Optimal Procedures Based on Interdirections and pseudo-mahalanobis Ranks for Testing multivariate elliptic White Noise against ARMA Dependence

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1 Optimal Procedures Based on Interdirections and pseudo-mahalanobis Rans for Testing multivariate elliptic White Noise against ARMA Dependence Marc Hallin and Davy Paindaveine Université Libre de Bruxelles, Belgium Abstract We propose a multivariate generalization of signed-ran tests for testing elliptically symmetric white noise against ARMA serial dependence. These tests are based on Randles 1989) s concept of interdirections and the rans of pseudo-mahalanobis distances. They are affine-invariant, and asymptotically equivalent to strictly distribution-free statistics. Depending on the score function considered van der Waerden, Laplace,... ), they allow for locally asymptotically maximin tests at selected densities multivariate normal, multivariate double-exponential,... ). Local powers and asymptotic relative efficiencies with respect to the Gaussian procedure are derived. We extend to the multivariate serial context the Chernoff- Savage result, showing that classical correlogram-based procedures are uniformly dominated by the van der Waerden version of our tests, so that correlogram methods are not admissible in the Pitman sense. We also prove an extension of the celebrated Hodges-Lehmann.864 result, providing, for any fixed space dimension, the lower bound for the asymptotic relative efficiency of the proposed multivariate Spearman type tests with respect to the Gaussian tests. These asymptotic results are confirmed by a Monte-Carlo investigation. 1 Introduction. Much attention has been devoted in the past ten years to multivariate extensions of the classical, univariate theory of ran and signed-ran tests; see Oja 1999) for an insightful review of the abundant literature on this subject. Emphasis in this literature however has been put, essentially if not exclusively, on invariance and robustness rather than on asymptotic optimality issues. Hallin and Paindaveine 22) recently showed that invariance and asymptotic efficiency, in this context, are not irreconciliable objectives, and that locally asymptotically optimal procedures in the Le Cam sense), in the context of multivariate one-sample location models, can be based on the robust tools that have been developed in the area, such as Randles interdirections, and pseudo-)mahalanobis rans. Although the need for non-gaussian and robust procedures in multivariate time series problems is certainly as strong as in the context of independent observations, little has been done to extend this strand of statistical investigation beyond the traditional case of linear models with Research supported by a P.A.I. contract of the Belgian federal Government and an A.R.C. contract of the Communauté française de Belgique. 1

2 independent errors. A similar phenomenon has been observed in the development of univariate ran-based methods which, for quite a long period, were restricted to nonserial models involving independent observations) despite the serial nature of some of the earliest nonparametric and ran-based procedures, such as the tests based on runs or turning points. A ran-based test for randomness against multivariate serial dependence of the ARMA type was proposed by Hallin, Ingenblee, and Puri 1989), but suffers the same lac of invariance with respect to affine transformations as all methods based on componentwise ranings the reader is referred to the monograph by Puri and Sen 1971) for an extensive description, in the nonserial case, of this approach). The purpose of this paper is to attac the same problem from an entirely different perspective, in the light of the nonserial contributions of Möttönen et al. 1995, 1997, and 1998), Hettmansperger et al. 1994, 1997), Randles 1989), Peters and Randles 199), or Jan and Randles 1994), to quote only a few. Basically, we show that tests of randomness that are locally and asymptotically optimal, in the Le Cam sense, against ARMA dependence can be based on the tools developed in some of these papers, namely, interdirections and pseudo-mahalanobis rans, which jointly provide a multivariate extension of univariate) signed rans. Testing for randomness or white noise of course is the simplest of all serial problems one can thin of. In view of the crucial role of white noise in most time series models, it is also an essential one, as most hypothesis testing problems, in time-series analysis, more or less reduce to testing that some transformation possibly involving nuisances) of the observed process is white noise. The paper is organized as follows. Section 2 introduces the main technical assumptions, states the LAN result to be used throughout, and provides the locally and asymptotically maximin parametric procedures for the problem under study. Section 3 presents the class of multivariate serial ran statistics to be used in the sequel. Section 4 provides the asymptotic relative efficiencies of the proposed procedures with respect to their Gaussian counterparts, and multivariate serial extensions of two classical results by Chernoff-Savage 1958) and by Hodges-Lehmann 1956), respectively. In Section 5, we investigate finite-sample performance via a Monte-Carlo study. Proofs are concentrated in the appendix. 2 Local asymptotic normality, and parametric optimality. 2.1 Main assumptions. The testing procedures we are proposing here constitute a multivariate generalization of the classical univariate signed-ran methods. So, they require some symmetry condition : throughout, we will restrict our attention to elliptically symmetric white noise. Let X n) := X n) ) 1,..., Xn) n be an observed -dimensional series of length n. Denote by Σ a symmetric positive definite matrix, and by f : R + R+ a nonnegative function. We say that X n) is a finite realization of an elliptic white noise process with scatter matrix Σ and radial density f, if and only if its probability density at x = x 1,..., x n ) R n is of the form n t=1 fx t ; Σ, f), where fx 1 ; Σ, f) := c,f 1 det Σ) 1/2 f x 1 Σ ), x 1 R. 1) Here, x Σ := x Σ 1 x) 1/2 denotes the norm of x in the metric associated with Σ. The constant c,f is the normalization factor ω µ 1;f ) 1, where ω stands for the 1)-dimensional 2

3 Lebesgue measure of the unit sphere S 1 R, and µ l;f := r l fr) dr. The following assumption is thus required on f : Assumption A1). The radial density f satisfies f > a.e. and µ 1;f <. Note that the scatter matrix Σ in 1) needs not be a multiple of) the covariance matrix of the observations, which may not exist, and that f is not, strictly speaing, a probability density; see Hallin and Paindaveine 22) for a discussion. Moreover, Σ and f are identified up to an arbitrary scale transformation only. More precisely, for any a >, letting Σ a := a 2 Σ and f a r) := far), we have fx; Σ, f) = fx; Σ a, f a ). This will not be a problem in the sequel, where estimated scatter matrices are always defined up to a positive factor a see Assumption A4) below). Under A1), f r) := µ 1;f ) 1 r 1 fr) is a probability density over R +. More precisely, f is the density of X, where X is a random -vector with density f. ; I, f) I denotes the -dimensional identity matrix). In the sequel, we denote by F the distribution function associated with f. Whenever local asymptotic normality is needed, or when Gaussian procedures are to be used, or, more generally, whenever finite second-order moments are required, Assumption A1) has to be strengthened into Assumption A1 ). Same as Assumption A1), but we further assume that µ +1;f <. Null hypotheses of elliptical white noise will be tested against alternatives of multivariate ARMAp, q) dependence. As usual, denoting by L the lag operator, consider the multivariate ARMAp, q) model AL) X t = BL) ε t, 2) where AL) := I p A il i and BL) := I + q B il i for some real matrices A 1,..., A p, B 1,..., B q such that A p B q. Writing θ := vec A 1 ),..., vec A p ), vec B 1 ),..., vec B q ) ) R 2 p+q), we denote by H n) θ, Σ, f) the hypothesis under which the observed n-tuple X n) 1,..., Xn) n is a finite realization of some solution of 2), where {ε t, t Z} is elliptic white noise with scatter f H n) θ, Σ, f) parameter Σ and radial density f. Writing H n) θ,.,.) for the hypothesis Σ where unions are taen over the largest sets that are compatible with the assumptions), our objective is to test H n),.,.) against θ H n) θ,.,.). Under some further regularity assumptions on the radial density f, the local asymptotic normality for fixed Σ and f) of ARMA models follows from a more general result in Garel and Hallin 1995). The elliptic version of these assumptions, which essentially guarantee the quadratic mean differentiability of f 1/2, taes the following form see Hallin and Paindaveine 22) for a discussion). Considering the space L 2 R +, µ l) of all functions that are square-integrable w.r.t. the Lebesgue measure with weight r l on R + i.e. the space of measurable functions h : R+ R satisfying [hr)]2 r l dr < ), denote by W 1,2 R +, µ l) the subspace containing all functions of L 2 R +, µ l) admitting a wea derivative that also belongs to L 2 R +, µ l). The following assumption is strictly equivalent to the quadratic mean differentiability of f 1/2 see Hallin and Paindaveine 22), Proposition 1), but has the important advantage of involving univariate quadratic mean differentiability only. 3

4 Assumption A2). The square root f 1/2 of the radial density f is in W 1,2 R +, µ 1); denote by f 1/2 ) its wea derivative in L 2 R +, µ 1), and let ϕ f := 2 f 1/2 ) f 1/2. Assumption A2) guarantees the finiteness of the radial Fisher information I,f := µ 1;f ) 1 [ϕ f r)] 2 r 1 fr) dr. Whenever rans and ran-based statistics come into the picture, they will be defined from score functions 1 and 2 ; suitable score functions will be required to satisfy the following conditions : Assumption A3). The score functions l : ], 1[ R, l = 1, 2, are continuous, satisfy 1 lu) 2+δ du < for some δ >, and can be expressed as differences of two monotone increasing functions. The score functions yielding locally and asymptotically optimal procedures, as we shall see, are 1 of the form 1 := ϕ f F and 1 2 := F, for some radial density f. Assumption A3) then taes the form of an assumption on f : Assumption A3 ). The radial density f satisfies Assumption A2), and µ +1+δ;f < for some δ >. The associated function ϕ f is continuous, satisfies ϕ f r) 2+δ r 1 f r) dr < for some δ >, and can be expressed as the difference of two monotone increasing functions. The assumption of being the difference of two monotone functions, which characterizes the functions with bounded variation, is extremely mild. In most cases f normal, double exponential,... ), ϕ f itself is monotone increasing, and, without loss of generality, this will be assumed to hold for the proofs. The multivariate t-distributions considered in the sequel however are an example of non-monotone score functions ϕ f satisfying Assumption A3). Finally, the matrix Σ in practice is never nown, and has to be estimated from the observations. We assume that a sequence of statistics Σ n) is available, with the following properties. Assumption A4). The sequence Σ n) is invariant under permutations and reflections with respect to the origin in R of the observations; n Σ n) a Σ) is O P 1) as n under H n), Σ, f) for some a >. The corresponding pseudo-mahalanobis distances X n) i Σ n) ) 1 X n) i ) 1/2 are quasi-affine-invariant in the sense that, if Y n) i = MX n) i for all i, where M is an n) n) arbitrary non-singular matrix, denoting by Σ X and Σ Y the estimators computed from X n) 1,..., Xn) n ) and Y n) 1,..., Y n n) ), respectively, we have Y n) i Σ n) Y ) 1 Y n) i ) 1/2 = d X n) i Σ n) X ) 1 X n) i ) 1/2, for some positive scalar d that may depend on M and the sample X n) 1,..., Xn) n ). In the sequel, we write Σ and X i instead of Σ n) and X n) i. Note that quasi-affine-invariance of pseudo-mahalanobis distances implies strict affine-invariance of their rans. Under Assumption A1 ), f has finite second moments, and the empirical covariance matrix n 1 n X i X i satisfies Assumption A4). The resulting pseudo-mahalanobis distances are then equivalent to the classical ones, and are of course strictly affine-invariant. However, if as in Assumption A1)) no assumption is made about the moments of the radial density, the empirical covariance matrix may not be root-n consistent. Other affine-equivariant estimators of scatter then have to be considered, such as the one proposed by Tyler 1987). For the -dimensional 4

5 sample X 1, X 2,..., X n ), this estimator is defined as := Cn) Tyl ) C n) Tyl ) 1, where C n) Tyl is the unique for n > 1)) upper triangular matrix with positive diagonal elements and a 1 in the upper left corner that satisfies 1 n n Σ n) Tyl n) C Tyl X ) n) i C Tyl X ) i C n) Tyl X i C n) Tyl X = 1 i I. 3) See Tyler 1987) and Randles 2) for the invariance and consistency properties of this empirical measure of scatter. Unless otherwise stated, we always use this estimator in the sequel under the notation Σ n) or Σ) to compute pseudo-mahalanobis distances. 2.2 Local asymptotic normality LAN). Writing A n) L) := I p i L i and B n) L) := I + q i L i, consider the sequence of experiments associated with the sequence of) stochastic difference equations n 1/2 A n) A n) L) X t = B n) L) ε t, n 1/2 B n) where the parameter vector τ n) := vec A n) 1 ),..., vec A n) p ), vec B n) 1 ),..., vec B n) q ) ) R 2 p+q) is such that sup n τ n) ) τ n) <. With the notation above, this sequence of parameters characterizes a sequence of local alternatives H n) n 1 2 τ n), Σ, f). Let d t Σ) = d n) t Σ) := X t Σ and U t Σ) = U n) t Σ) := Σ 1/2 X t /d t Σ), where Σ 1/2 denotes an arbitrary symmetric square root of Σ 1. Writing ϕ f for 2 Df 1/2 )/f 1/2, where Df 1/2 denotes the quadratic mean gradient of f 1/2, define, as in Garel and Hallin 1995), the f-cross covariance matrix of lag i as Γ n) i;σ,f := n i) 1 n t=i+1 = n i) 1 Σ 1/2 ϕ f X t ) X t i n t=i+1 ϕ f d t Σ)) d t i Σ) U t Σ)U t iσ) Σ 1/2. The maximum lag we will need is π := maxp, q). Considering the 2 π 2 p + q) matrix M := D n) i := I 2 p I 2 q 2 π p) 2 p) 2 π q) 2 q) let d n) := M τ n). Note that d n) = vec D n) 1 ),..., vec D n) π ) ) R 2π, where A n) i + B n) i if i = 1,..., minp, q) A n) i if i = q + 1,..., π B n) i if i = p + 1,..., π. When A n) = A, B n) = B, hence τ n) = τ, are constant sequences, we also write D i instead of D n) i. Local asymptotic normality, for given Σ and f, then taes the following form. ), 5

6 Proposition 1 Assume that Assumptions A1 ) and A2) hold. Then, the logarithm L n) n 1 2 τ n) /;Σ,f of the lielihood ratio associated with the sequence of local alternatives H n) n 1 2 τ n), Σ, f) with respect to H n), Σ, f) is such that L n) n 1 2 τ n) /;Σ,f X) = dn) ) n) Σ,f 1 2 dn) ) Γ Σ,f d n) + o P 1), as n, under H n), Σ, f), with the central sequence where and n) Σ,f := n) Σ,f ) 1,..., n) Σ,f ) π ), n) Σ,f ) i := n i) 1/2 vec Γ n) i;σ,f, Γ n) Σ,f := µ +1;f I,f 2 µ 1;f I π Σ Σ 1 ) Moreover, n) Σ,f, still under Hn), Σ, f), is asymptotically N 2 π, Γ Σ,f ). Proof. The result is a very particular case of the LAN result in Garel and Hallin 1995). The required quadratic mean differentiability of x f 1/2 x ) follows from Assumption A2) see Hallin and Paindaveine 22). Note that the causality and invertibility conditions are trivially satisfied. One can also verify that LAN in this purely serial model only requires the finiteness of radial Fisher information, rather than the stronger condition 1 [ϕ 1 f F u))]4 du < as for the more general model considered in Garel and Hallin 1995), which includes a linear trend. 2.3 Parametric optimality. The above LAN property straightforwardly allows for building locally and asymptotically optimal testing procedures, under fixed Σ and f, for the problem under study. More precisely, the test that rejects the null hypothesis H n), Σ, f ) whenever Q par Σ,f := n) Σ,f Γ 1 Σ,f n) Σ,f = 2 µ 1;f µ +1;f I,f π n i) 1 n ϕ f d t Σ)) ϕ f d t Σ)) 4) d t i Σ) d t i Σ) U tσ)u t Σ) U t iσ)u t i Σ) exceeds the α-upper quantile χ 2 2 π,1 α of a chi-square distribution with 2 π degrees of freedom, is locally asymptotically maximin, at asymptotic level α, for H n), Σ, f ) against alternatives of the form θ Hn) θ, Σ, f ) : see Le Cam 1986), Section This procedure is of course highly parametric; in particular, it is only valid if the underlying radial density is f. This can be improved by replacing the exact asymptotic variance of n) Σ,f with an estimation. Consider, for instance, the Gaussian case f z) = φz) := exp z 2 /2), and the test statistic where Q par N 1 := n) I,φ ˆΓ I,φ n) I,φ, 5) ˆΓ I,φ := I π 6 ˆΓ 1) I,φ,

7 with ˆΓ 1) n I,φ := n 1) 1 vec X t X t 1) vec Xt X t 1)). t=2 Note that Q par N is affine-invariant. The ergodic theorem see e.g., Hannan 197), Theorem 2, p. 23) yields ˆΓ I,φ = Γ I,φ;f + o P 1) under H n), I, f), where, denoting by U a random variable uniform over ], 1[, Γ I,φ;f := 1 2 E2 [ 1 F U))2 ] I 2 π is the asymptotic variance of n) I,φ under Hn), I, f), so that Q par N is asymptotically equivalent, under H n),., φ) and under contiguous alternatives, to the Gaussian version of the parametric statistic 4). The following result then is easy to derive. Proposition 2 Assume that Assumptions A1 ) and A2) hold. Consider the test φ n) N that rejects the null hypothesis H n),.,.) whenever Q par N exceeds the α-upper quantile χ2 2 π,1 α of a chi-square distribution with 2 π degrees of freedom. Then, i) Q par N is asymptotically chi-square with 2 π degrees of freedom under H n),.,.), and asymptotically noncentral chi-square, still with 2 π degrees of freedom but with noncentrality parameter 1 2 E2 1 [ F U) ϕ 1 f F π U))] tr Σ 1/2 D i Σ 1 D i Σ 1/2), here and in the sequel, U stands for a random variable uniform over ], 1[) under H n) n 1/2 τ, Σ, f). ii) The sequence of tests φ n) N has asymptotic level α. iii) The sequence of tests φ n) N is locally asymptotically maximin, at asymptotic level α, for H n),.,.) against alternatives of the form θ Hn) θ,., φ). 3 Test statistics and their asymptotic distributions. 3.1 Group invariance, interdirections, and pseudo-mahalanobis rans. In this short section, we briefly review the invariance features of the problem under study justifying the invariant) statistics to be used in the sequel : interdirections, and pseudo- Mahalanobis rans. Denote by Z t Σ) = Z n) t Σ) := Σ 1/2 X t, t = 1,..., n the standardized residuals associated with the null hypothesis of randomness. Under H n), Σ,.), the vectors U t Σ) = U n) t Σ) = Z n) t Σ)/ Z n) t Σ) are independent and uniformly distributed over the Σ n) ) associated with the unit sphere S 1. The notation Ẑt will be used for the residuals Z n) t estimator Σ n) considered in A4). The interdirection c n) associated with the pair Ẑt, t, t Ẑ t ) in the n-tuple of residuals Ẑ1,..., Ẑn is defined Randles 1989) as the number of hyperplanes in R passing through the origin and 7

8 1) out of the n 2) points Ẑ1,..., Ẑt 1, Ẑt+1,..., Ẑ t 1, Ẑ t+1,..., Ẑn, that are separating Ẑt and Ẑ t : obviously, cn) n 2 t, t 1). Interdirections are invariant under linear transformations, so that they indifferently can be computed from the residuals Ẑt as from the residuals Z t Σ), or from the observations X t themselves. Finally, let p = t, t pn) := c n) t, t t, t / n 2 1) for t t, and p t,t :=. Interdirections provide affine-invariant estimations of the Euclidean angles between the unobserved residuals Z t Σ), that is, they estimate the quantities π 1 arccos U tσ)u t Σ)). The following consistency result is proved in Hallin and Paindaveine 22), by using a U-statistic representation. Lemma 1 Let X 1, X 2,...) be an i.i.d. process of -variate random vectors with spherically symmetric density. For any fixed v and w in R, denote by αv, w) := arccosv w/ v w )) the angle between v and w, and by c n) v, w) the interdirection associated with v and w in the sample X 1, X 2,..., X n. Then, c n) v, w)/ n 2 1) converges in quadratic mean to π 1 αv, w) as n. The rans of pseudo-mahalanobis distances between the observations and the origin in R are the other main tool used in this paper. Let R t Σ) = R n) t Σ) denote the ran of d t Σ) among the distances d 1 Σ),..., d n Σ); write ˆR t and ˆd t for R t Σ) and d t Σ), respectively, where Σ is the estimator considered in A4). It will be convenient to refer to ˆR t as the pseudo-mahalanobis ran of X t. The following result is proved in Peters and Randles 199). Lemma 2 Peters and Randles 199) For all t N, under H n), Σ,.). ˆRt R t Σ)) /) is o P 1) as n, For each Σ and n, consider the group of transformations G n) Σ = {Gn) g }, acting on R ) n, such that GX 1,..., X n ) := gd 1 Σ)) Σ 1/2 U 1 Σ),..., gd n Σ)) Σ 1/2 U n Σ)), where g : R + R + is continuous, monotone increasing, and such that g) = and lim r gr) =. The group G n) Σ is a generating group for the submodel H n), Σ,.). Interdirections clearly are invariant under the action of G n) Σ, and so are the rans R tσ). Lemma 2 thus entails the asymptotic invariance of the pseudo-mahalanobis rans ˆR t /). Another group of interest is the group of affine transformations acting on R ) n more precisely, the group G n) = {G n) M }, where M >, and G MX 1,..., X n ) := MX 1,..., MX n ). This group of affine transformations is a generating group for the submodel H n),., f); unlie the componentwise rans considered in Hallin, Ingenblee, and Puri 1989), interdirections and pseudo-mahalanobis rans clearly are invariant under this group see Assumption A4)). For = 1, interdirections or more precisely, the cosines cosπp t, t )) reduce to signs, and pseudo-mahalanobis rans to the rans of absolute values. The statistics we are considering in the next section thus are a multivariate generalization of the signed ran statistics of the serial type considered in Hallin and Puri 1991). 8

9 3.2 A class of statistics based on interdirections and pseudo-mahalanobis rans. Denoting by 1 and 2 : ], 1[ R two score functions, consider quadratic test statistics of the form Q n) := 2 π E[1 2U)]E[2 2 U)] n i) 1 n ˆRt ) ) ˆR t 1 1 6) ˆRt i ) ) ˆR t i 2 2 cosπp ) cosπp t, t t i, t i ), where U is uniform over ], 1[. The form of Q n) of course is closely related to that of 4). Of course, exact variances σ n) )2 can be substituted, in 6), for the asymptotic ones; see Hallin and Puri 1991, page 12) for an explicit form of σ n) )2. Specific choices of the scores 1 and 2 yield a variety of statistics generalizing some wellnown univariate ones. If we let 1 u) = 2 u) = 1 for all u, 6) reduces to the quadratic multivariate sign test statistic Q n) S := 2 π n i) 1 n cosπp ) cosπp t, t t i, t i ), 7) a serial version of Randles multivariate sign test statistic Randles 1989) but also a multivariate extension of the quadratic generalized runs statistics proposed in Dufour, Hallin, and Mizera 1998). Linear scores 1, 2 yield Q n) SP := 92 ) 4 π n i) 1 n ˆR t ˆR t ˆR t i ˆR t i cosπp t, t ) cosπp t i, t i ), 8) a multivariate version of the Spearman-autocorrelation type test statistics. This is the serial version of Peters and Randles Wilcoxon-type test statistic see Peters and Randles 199). When local asymptotic optimality is required under some specified radial density f, the adequate score functions 1 and 2 are 1 = J,f := ϕ f Q n) f := 2 µ 1;f µ +1;f I,f Particular cases are π n i) 1 n ˆRt J,f F 1 ) J,f and 2 = ˆR t ) F 1, yielding F 1 ˆRt i ) ) F 1 ˆR t i cosπp ) cosπp t, t t i, t i ). the van der Waerden test statistic, associated with Gaussian densities f r) := φr)), Q n) π vdw := n n i) 1 Ψ 1 ˆRt ) ) Ψ 1 ˆR t 1) Ψ 1 ˆRt i ) ) Ψ 1 ˆR t i cosπp ) cosπp t, t t i, t i ), where Ψ stands for the chi-square distribution function with degrees of freedom, or 9 9)

10 the Laplace statistic, Q n) L := + 1 π n i) 1 n F 1 ˆRt i ) ) F 1 ˆR t i cosπp ) cosπp t, t t i, t i ), 11) associated with double-exponential radial densities f r) := exp r); F r) = Γ,r) Γ), where Γ stands for the incomplete gamma function, defined by Γ, r) := r s 1 exp s) ds). Note that, as usual in time series problems, the Laplace statistic 11) does not coincide with the sign test statistic 7). Before investigating the asymptotic behavior of 6) and describing the associated asymptotic tests, let us point out that, for small samples, Q n) allows for particularly pleasant permutational procedures. Exact permutational critical values indeed can be obtained from enumerating the 2 n possible values s 1 X 1,..., s n X n s := s 1,..., s n ) { 1, 1} n ), which are equally probable under the null. What maes these exact procedures so pleasant is that the at most) 2 n corresponding possible values of the test statistics can be based on a unique evaluation of the interdirections and the pseudo-)mahalanobis rans. Indeed, denoting by p t, t s) the interdirection associated with the pair s t X t, s t X t ) in the n-tuple s 1X 1,..., s n X n, it can be easily checed that cosπ p s)) = s t, t t s t cosπ p t, t ). 12) It follows that the value of the test statistic Q n) s) computed at s 1X 1,..., s n X n is simply 2 π E[1 2U)]E[2 2 U)] n i) 1 n s t s t s t i s t i ˆRt ) ) ˆR t 1 1 ˆRt i ) ) ˆR t i 2 2 cosπp ) cosπp t, t t i, t i ). 3.3 Asymptotic behavior of statistics based on interdirections and pseudo- Mahalanobis rans. We now turn to the asymptotic behavior of Q n) as n, both under the null hypothesis of randomness as under local alternatives of ARMA dependence. Proofs are given in the Appendix. The following lemma provides an asymptotic representation result for Q n). Lemma 3 Assume that A1) through A4) hold. Then, under H n), Σ, f), as n, where Q n) ;Σ,f := 2 E[ 2 1 U)]E[2 2 U)] π Q n) n) = Q ;Σ,f + o P1) n i) 1 n 1 F d t Σ))) 1 F d t Σ))) 2 F d t i Σ))) 2 F d t i Σ))) U t Σ)U t Σ) U t i Σ)U t iσ). 1

11 Let D ; f) := 1 1 u) F u) du and C ; f) := 1 u) J,fu) du, where denotes some score function defined over ], 1[. When is a density over R + rather than a score function, 1 we write D f 1, f 2 ) and C f 1, f 2 ) for D F 1 ; f 2) and C J,f1 ; f 2 ) respectively; for simplicity, we also write C f) and D f) instead of C f, f) and D f, f). We then have the following results. Proposition 3 Assume that A1) through A4) hold. Then, under H n),.,.), Q n) is asymptotically chi-square with 2 π degrees of freedom, as n. Under H n) n 1/2 τ, Σ, f), and provided that A1) is reinforced into A1 ), Q n) is asymptotically noncentral chi-square, still with 2 π degrees of freedom but with noncentrality parameter 1 D 2 2; f) C 2 1; f) 2 E[2 2U)] E[1 2U)] π tr Σ 1/2 D iσ 1 D i Σ 1/2). Proposition 4 Assume that A1) through A4) hold. Consider the test φ n) rejects the null hypothesis H n),.,.) whenever Q n) resp. Qn) χ 2 2 π,1 α of a chi-square distribution with 2 π degrees of freedom. Then, i) the sequences of tests φ n) and φn) f have asymptotic level α; resp. φn) f ) that f ) exceeds the 1 α) quantile ii) provided that A1) is reinforced into A1 ), the sequence of tests φ n) f is locally asymptotically maximin, at asymptotic level α, for H n),.,.) against alternatives of the form θ Hn) θ,., f ). 4 Asymptotic performances. 4.1 Asymptotic relative efficiencies. We now turn to asymptotic relative efficiencies of the tests φ n) with respect to the Gaussian test φ n) N. For the sae of simplicity, we drop useless superscripts, writing φ, φ N, etc. for φ n), φ n) N, etc. Proposition 5 Assume that A1 ) through A4) hold. Then, the asymptotic relative efficiency of φ with respect to the Gaussian test φ N, under radial density f, is For the f -scores procedures, this yields ARE ser),f φ /φ N ) = 1 2 D 2 2; f) E[ 2 2 U)] C 2 1; f) E[ 2 1 U)]. ARE ser),f φ f /φ N ) = 1 D 2f, f) 2 D f ) C 2 f, f) C f ). These ARE values directly follow as the ratios of the corresponding noncentrality parameters in the asymptotic distributions of φ φ f ) and φ N under local alternatives see Propositions 2 and 3). 11

12 4.2 A generalized Chernoff-Savage result. Denote by ARE loc),f φ f /φ N ) the asymptotic relative efficiency, under radial density f, for the multivariate one-sample location problem, of the generalized signed-ran test associated with radial density f, with respect to the corresponding Gaussian procedure φ N, namely the Hotelling test T 2. Then Hallin and Paindaveine 22) show that ARE loc),f φ f /φ N ) = D f) C 2f, f) 2 C f ). It directly follows from the Cauchy-Schwarz inequality that ARE ser),f φ f /φ N ) ARE loc),f φ f /φ N ), 13) with equality iff the radial densities f and f are of the same density type, that is, iff fr) = λf ar) for some λ, a >. Lie in the univariate case see Hallin 1994), the van der Waerden procedure is uniformly more efficient than the Gaussian procedure. More precisely, we show the following generalization of the serial Chernoff-Savage result of Hallin 1994). Proposition 6 Denote by φ vdw and φ N the van der Waerden test, based on the test statistic 1), and the Gaussian test based on 5), respectively. For any f satisfying Assumptions A1 ) and A2), ARE ser),f φ vdw /φ N ) 1, where equality holds if and only if f is normal. Some numerical values of ARE ser),f φ vdw /φ N ) are provided in Table 2, where it appears that the advantage of the van der Waerden procedure over the Gaussian parametric grows with the dimension of the observation, and with the importance of the tails of underlying densities an ARE value of is reached for a 1-variate Student density with 3 degrees of freedom). Note that the multivariate extension of the Chernoff-Savage 1958) theorem presented in Hallin and Paindaveine 22) for the location problem appears, via inequality 13), as a corollary of Proposition 6. Interestingly enough, although Proposition 6 is stronger than its location model counterpart, its proof is simpler than the direct proof of its weaer) location model counterpart see Appendix B for the proof). 4.3 A multivariate serial version of Hodges and Lehmann s.864 result. Although it is never optimal there is no density f such that Q f coincides with Q SP ), the Spearman-type procedure φ SP, based on 8), exhibits excellent asymptotic efficiency properties, especially for relatively small dimensions. To show this, we extend the.856 result of Hallin and Tribel 2) the serial analog of Hodges and Lehmann 1956) s celebrated.864 result ) by computing, for any dimension, the lower bound for the asymptotic relative efficiency of φ SP with respect to the Gaussian procedure φ N. More precisely, we prove the following result see Appendix B for the proof). 12

13 Proposition 7 Denote by φ SP the Spearman procedure based on the test statistic 8). Then, denoting by J r the first-ind Bessel function of order r, { } { cr) := min x > x J r x)) = = min x > x J r+1x) = r + 1 }, J r x) 2 the lower bound for the asymptotic relative efficiency of φ SP with respect to φ N is inf f AREser),f φ SP /φ N ) = 9 2 c 2 ) 4 2 1/2) c 4, 14) 2 1/2) where the infimum is taen over all radial densities f satisfying Assumptions A1 ) and A2). Some numerical values are presented in Table 1 and Figure 1, along with the corresponding bounds for the location model. Note that the sequence of lower bounds 14) is monotonically decreasing for 2; as the dimension tends to infinity, it tends to 9/16 = The reader is referred to the proof of Proposition 7 in Appendix B for an explicit form, and a graph, of the densities achieving the infimum cf. Figure 2). 4.4 Asymptotic performance under heavy-tailed densities. The tests we are proposing can be expected to exhibit better performances than the traditional Gaussian ones under heavy-tailed densities. In order to evaluate the impact of heavy tails on asymptotic performances, we consider the particular case of a multivariate Student density with ν degrees of freedom. Recall that a -dimensional random vector X is multivariate Student with ν degrees of freedom if and only if there exist a vector µ R and a symmetric positive definite matrix Σ such that the density of X can be written as Γ + ν)/2) πν) /2 Γν/2) det Σ) 1/2 f ν x µ Σ ), with f ν r) := 1 + r 2 /ν) +ν)/2. Fixing ν > 2, consider the test φ fν associated with the radial density f ν. Since ϕ fν r) = + ν )r/ν + r 2 ), and since the distribution of X 2 / under H n), I, f ν ) is Fisher-Snedecor with and ν degrees of freedom, the test statistic Q fν is + ν ) + ν + 2)ν 2) ν π n i) 1 n T t ν + T 2 t T t ν + T 2 t T t i T t i cosπp t, t ) cosπp t i, t i ) where, denoting by G,ν the Fisher-Snedecor distribution function and ν degrees of freedom), T t := G 1 ˆRt ),ν. Table 2 reports the AREs of the tests φ f5, φ f8, and φ f15, as well as those of the van der Waerden tests φ vdw, with respect to the Gaussian test φ N, under -variate Student densities with various degrees of freedom ν, including the Gaussian density obtained for ν =. Inspection of Table 2 reveals that φ fν respectively, φ vdw ), as expected, performs best when the underlying density itself is Student with ν degrees of freedom respectively, normal). In that case, the AREs for the serial and nonserial cases coincide. All tests however exhibit rather good performance, 13

14 particularly under heavy tailed densities. Note that the van der Waerden test performs uniformly better than the Gaussian test, which provides an empirical confirmation of Proposition 6. Since D f ν ) = ν /ν 2) and C f ν ) = + ν )/ + ν + 2), we obtain that ARE ser),f ν [φ fν /φ N ] = + ν ) ν + ν + 2)ν 2), 15) a quantity that increases with, and tends to ν /ν 2) as. The advantage of φ fν over the Gaussian test thus increases with the dimension of the observations. Table 3 presents some of these limiting ARE values. 4.5 The multivariate sign and Spearman tests. In view of their simplicity, the multivariate sign test against randomness S) and the multivariate Spearman SP) test, which are the serial counterparts of Randles multivariate sign test and Peters and Randles Wilcoxon-type multivariate signed-ran test, respectively, deserve special interest. Table 4 provides the asymptotic relative efficiencies, still with respect to the Gaussian tests, and under the same densities as in Table 2, of these tests, based on the statistics 7), 8) and 11), respectively. Because of its relation to the sign test S), the multivariate Laplace test L) also has been included in this study. For the multivariate sign test S), the following closed-form expressions are obtained : ARE ser),f ν [S/φ N ] = 16 2 ν 1) 2 ARE ser),φ [S/φ N ] = 4 2 ) ) Γ +1 2 Γ ν+1 2 ) Γ Γ ν ) 2 2 ) Γ +1 2 ) Γ 2 4 4, 16). 17) The asymptotic relative efficiencies of the corresponding one-sample location tests with respect to Hotelling s test, namely Randles multivariate sign test S loc) ) and Peters and Randles Wilcoxon-type multivariate signed-ran test W ), are also provided in Table 4, thus allowing for a comparison between the serial and the nonserial cases. 5 Simulations. The following Monte-Carlo experiment was conducted in order to investigate the finite-sample behavior of the tests proposed in Section 3 for = 2 : N = 2, 5 independent samples ε 1,..., ε 4 ) of size n = 4 have been generated from bivariate standard Student densities with 3, 8, and 15 degrees of freedom, and from the bivariate standard normal distribution. The simulation of bivariate Student variables ε i was based on the fact that for ν degrees of freedom; = d stands for equality in distribution) ε i = d Z i / Y i /ν, where Z i N 2, I 2 ) and Y i χ 2 ν are independent. Autoregressive alternatives of the form X t ma) X t 1 = ε t, X =, 18) 14

15 ).5.2 were considered, with A =, and m =, 1, 2, 3. For each replication, the.1.4 following seven tests were performed at nominal asymptotic probability level α = 5% : the Gaussian test φ N, φ f5, φ f8, φ f15, φ vdw, the sign test for randomness S), and the Spearman type test SP). Tyler s estimator of scatter was used whenever pseudo-mahalanobis rans had to be computed. The estimator was obtained via the iterative scheme described in Randles 2). Iterations were stopped as soon as the Frobenius distance between the left and right hand sides of 3) fell below 1 6. Rejection frequencies are reported in Table 5. Note that the corresponding standard errors are for N = 2, 5 replications).44,.8, and.1 for frequencies size or power) of the order of p =.5 p =.95), p =.2 p =.8), and p =.5, respectively. All tests apparently satisfy the 5% probability level constraint a 95%-confidence interval has approximate half-length.1). Power ranings are essentially consistent with the ARE values given in Tables 2 and 4. For instance, under Gaussian densities, the powers of the φ fν tests are increasing with ν, as expected, whereas the asymptotic optimality of φ fν under the Student distribution with ν degrees of freedom is confirmed. 6 Appendix : proofs. 6.1 Appendix A : Proofs of Section 3. The main tas here consists in proving Lemma 3. Let us first establish the following serial result about interdirections. Lemma 4 Let i {1,..., π} and s, s, t, t {i + 1,..., n} be such that at least one out of the eight indices t i, t i, t, t, s i, s i, s, and s is distinct of the seven other ones. Let g : X gx 1,..., X n ) be even in all its arguments. Then, letting C t, t;i := cosπp t, t ) cosπp t i, t i ) U t I )U t I ) U t i I )U t i I ), 19) we have E[gX) C s, s;i C t, t;i ] = under Hn), I,.), provided that this expectation exists. Similarly, defining D t, t;i := U ti )U t I ) U t ii )U t i I ), E[gX) D t, t;i ] = for t t under H n), I,.), provided that this expectation exists Proof. Define b t := sgnx t1 ) and Y t := b t X t, where sgnz) := I[z > ] I[z < ] stands for the sign function and X t1 denotes the first component of X t. Under H n), I,.), the b t s are i.i.d. Bernoulli with P [b t = ±1] = 1/2, and are independent of the Y t s. Denote by p Y t, t the interdirection associated with Y t, Y t ) within the Y-sample; also write U t and U Y t for U t I ) = X t / X t and Y t / Y t, respectively. Finally, let C Y t, t;i := cosπpy t, t ) cosπpy t i, t i ) U Y t ) U Ỹ U Y t t i ) U Ỹ t i. Without loss of generality, suppose that t is distinct of t i, t i, t, s i, s i, s, and s. Then, using the fact that gx) is a function of the Y m s, [ ] E gx) C s, s;i C t, t;i X 1,..., X t 1, Y t, X t+1,..., X n = gx) b s b s b s i b s i C Y s, s;i b t b t i b t i CY t, t;i E [ b t Y t ] =, 15

16 in view of the symmetry properties 12) of interdirections. The second assertion is proved in the same way. We are now ready to prove Lemma 3. Proof of Lemma 3. Without loss of generality, we may assume that Σ = I. In this proof, we will write d t, R t and U t for d t I ), R t I ) and U t I ), respectively. Decompose Q n) n) Q ;I,f into 2 π E[1 2U)]E[2 2 U)] T n) 1;i + T n) 2 ), where T n) 1;i := n i) 1 n ˆRt ) 1 1 ˆR t ) ˆRt i ) ) ˆR t i 2 2 cosπp ) cosπp t, t t i, t i ) U tu t U t iu t i ), and T n) 2 := π n n i) 1 ˆRt ) 1 1 ˆR t ) ˆRt i ) ) ˆR t i F d t )) 1 F d t )) 2 F d t i )) 2 F d t i )) ) U t U t U t i U t i. Let us show that, under H n), I, f) throughout this proof, all convergences and mathematical expectations are taen under H n), I, f)), there exists s > such that T n) 1;i and T n) L s 2 for all i as n. Slutzy s classical argument then concludes the proof. Let us start with T n) 2. Define where and Note that S n) Tn) T n) ;f ) i := n i) 1/2 S n) ) i := n i) 1/2 Ŝn) ) i := n i) 1/2 ;f 2 L 2= π n c n) t;i )2 E t=i+1 T n) ;f := T n) ;f ) 1,..., Tn) ;f ) π ), S n) := S n) ) 1,..., Sn) ) π ), Ŝ n) := Ŝn) ) 1,..., Ŝn) ) π ), n t=i+1 n t=i+1 n t=i+1 vec vec 1 F d t )) 2 F ) d t i )) U t U t i, 1 Rt ) Rt i ) ) 2 U t U t i, ) ˆRt ) ˆRt i ) vec 1 2 U t U t i. [ Rt 1 16 ) Rt i ) ) ] F d t )) 2 F d t i )),

17 where c n) t;i = n i) 1/2 for all t = i + 1,..., n. Proposition 2.1 in Hallin and Puri 1991) thus implies that S n) Tn) ;f L2 = o1) as n incidentally, the same result also implies that, for all i = 1,..., π and for all t = i + 1,..., n, [ Rt ) Rt i ) ) ] 2 E F d t )) 2 F d t i )) = o1) 2) as n. Using Lemma 4, we obtain similarly Ŝn) Sn) 2 π n L 2= n i) 1 ˆRt E 1 t=i+1 Consequently, Ŝn) Sn) L 2 is o1) if ˆRt ) ˆRt i ) Rt ) ˆRt i ) 2 1 Rt ) Rt i ) 2 2. ) Rt i ) L 2 2, as n. 21) Lemma 2 establishes the same convergence as in 21), but in probability. We have seen above that 1 R t /n+1)) 2 R t i /n+1)) 1 F d t )) 2 F d t i )) tends to zero in quadratic mean, so that [ 1 R t /)) 2 R t i /))] 2 is uniformly integrable. In view of Assumption A4), the same conclusion holds for [ 1 ˆR t /n+1)) 2 ˆR t i /n+1))] 2, and 21) follows. Consequently, Ŝn) Tn) ;f L2 = o1) as n. On the other hand, T n) ;f 2 L = π E[ U)] E[2 2 is bounded. Finally, in view of Cauchy-Schwarz, T n) 2 L 1 = Ŝn) ) Ŝ n) Tn) ;f ) T n) ;f L 1 Ŝn) U)] for all n, so that the sequence Ŝn) L 2 + Tn) ;f L 2 Ŝn) Tn) ;f L2 = o1) as n. Turning to T n) n) 1;i, write T 1;i = n i) 1 n r t, t;i, where r := ˆRt ) t, t;i 1 1 ˆR t and C t, t;i is defined in 19). It follows from Lemma 4 that 2 n i)t n) 1;i 2 L 2 = E n r t, t;i = 4 E so that it is sufficient to prove that ) ˆRt i ) ) ˆR t i 2 2 C t, t;i n t< t r 2 t, t;i + n s, s=i+1 s< s n t< t,s, s) t, t) r s, s;i r t, t;i [ ) ] n i)n i 1) = 4 n 2i) E[r 2i+1,3i+1;i 2 ] + n 2i) E[r2i+1,2i+1;i ], E[ri+1,3i+1;i 2 ] = o1) and E[r2 i+1,2i+1;i ] = on) 22) 17

18 as n. Writing l;t for l ˆR t n+1 ) l = 1, 2), Hölder s inequality yields and E[r 2 i+1,3i+1;i ] E[ 1;i+1 1;3i+1 2;1 2;2i+1 2+δ] ) 2/2+δ) E [ C i+1,3i+1;i 22+δ)/δ] ) δ/2+δ), E[r 2 i+1,2i+1;i ] E[ 1;i+1 1;2i+1 2;1 2;i+1 2+δ] ) 2/2+δ) E [ C i+1,2i+1;i 22+δ)/δ] ) δ/2+δ), where δ > is as in Assumption A3). Now, Lemma 1 and the boundedness of C i+1,3i+1;i yield that E [ C i+1,3i+1;i 22+δ)/δ] = o1) as n. On the other hand, since the ˆR t s are the rans of an exchangeable n-tuple see Assumption A4)), we obtain that nn 1)n 2)n 3) ) 4 E [ 1;i+1 1;3i+1 2;1 2;2i+1 2+δ] = 1 ) 4 1 n j 1,j 2,j 3,j 4 =1 all n+1 1 j=1 1 j1 ) ) j 2+δ 1 j2 ) n+1 2 j=1 j3 ) 2 ) j 2+δ 2 j4 ) 2+δ = O1) 23) as n. Indeed, the two sums in the upper bound 23) are Riemann sums, for 1 1u) 2+δ du and 1 2u) 2+δ du, respectively, and these two integrals are finite from Assumption A3). Consequently, E [ 1;i+1 1;3i+1 2;1 2;2i+1 2+δ] = O1) as n. Woring along the same lines, one can show that E [ C i+1,2i+1;i 22+δ)/δ] = o1) and E [ 1;i+1 1;2i+1 2;1 2;i+1 2+δ] = On) as n ; 22) follows. Proof of Proposition 3. From Lemma 3, we have, under H n), Σ, f), where T n) and Q n) ;Σ,f ) i := n i) 1/2 vec = Tn) ;Σ,f ) Γ ;Σ,f ) 1 T n) T n) ;Σ,f n t=i+1 ;Σ,f + on) P 1) = Q n) := Tn) ;Σ,f ) 1,..., T n) ;Σ,f ) π), ;Σ,f + on) P 1), 1 F d t Σ))) 2 F d t i Σ))) Σ 1/2 U t Σ)U t i Σ)Σ1/2, Γ ;Σ,f := 1 2 E[2 1 U)]E[2 2 U)] I π Σ Σ 1 ). The proof of the first part of Proposition 3 follows, since T n) asymptotically N 2 π, Γ ;Σ,f ). It is also easy to see that, still under H n), Σ, f), T n) ;Σ,f are jointly multinormal, with asymptotic covariance L n) n 1 2 τ/;σ,f ;Σ,f 1 2 D 2 ; f) C 1 ; f) [I π Σ Σ 1 )] M τ; 18 under Hn), Σ, f) is and the local log-lielihood

19 Le Cam s third Lemma thus implies that T n) ;Σ,f under Hn) n 1 2 τ, Σ, f) is asymptotically N 2 π 1 D 2 2 ; f) C 1 ; f) [I π Σ Σ 1 )] M τ, Γ ;Σ,f ). This establishes the second part of Proposition Appendix B : Pitman non-admissibility of correlogram-based methods and lower bounds for the efficiency of Spearman procedures. Proof of Proposition 6. The asymptotic relative efficiency of the van der Waerden test, with respect to the Gaussian procedure, under radial density f, is ARE ser),f φ vdw /φ N ) = 1 4 D2 φ, f) E2 [ Φ 1 U) J,f U)], where, letting φr) := exp r 2 /2), Φ stands for the distribution function associated with φ r) := µ 1;φ ) 1 r 1 φr) I [r>]. Without loss of generality, we restrict ourselves to the radial densities f satisfying D φ, f) = E[ Φ 1 1 U) F U)] =. Indeed, writing f ar) := far), a >, 1 we have F a u) = a 1 1 F u) and ϕ f a r) = a ϕ f ar), so that D φ, f a ) = a 1 D φ, f) and ARE ser),f a φ vdw /φ N ) = ARE ser),f φ vdw /φ N ). Thus, we only have to show that, for any N and any f such that D φ, f) =, H f) := E[ Φ 1 U) J,f U)], with equality at f = φ only. This variational problem taes a simpler form after the following change of notation. First rewrite the functional H as H f) = = = 1 Φ 1 F r)) ϕ f r) f r) dr µ 1;f [ 1 Φ 1 F r)) f r)) r 1 dr φ Φ 1 F r))) f r) + 1 r Φ 1 F r))] f r) dr. 1 For any radial density f satisfying Assumption A1), the function R : z F Φ z) and its inverse R 1 1 : r Φ F r) are continuous monotone increasing transformations, mapping R + onto itself, and satisfying lim z Rz) = lim r R 1 r) = and lim z Rz) = lim r R 1 r) =. Similarly, any continuous monotone increasing transformation R of R + such that lim Rz) = and lim Rz) = 24) z z characterizes a nonvanishing radial density f over R + 1 via the relation R = F Φ. The variational problem just described thus consists in minimizing H R) = = [ 1 φ z) φ z) R z) + 1 Rz) z ] φ z) dz 25) [ 1 R z) + 1 Rz) z ] φ z) dz, 19

20 with respect to R : R + R + continuous and monotone increasing, under the constraints 24), since f r) = d F dr r) = φ z)/ d dz R), and f r) dr = d F r) = φ z) dz. The constraint D φ, f) = now taes the form D φ, R) = z Rz) φ z) dz =. 26) This problem is very similar to its one sample location counterpart, which is solved in Hallin and Paindaveine 22). While both problems share the same functional H R), the situation here is a lot simpler, due to the fact that the constraint 26) is linear in R while the associated constraint in the location case is quadratic in R). This allows for the following simple solution. Let R be the class of monotone increasing and continuous functions R : R + R+ such that 24) holds and D φ, R) =. Then the following lemma clearly follows from the convexity of R and H R). Lemma 5 Let R 1 belong to the class R. Then R 1 is the unique solution of the minimization problem under study if and only if H d ) := dw H 1 w)r 1 +wr 2 )) w=, for any R 2 R. Now, it is easy to chec that H ) = = so that, if R 1 z) := z for all z >, [ R 2 z) R 1 z) R 1 z))2 1)zR 2z) R 1 z)) R 1 z)) 2 ] φ z) dz [ φ R 2 z) R 1 z)) z) R 1 2 φ z)r 1 z) z))2 R 1 1)z φ z) ] z))3 R 1 z)) 2 dz, [ H ) = R 2 z) z) φ z) 1) φ z) ] dz z = = 1 µ 1;φ 1 µ 1;φ R 2 z) z) z 1 φ z) dz R 2 z) z) z 1[ ] zφz) dz = D φ, R 2 ), which equals zero if R 2 belongs to R. Lemma 5 therefore establishes the result. We now turn to the proof of the multivariate extension of the Hallin and Tribel 2) result. Proof of Proposition 7. First note that, from Proposition 5, ARE ser),f φ SP /φ N ) = 9 2 E2 [U 1 F U)] E2 [UJ,f U)]. As in the proof of Proposition 6, it is clear by considering f a r) := far), a, r > ) that we may assume that E[U U)] = 1. Therefore, the problem reduces to the variational problem F 1 inf E[UJ,fU)], with C := f C { f } 1 E[U F U)] = 1. 27) 2

21 Integrating by parts, we obtain E[UJ,f U)] = so that 27) in turn is equivalent to F r) ϕ f r) f r) dr = inf f C [ f r)) r [ f r)) r ] F r) f r) dr, ] F r) f r) dr, 28) where C is the set of all f defined on R + such that f r) dr = r f r) F r) dr = 1. 29) Substituting y, ẏ, and t for F, f, and r, respectively, the Euler-Lagrange equation associated with the variational problem 28)-29) taes the form 2t 2 ÿ 1 λ 2 t 2 )y =, 3) where λ 2 stands for the Lagrange multiplier associated with the second constraint in 29). Letting y = t 1/2 u, equation 3) reduces to the Bessel Equation t 2 λ2 ü + t u + 2 t2 2 1 ) u =, 4 so that, denoting by J r.) and Y r.) the first and second type Bessel functions of order r respectively, the general solution of 3) is given by yt) = α t 1/2 J r ωt) + β t 1/2 Y r ωt), where r := 2 1/2 and ω := λ 2 /2. Since y+) =, it is clear that β =. On the other hand, ẏ implies that ẏ is compactly supported in R +, with support [, a], say. It follows from the constraints 29) and the continuity of ẏ that the extremals of the variational problem under study are the solutions of 3) that satisfy ya) = 1, ẏa) = and a t yt)ẏt) dt = a a yt)) 2 dt = 1. 31) By using the identities xj rx) = rj r x) xj r+1 x) and xj rx) = rj r x)+xj r 1 x), it is easily checed that the constraints 31) tae the form α a 1/2 J r = 1, 32) [ α a 1/2 r + 1 ) ] J r ωa)j r +1 = 2 33) [ a α J r ) 2 + a2 2 J r +1) 2 a ] ω r J r J r +1 = a 2, 34) where all Bessel functions are evaluated in ωa. 21

22 Equations 32) and 33) allow to compute J r ωa) and J r +1ωa), with respect to a, α, ω and r. Substituting these values in 34) yields 2 1 = 4 r 2 = ω2 a 4 ω 2 a 2, or a = 8ωa) 2 2ωa) Since 33) implies that ωa = cr ) where cr ) is defined in Proposition 7), we obtain ω = ωa a = 2ωa) ωa) To conclude, note that, integrating by parts and using 3), inf f E[UJ,fU)] = = [ ẏt)) ] yt) ẏt) dt t [ 2 t ẏt)ÿt) + 1 ] yt) ẏt) dt = t = 2 cr ) ) 8 cr ) λ 2 t yt)ẏt) dt = λ 2 = 2 ω 2, so that inf f ARE ser),f φ SP /φ N ) = 36 2 ω 4. This completes the proof of Proposition 7. Remar. As an immediate corollary, we also obtain that the infimum in Proposition 7 is reached for fixed ) at the collection of radial densities f for which F is in { F,σ r) := F,1 σ 1 r)}, with [ ωr J r ωr) F,1 r) := cr ) J r cr )) I < r cr ] [ ) + I r > cr ] ), ω ω where ω has been obtained in 35). Recall that r := 2 1/2. This also justifies the somewhat mysterious definition of cr ) in Proposition 7. See Figure 2 for the graphs of the associated densities f,1 for several values of the space dimension. References [1] Blumen, I. 1958). A new bivariate sign test, J. Amer. Statist. Assoc. 53, [2] Brown, B. M., T. P. Hettmansperger, J. Nyblom, and H. Oja 1992). On certain bivariate sign tests and medians, J. Amer. Statist. Assoc. 87, [3] Chernoff, H., and I. R. Savage 1958). Asymptotic normality and efficiency of certain nonparametric tests, Ann. Math. Statist. 29, [4] Dufour, J.-M., Hallin, M., and Mizera, I. 1998). Generalized runs tests for heteroscedastic time series, J. Nonparam. Statist. 9, [5] Garel, B., and M. Hallin 1995). Local asymptotic normality of multivariate ARMA processes with a linear trend, Ann. Inst. Statist. Math. 47, [6] Hallin, M. 1994). On the Pitman-nonadmissibility of correlogram-based methods, Journal of Times Series Analysis 15, J. Multivariate Anal. 3, [7] Hallin, M., J.-F. Ingenblee, and M. L. Puri 1989). Asymptotically most powerful ran tests for multivariate randomness against serial dependence, J. Multivariate Anal. 3,