i=1 sin(k(θi 1/2 Γf0 ,k;22

− µ))|

(3.5)

exceeds zα/2 , the α/2 upper quantile of the standard normal distribution (tests for reflective symmetry against one-sided alternatives of the form λ > 0 or λ < 0 are built similarly). It follows from the Le Cam theory that this test is locally and asymptotically maximin for testing (n) µ µ the null H0;f against H1;f := ∪λ6=0∈(−1,1) P(µ,λ)0 ;f0 ,k . Note that this optimality does not hold 0 0 ,k against k 0 -sine-skewed laws with k 0 6= k , each value of k leads to a distinct optimal test. (n) (n) Now consider g0 ∈ F . Under P(µ,0)0 ;g0 , ∆k;2 (µ) is asymptotically normal with mean 0 and ∗(n);µ

variance Γg0 ,k;22 6= Γf0 ,k;22 . It is therefore natural to consider the studentized test φk rejects (at asymptotic level α) the null of circular reflective symmetry H0µ when ∗(n);µ Qk

P | ni=1 sin(k(θi − µ))| := P 1/2 n 2 i=1 sin (k(θi − µ))

that

(3.6)

exceeds zα/2 . We attract the reader’s attention to the fact that this very simple test statistic ∗(n);µ

no more depends on f0 (hence the omission of the index f0 in φk ). This is due to the fact (n) that the central sequence for skewness, ∆k;2 (µ), does not depend on f0 . This remarkable fact ∗(n);µ

(n);µ

, implies that all parametric tests φf0 ;k , k ∈ N, lead to the same studentized test statistic φk which therefore inherits optimality from its parametric antecedents under any base symmetric distribution! This nice property as well as the asymptotics of such tests, under any f0 ∈ F , follow from the ULAN property in Theorem 2.1 and are summarized in the following result (see the Appendix for a proof). Theorem 3.1. Let k ∈ N0 . Then, (n) ∗(n);µ D ∗(n);µ (i) under ∪f0 ∈F P(µ,0)0 ;f0 , Qk → N (0, 1) as n → ∞, so that the test φk has asymptotic level α under the same hypothesis; (n) ∗(n);µ (ii) under P −1/2 (n) 0 with f0 ∈ F and k 0 ∈ N0 , Qk is asymptotically normal 0 (µ,n τ2 ) ;f0 ,k −1/2 (n) with mean Γf0 ,k;22 Cf0 (k, k 0 )τ2 and variance 1, where τ2 = limn→∞ τ2 and Cf0 (k, k 0 ) := Rπ 0 −π sin(kθ) sin(k θ)f0 (θ) dθ (which is finite); ∗(n);µ (n);µ (n) (iii) for all f0 ∈ F , Qk = Qf0 ;k + oP (1) as n → ∞ under P(µ,0)0 ;f0 , so that the studentized ∗(n);µ test φk is locally and asymptotically maximin, at asymptotic level α, when testing H0µ (n) against alternatives of the form ∪λ6=0∈(−1,1) ∪f0 ∈F P(µ,λ)0 ;f0 ,k . ∗(n);µ

Theorem 3.1(i) shows that the studentized test φk is indeed valid under the entire null µ hypothesis H0 , hence is asymptotically distribution-free. Note the uniform (in f0 , not in k ) optimality of our studentized test. 8

∗(n);µ

Figure 1: Power curves, as a function of τ2 , of the studentized test φk for k = 2 against (n) local alternatives P −1/2 (n) 0 for f0 the von Mises density with concentration parameter 0 (µ,n

τ2

) ;f0 ,k

1 and for k 0 equal to 1 (blue line), 2 (red line) and 3 (yellow line). For the sake of generality, we have also considered above alternatives where k ∈ N is replaced by some k 0 ∈ N possibly different from the k used in the construction of our tests. Point (ii) ∗(n);µ of Theorem 3.1 allows us to give the explicit asymptotic power of φk against the local (n) alternatives P −1/2 (n) 0 : 0 (µ,n

τ2

) ;f0 ,k

1 − Φ zα/2 − (Γf0 ,k;22 )−1/2 Cf0 (k, k 0 )τ2 + Φ −zα/2 − (Γf0 ,k;22 )−1/2 Cf0 (k, k 0 )τ2 , where Φ stands for the cdf of the standard Gaussian distribution. In Figure 1, we have plotted this power as a function of τ2 for f0 the von Mises density with concentration parameter 1 and for k = 2 and k 0 = 1, 2, 3. The plot shows that the power of the test becomes lower if k 0 is not correctly chosen, a result which we shall also see in the simulation study of the next section. We finally stress several important and interesting facts. As described in Section 2.1, for 1 f0 (θ) = 2π , the uniform density, (2.3) with k = 1 corresponds to the cardioid density with mode at µ + π/2 ∈ [−π, π). Hence, for fixed µ, an optimal test for testing the null hypothesis of uniformity against cardioid alternatives shall be based on (3.5) with k = 1 and Γf0 ,k;22 = 12 . It can be easily shown that the latter statistic coincides with the Rayleigh (1919) test statistic which is indeed known to be optimal for the null hypothesis of uniformity against cardioid alternatives ∗(n);µ (see Jammalamadaka and SenGupta 2001, p. 133). Now, when k = 2, the statistic φ2 coincides exactly with the so-called “b2-star” test proposed in Pewsey (2004). We have thus shown that that test enjoys maximin optimality features against 2-sine-skewed alternatives, and provided its asymptotic powers against contiguous alternatives. This not only complements, but also gives further insight into the b2-star test. Finally, the very simple tests we have obtained are also easy to interpret as they are based on sine moments, which are classical measures of skewness for circular data (see, e.g., Batschelet 1965).

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4

Monte Carlo simulation study

In this section we investigate the finite-sample properties of the proposed testing procedures for reflective symmetry. More precisely, we check the nominal level constraint under distinct forms of reflective symmetry and determine the power properties under various forms of asymmetry. To this end, we have generated N = 10, 000 independent samples of small size n = 30 and moderate size n = 100 from reflectively symmetric and increasingly skewed (λ > 0) circular distributions, and run our tests (which contain Pewsey’s b2star test) as well as the modified runs test of Pewsey (2004), under two-sided form at the asymptotic level α = 5%. Without loss of generality, we fix the center of symmetry µ to 0. We have performed ∗(n);0 ∗(n);0 ∗(n);0 (n) our tests φ1 , φ2 and φ3 as well as the modified runs test φmodrun (with p = 0.6, see Pewsey 2004). Of course, we here choose settings other than those in Pewsey (2004), and hence also add new simulation-based information on Pewsey’s b2star test. We also remark that we consider here k = 1, 2, 3 for our tests because these values are able to capture both skew unimodality (k = 1) and multimodality, but do not lead to too many oscillations of the sines within [−π, π) which can lead to wrong numerical calculations. As reflectively symmetric distributions representing the null hypothesis, we have considered the von Mises laws fVM1 and fVM10 , the cardioid fCA0.5 , the wrapped Cauchy fWC0.5 as well as a mixture of two fVM1 and two fVM10 von Mises laws with, in each case, respective centers at −π/4 and π/4 and mixing probability 0.5. The latter mixture is used in order to assess the performances of our tests under bimodality. The densities fVM1 and fVM10 have then been turned into their 1-,2- and 3-sine-skewed versions, whereas fCA0.5 and fWC0.5 have become 1and 2-sine-skewed. In each case the skewness parameter λ increases from zero to successively positive values. The bimodal mixture of von Mises laws has been skewed by simply shifting the center π/4 to π/4 + λ. In order to also investigate other forms of perturbation of symmetry, we have applied the Moebius transform of Kato and Jones (2010) fVM10 with r = 0.5, to fVM1 and θi −λ with ωr = 1−r which transforms each θi , i = 1, . . . , n, into λ + 2 arctan ωr tan 2 1+r . The empirical rejection probabilities are reported in Table 1 for 1-sine-skewed alternatives, Table 2 for 2-sine-skewed alternatives and Table 3 for Moebius, von Mises mixtures and 3-sine-skewed alternatives.

Whatever sample size considered, all four tests meet the 5% nominal level constraint under each reflectively symmetric density considered, even under bimodality, and seem to be unbiased. ∗(n);0 Under k -sine-skewed alternatives, the theoretical optimality features of our tests φk are ∗(n);0 0 0 confirmed, whereas under certain k -sine-skewed densities the test φk for k 6= k exhibits low powers (especially when combining the indices 1 and 3). When the observations are highly concentrated (fVM10 case), the differences in performance between the three tests vanish. This is maybe due to the fact that sin(kθ) is always quite small in concentrated settings, whatever the value of k . We also note that all our tests are powerful under the Moebius transformed skew densities and even under skewed von Mises mixture distributions with high concentration parameter κ, which shows that the proposed tests also perform well under other skew laws then the sine-skewed ones. As an overall summary, we see that our three tests generally outperform 10

Table 1: Empirical rejection probabilities, out of N = 10, 000 replications and for the sample sizes n = 30 and n = 100, under various reflectively symmetric and 1-sine-skewed distributions, ∗(n);0 ∗(n);0 ∗(n);0 (n) of the optimal tests φ1 , φ2 and φ3 as well as of the modified runs test φmodrun with p = 0.6. The tests are performed at level α = 5%. Test

∗(n);0 φ1 ∗(n);0 φ2 ∗(n);0 φ3 (n) φmodrun

∗(n);0

φ1 ∗(n);0 φ2 ∗(n);0 φ3 (n) φmodrun

∗(n);0

φ1 ∗(n);0 φ2 ∗(n);0 φ3 (n) φmodrun

∗(n);0

φ1 ∗(n);0 φ2 ∗(n);0 φ3 (n) φmodrun

n = 30/n = 100 λ=0 .047/.048 .051/.053 .046/.054 .054/.054 λ=0 .046/.048 .047/.048 .048/.047 .053/.051 λ=0 .051/.051 .049/.046 .052/.048 .043/.051 λ=0 .048/.049 .050/.052 .050/.049 .051/.052

n = 30/n = 100 n = 30/n = 100 1-sine-skewed fVM1 λ = 0.2 λ = 0.4 .110/.266 .311/.779 .063/.090 .101/.228 .047/.053 .051/.062 .064/.072 .112/.156 1-sine-skewed fVM10 λ = 0.2 λ = 0.4 .058/.095 .091/.235 .059/.093 .090/.234 .058/.087 .086/.212 .054/.060 .070/.089 1-sine-skewed fCA0.5 λ = 0.2 λ = 0.4 .111/.292 .341/.824 .054/.065 .070/.107 .046/.051 .052/.050 .061/.064 .094/.128 1-sine-skewed fWC0.5 λ = 0.2 λ = 0.4 .100/.227 .270/.693 .060/.083 .091/.197 .048/.056 .060/.085 .061/.074 .106/.153

11

n = 30/n = 100 λ = 0.6 .608/.988 .156/.449 .056/.074 .204/.375 λ = 0.6 .164/.458 .164/.451 .151/.408 .096/.150 λ = 0.6 .666/.994 .089/.185 .049/.048 .163/.262 λ = 0.6 .539/.972 .143/.382 .067/.125 .200/.345

Table 2: Empirical rejection probabilities, out of N = 10, 000 replications and for the sample sizes n = 30 and n = 100, under various reflectively symmetric and 2-sine-skewed distributions, ∗(n);0 ∗(n);0 ∗(n);0 (n) of the optimal tests φ1 , φ2 and φ3 as well as of the modified runs test φmodrun with p = 0.6. The tests are performed at level α = 5%. Test

n = 30/n = 100

∗(n);0 φ1 ∗(n);0 φ2 ∗(n);0 φ3 (n) φmodrun

λ=0 .048/.048 .049/.049 .047/.047 .048/.048

Test ∗(n);0 φ1 ∗(n);0 φ2 ∗(n);0 φ3 (n) φmodrun

λ=0 .049/.049 .050/.050 .050/.049 .049/.054


λ=0 .049/.048 .048/.049 .047/.051 .046/.050


λ=0 .051/.051 .053/.055 .046/.048 .051/.047

n = 30/n = 100 n = 30/n = 100 2-sine-skewed fVM1 λ = 0.2 λ = 0.4 .064/.103 .107/.254 .115/.295 .347/.825 .060/.097 .102/.242 .056/.067 .081/.122 2-sine-skewed fVM10 λ = 0.2 λ = 0.4 .084/.175 .201/.572 .086/.179 .205/.577 .083/.173 .201/.553 .065/.068 .117/.176 2-sine-skewed fCA0.5 λ = 0.2 λ = 0.4 .051/.060 .067/.106 .121/.292 .336/.820 .055/.065 .066/.113 .059/.071 .097/.137 2-sine-skewed fWC0.5 λ = 0.2 λ = 0.4 .066/.096 .105/.223 .113/.277 .315/.788 .066/.094 .109/.264 .056/.067 .082/.123

12

n = 30/n = 100 λ = 0.6 .186/.491 .669/.994 .173/.482 .131/.244 λ = 0.6 .414/.901 .419/.907 .405/.890 .211/.405 λ = 0.6 .086/.186 .670/.993 .089/.187 .171/.289 λ = 0.6 .165/.456 .642/.992 .192/.524 .139/.262

Table 3: Empirical rejection probabilities, out of N = 10, 000 replications and for the sample sizes n = 30 and n = 100, under various reflectively symmetric and various skewed distributions, ∗(n);0 ∗(n);0 ∗(n);0 (n) of the optimal tests φ1 , φ2 and φ3 as well as of the modified runs test φmodrun with p = 0.6. The tests are performed at level α = 5%. Test

n = 30/n = 100

φ1 ∗(n);0 φ2 ∗(n);0 φ3 (n) φmodrun

λ=0 .051/.051 .050/.049 .047/.048 .049/.053


λ=0 .046/.046 .047/.048 .047/.048 .050/.052


λ=0 .048/.053 .048/.049 .050/.048 .049/.049


λ=0 .049/.050 .053/.051 .050/.050 .051/.053


λ=0 .046/.052 .051/.050 .047/.048 .051/.046


λ=0 .049/.046 .050/.047 .049/.049 .052/.050

∗(n);0

n = 30/n = 100 n = 30/n = 100 Moebius transformed fVM1 λ = 0.2/3 λ = 0.4/3 .074/.127 .142/.351 .082/.153 .169/.453 .082/.154 .163/.460 .057/.059 .074/.074 Moebius transformed fVM10 λ = 0.02 λ = 0.04 .092/.215 .244/.641 .092/.215 .245/.644 .093/.215 .247/.646 .069/.081 .133/.204 Skewed fVM1 mixtures λ = 0.4 λ = 0.8 .067/.102 .072/.149 .052/.054 .072/.123 .050/.051 .048/.052 .053/.057 .073/.078 Skewed fVM10 mixtures λ = 0.2 λ = 0.4 .066/.121 .103/.253 .055/.065 .138/.361 .255/.676 .835/.999 .132/.238 .407/.802 3-sine-skewed fVM1 λ = 0.2 λ = 0.4 .050/.052 .057/.063 .060/.098 .104/.240 .117/.290 .340/.828 .059/.065 .083/.132 3-sine-skewed fVM10 λ = 0.2 λ = 0.4 .099/.230 .266/.689 .103/.245 .280/.722 .105/.255 .287/.743 .067/.079 .132/.201

13

n = 30/n = 100 λ = 0.2 .239/.639 .304/.776 .302/.771 .086/.116 λ = 0.06 .464/.937 .466/.938 .469/.940 .237/.457 λ = 1.2 .066/.095 .107/.256 .047/.053 .066/.086 λ = 0.6 .130/.341 .502/.957 .990/1.00 .734/.995 λ = 0.6 .058/.080 .176/.482 .676/.995 .128/.280 λ = 0.6 .548/.970 .574/.980 .593/.984 .264/.511

the modified runs test.

5

A real data application

In this section, we apply our optimal tests for reflective symmetry to a well-known data set from an animal orientation experiment. This data set stems from an experiment with 730 red wood ants (Formica rufa L.) described in Jander (1957). Each ant was individually placed in the center of an arena with a black target positioned at an angle of 180◦ from the zero direction, and the initial direction in which each ant moved upon release was recorded to the nearest 10◦ . Thus it is clear that the experimental design suggests the black target as natural median direction, a fact that is clearly corroborated by the graphical representation of the data in Figure 2. The question of interest is whether the directions chosen by the ants are symmetrically distributed around the median direction representing the black target, allowing to know inter alia whether the classical symmetric or rather the more recent skew distributions better model this data set. Papers having formerly investigated this problem include Pewsey (2004), Umbach and Jammalamadaka (2009) and Abe and Pewsey (2011). From the description of the experimental design, this real data set happens to be a very good candidate for testing circular symmetry about a known median direction. Assuming the median direction unknown might even be inappropriate here and the corresponding tests for symmetry will not be as powerful as tests for symmetry about a fixed direction. The data plot in Figure 2 indicates that the underlying density might be multimodal rather than unimodal, indicating ∗(n);0 ∗(n);0 ∗(n);0 that the tests φ2 and φ3 might be more powerful in the present situation than φ1 (we refer to Abe and Pewsey 2011 for a discussion on the conditions under which 1-sine-skewed ∗(n);0 ∗(n);0 distributions are unimodal or multimodal). Indeed, φ1 yields a p-value of 0.778, while φ2 ∗(n);0 and φ3 respectively give p-values 0.011 and 0.013. The latter two p-values provide evidence that the data are in fact not symmetrically distributed around the median direction of 180◦ . ∗(n);0 Pewsey (2004) obtained the same conclusion with his b2star test, but our combination of φ2 ∗(n);0 with the test φ3 provides yet further information and evidence. In Abe and Pewsey (2011), the authors notice that neither the symmetric nor the 1-sine-skewed distributions they have considered provide an adequate fit to this data. Their findings are not a surprise: according ∗(n);0 to φ1 , 1-sine-skewed densities are not preferable over symmetric ones, while our other tests reject the hypothesis of reflective symmetry at just above the 1% level. This shows that, most probably, the ant data are best fitted by 2- or 3-sine-skewed distributions.

6

Singularity of the location-skewness Fisher information matrix

Besides its numerous favorable properties, the skew-normal distribution of Azzalini (1985) is also famous for having a singular Fisher information matrix in the vicinity of symmetry, due to the collinearity of the scores for location and skewness in its initial parameterization. A vast literature has been devoted to the analysis of the reasons for this singularity, to its negative impact on inferential procedures, to possible cures (reparameterizations) and to the 14

Figure 2: Raw circular plot of the Jander (1957) data set recorded during an orientation experiment with 730 red wood ants. Each dot represents the direction chosen by five ants. study of which other skew-symmetric distributions suffer from the same drawback. For a recent overview on this issue and for further references, we refer the reader to Hallin and Ley (2012), where the class of skew-symmetric distributions suffering from the Fisher singularity are exactly determined. The present section can be inscribed into this stream of literature, as it discusses and solves the same problem for k -sine-skewed circular distributions. Moreover, our results are very important when one considers the construction of optimal tests about an unknown center µ, as will be briefly discussed in the final section. Now, recall that the information matrix in the vicinity of symmetry is given by ! Rπ 2 Rπ ϕ sin(kθ)ϕ (θ)f (θ)dθ (θ)f (θ)dθ 0 0 f 0 −π Rπ R π −π f0 . Γ f0 ,k = 2 (kθ)f sin sin(kθ)ϕ (θ)f (θ)dθ 0 (θ)dθ 0 f 0 −π −π This matrix is singular if and only if Z

π

−π

ϕ2f0 (θ)f0 (θ)dθ

Z

π

2

sin (kθ)f0 (θ)dθ −π

Z

2

π

= −π

sin(kθ)ϕf0 (θ)f0 (θ)dθ

.

(6.7)

The Cauchy-Schwarz inequality readily yields that the equality sign “=” in (6.7) can be replaced by “≥” with equality holding if and only if ϕf0 (θ) = a sin(kθ) for some real constant a. The latter easy-to-solve first-order differential equation then shows that an information singularity can only occur for base symmetric densities f0 of the form c exp( ka cos(kθ)) for a ∈ R and c > 0 a normalizing constant. Now, bare in mind that the class of base densities F we consider contains the condition of unimodality on f0 , which directly rules out all values k ≥ 2 and forces a to be positive. Hence, the only base symmetric density for which the Fisher information matrix Γ f0 ,k is singular corresponds to f0 (θ) = c exp(κ cos(θ)) with κ = ka > 0 a concentration parameter, hence to the famous von Mises circular density. We formalize this result in the following proposition.

15

Proposition 6.1. Let f0 be a symmetric base density belonging to F and satisfying Assumption (A), and consider k -sine-skewed densities of the form f0 (θ−µ)(1+λ sin(k(θ−µ))). Then the Fisher information matrix associated with the parameters µ ∈ [−π, π) and λ ∈ (−1, 1) is singular in the vicinity of symmetry (that is, at λ = 0) if and only if k = 1 and f0 (θ) = c exp(κ cos(θ)) with κ > 0 a concentration parameter and c > 0 the normalizing constant, that is, if and only if one is considering sine-skewed von Mises densities. A referee raised the important question of the existence of a parameterization which avoids this singularity, as is the case for skew-normal distributions with the Centered Parameterization (Azzalini 1985) or the parameterization recently proposed in Hallin and Ley (2013). Mimicking these constructions, one obtains such a singularity-free parameterization, but this is beyond the scope of the present paper.

7

Final comments

In this paper we have tackled the problem of testing circular reflective symmetry about a specified center. The tests we propose are uniformly (over the null hypothesis) locally and asymptotically maximin against k -sine-skewed alternatives, asymptotically distribution-free and moreover of a very simple form. They furthermore exhibit nice finite sample behaviors. Now, as already mentioned before, it would also be of interest to adapt our procedures to the case of an unspecified center, and our general ULAN property provides the required theoretical background for constructing such tests. The crucial difference from the tests of the present paper, of course, lies in the fact that we will need to replace the unknown location µ with an estimator µ ˆ. Γ If the information matrix f0 ,k were diagonal, then the substitution of µ ˆ for µ would have (n) no influence, asymptotically, on the behavior of the central sequence for skewness ∆k;2 (µ). However, the covariance Γf0 ,k;12 only rarely equals zero, hence a local perturbation of µ has the (n) same asymptotic impact on ∆k;2 (µ) as a local perturbation of λ = 0. It follows that the cost of not knowing the actual value of the location µ is strictly positive when performing inference on λ; the stronger the correlation between µ and λ, the larger that cost. The worst case occurs of course when the information matrix is singular (see Section 6), which leads to asymptotic local powers equal to the nominal level α; more precisely, this situation entails that the best possible test is the trivial test, that is, the test discarding the observations and rejecting the null of reflective symmetry at level α whenever an auxiliary Bernoulli variable with parameter α takes value one. Now, in order to take into account the aforementioned cost of not knowing µ, one can replace (n) the central sequence ∆k;2 (µ) with the, in Le Cam terminology (see, e.g., Le Cam 1986), efficient central sequence (n)ef f

Γf0 ,k;12 (n) ∆ (µ) Γf0 ,k;11 f0 ,k;1 n X Γf ,k;12 n−1/2 sin(k(θi − µ)) − 0 ϕf0 (θi − µ) . Γf0 ,k;11 (n)

∆f0 ,k;2 (µ) := ∆k;2 (µ) − =

i=1

16

(n)

This efficient central sequence can be seen as the orthogonal projection of ∆k;2 (µ) onto (n)

(n)ef f

(n)

the subspace orthogonal to ∆f0 ,k;1 (µ), which ensures that ∆f0 ,k;2 (µ) and ∆f0 ,k;1 (µ) are asymptotically uncorrelated. An asymptotic test can then be easily obtained by considering (n)ef f a studentized version of ∆f0 ,k;2 (ˆ µ). Unfortunately, by doing so, it can be shown that, only under f0 , there is no asymptotic effect if µ is replaced with µ ˆ (this fails to hold for g0 6= f0 ). Therefore, rather than having as in the present paper a test that is valid under any density f0 ∈ F with a fixed location µ, we would obtain a test which is valid for any value of µ but only under a single f0 (complete parametric test). Constructing tests that are completely distribution-free (with respect to both the underlying base density and the location parameter) is an ongoing research project. Finally, we briefly discuss the choice of k in the statistic (3.6). We showed that, for a ∗(n);µ fixed k , the test based on Qk is asymptotically optimal against k -sine-skewed alternatives with the same k . Now, if a practitioner does not have in mind a particular value of k , that is, if he/she does not have a particular alternative in mind, we suggest two possibilities. ∗(n);µ First, as done in our real data example, consider several tests φk and compare their outcomes. Second, a test could be performed using the asymptotic joint distribution of ∗(n);µ ∗(n);µ ∗(n);µ 0 Q∗(n);µ := (Q1 , Q2 , . . . , Qq ) for a certain q ∈ N0 . The asymptotic distribution ∗(n);µ of Q under the null can easily be derived using Theorem 3.1. However, an asymptotic test based on Q∗(n);µ will clearly lose the optimality property against all the alternatives considered here, contrary to the aim of this paper. Furthermore, one can also then raise the question of the choice of q ∈ N0 in Q∗(n);µ ; this issue is clearly beyond the scope of the present paper.

ACKNOWLEDGEMENTS Christophe Ley thanks the Fonds National de la Recherche Scientifique, Communauté fran¸caise de Belgique, for support via a Mandat de Chargé de Recherche.

A

Proof of Theorem 2.1

Our proof relies on Lemma 1 of Swensen (1985)—more precisely, on its extension in Garel and Hallin (1995). The sufficient conditions for ULAN in those results readily follow from standard k )1/2 (θ) (see (2.2)) arguments (hence are left to the reader), once it is shown that (µ, λ)0 7→ (fµ,λ is quadratic mean differentiable at any (µ, 0)0 , which we establish in the following lemma. Lemma A.1. Let f0 ∈ F and k ∈ N0 , and assume that Assumption (A) holds. Define k 1/2 Dθ (fµ,0 ) (θ) := −

1 f˙0 (θ − µ) , 2 f 1/2 (θ − µ) 0

17

and

1 1/2 k 1/2 Dλ (fµ,λ ) (θ)|λ=0 := f0 (θ − µ) sin(k(θ − µ)). 2 Then, for any µ ∈ [−π, π), we have that, as (t, `) → (0, 0), Rπ k k )1/2 (θ) − tD (f k )1/2 (θ) 2 dθ = o(t2 ), (i) −π (fµ+t,0 )1/2 (θ) − (fµ,0 µ µ,0 (ii)

Rπ

(iii)

Rπ

(iv)

Rπ

−π

−π

k k k (fµ+t,` )1/2 (θ) − (fµ+t,0 )1/2 (θ) − `Dλ (fµ+t,λ )1/2 (θ)|λ=0 k k )1/2 (θ)| Dλ (fµ+t,λ )1/2 (θ)|λ=0 − Dλ (fµ,λ λ=0

( k (fµ+t,` )1/2 (θ)

−π

−

k )1/2 (θ) (fµ,0

−

t `

!0

2

2

dθ = o(`2 ),

dθ = o(1),

k )1/2 (θ) Dµ (fµ,0 k Dλ (fµ,λ )1/2 (θ)|λ=0

!)2 dθ = o(||(t, `)0 ||2 ).

k we can rewrite the left-hand side of (i) under Proof of Lemma A.1. (i) By definition of fµ,0 the simpler form

Z

π

−π

1/2 f0 (θ

− µ − t)) −

1/2 f0 (θ

1 f˙0 (θ − µ) − µ) + t 1/2 2 f (θ − µ)

!2 dθ.

(A.8)

0

Next, the a.e.-differentiability of f0 (Assumption (A)) combined with the mean value theorem turns (A.8) into !2 1 f˙0 (θ − µ) 1 f˙0 (θ − µ∗ ) t − t dθ 2 f 1/2 (θ − µ∗ ) 2 f 1/2 (θ − µ) −π 0 0 !2 Z f˙0 (θ − µ∗ ) f˙0 (θ − µ) 1 2 π − 1/2 dθ = t 1/2 4 −π f0 (θ − µ∗ ) f0 (θ − µ)

Z

π

with µ∗ ∈ (µ, µ + t). Assumption (A) and the periodicity requirement ensure that

(A.9) f˙0 (θ) 1/2

f0

(θ)

is continuous over [−π, π], hence its square can be bounded by a sufficiently large constant; consequently, the Lebesgue dominated convergence theorem implies that (A.9) is o(t2 ). (ii) Similarly, the left-hand side integral in (ii) can be re-expressed as 2 1 1/2 f0 (θ − µ − t) (1 + ` sin(k(θ − µ − t))) − 1 − ` sin(k(θ − µ − t)) dθ. 2 −π

Z

π

Exactly as for (i), the differentiability of sin(kθ) allows us to re-write this integral under the form 2 Z 1 2 π 1 2 f0 (θ − µ − t) sin (k(θ − µ − t)) ` − 1 dθ 4 (1 + `∗ sin(k(θ − µ − t)))1/2 −π with `∗ ∈ (0, `). Since sin2 (kθ)f0 (θ) is integrable and (1 + `∗ sin(k(θ − µ − t)))−1 is bounded by a constant not depending on ` (indeed, we can take `∗ ≤ ` < 1/2 as ` → 0, hence 18

1 + `∗ sin(k(θ − µ − t)) ≥ 1/2 over [−π, π) which does not depend on `), Lebesgue’s dominated convergence theorem applies and yields the desired o(`2 ) quantity. (iii) The left-hand side in (iii) equals Z 2 1 π 1/2 1/2 (A.10) f0 (θ − (µ + t)) sin(k(θ − (µ + t))) − f0 (θ − µ) sin(k(θ − µ)) dθ. 4 −π 1/2

Since f0 (θ) sin(kθ) is square-integrable, the quadratic mean continuity entails that (A.10) tends to zero as t → 0, hence is an o(1) quantity. (iv) The left-hand side in (iv) is bounded by C(S1 + S2 + `2 S3 ), where Z π 2 k k 1/2 k 1/2 S1 = (fµ+t,0 )1/2 (θ) − (fµ,0 ) (θ) − tDµ (fµ,0 ) (θ) dθ, −π

Z

π

S2 =

−π

and

k k k )1/2 (θ)|λ=0 )1/2 (θ) − (fµ+t,0 )1/2 (θ) − `Dλ (fµ+t,λ (fµ+t,`

Z

π

S3 =

−π

k k 1/2 Dλ (fµ+t,λ )1/2 (θ)|λ=0 − Dλ (fµ,λ ) (θ)|λ=0

2

2

dθ,

dθ.

The result then follows from (i), (ii) and (iii).

B

Proof of Theorem 3.1.

Proofs of Theorem 3.1. Fix f0 ∈ F . Part (i) of the theorem trivially follows from the Central Limit Theorem combined with the fact that P P |n−1/2 ni=1 sin(k(θi − µ))| |n−1/2 ni=1 sin(k(θi − µ))| ∗(n);µ Qk = −1 Pn = + oP (1) (B.11) 2 1/2 (Γf0 ,k;22 )1/2 (n i=1 sin (k(θi − µ))) (n)

as n → ∞ under P(µ,0)0 ;f0 . Part (ii) is slightly more subtle but can also be readily handled by using the “Third Lemma of Le Cam” (see Le Cam 1986). Un(n) (n) 0 )τ der P −1/2 (n) 0 , the asymptotic normality of ∆ (µ) with mean C (k, k 2 f0 k;2 0 (µ,n

τ2

) ;f0 ,k

(n)

and variance Γf0 ,k;22 is obtained by establishing the joint normality of ∆k;2 (µ) and (n) (n) (n) log dP −1/2 (n) 0 /dP under P(µ,0)0 ;f0 and then applying Le Cam’s third Lemma 0 (µ,0) ;f 0 0 (µ,n

τ2

) ;f0 ,k

(which holds thanks to the ULAN property). Part (ii) follows immediately since (B.11) also (n) holds under P −1/2 (n) 0 by contiguity. Finally, Part (iii) trivially follows from (B.11) and 0 (µ,n

τ2

) ;f0 ,k

(n);µ

the optimality features of the parametric test φf0 ;k for all f0 ∈ F .

19

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