Fully distribution-free center-outward rank tests for multiple-output regression and MANOVA
SSubmitted to the Annals of Statistics
FULLY DISTRIBUTION-FREE CENTER-OUTWARD RANKTESTSFOR MULTIPLE-OUTPUT REGRESSION AND MANOVA
By Marc Hallin, Daniel Hlubinka, and ˇS´arka Hudecov´a
Universit´e libre de Bruxelles and Charles University, Prague
Extending rank-based inference to a multivariate setting such asmultiple-output regression or MANOVA with unspecified d -dimen-sional error density has remained an open problem for more thanhalf a century. None of the many solutions proposed so far is enjoy-ing the combination of distribution-freeness and efficiency that makesrank-based inference a successful tool in the univariate setting. A con-cept of center-outward multivariate ranks and signs based on measuretransportation ideas has been introduced recently. Center-outwardranks and signs are not only distribution-free but achieve in dimen-sion d >
1. Introduction.
Rank-based testing methods have been quite success-ful in testing problems for single-ouput regression and linear models such asANOVA (see the classical monographs by H´ajek and ˇSid´ak (1967), Randlesand Wolfe (1979) or Puri and Sen (1985)) and univariate linear time series(Hallin et al. (1985), Koul and Saleh (1993), Hallin and Puri (1994)). Beingdistribution-free, they remain valid over the full class of absolutely contin-uous distributions. In linear models (this includes testing for single-outputregression slopes, testing for treatment effects in analysis of variance, testingagainst location shifts in two-sample problems) and ARMA time series, theydo reach parametric or semiparametric efficiency bounds at given referencedensities, thus reconciling the objectives of robustness and efficiency.Extending those attractive features to a multivariate context has been along-standing open problem, for which many solutions have been proposed inthe literature. Puri and Sen (1971) for a variety of problems in multivariateanalysis (including multiple-output regression and MANOVA) and Hallin
MSC 2010 subject classifications:
Primary 62G30
Keywords and phrases:
Multivariate ranks; Multivariate signs; Multiple output regres-sion; MANOVA; Rank test; H´ajek representation. imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 a r X i v : . [ m a t h . S T ] J u l HALLIN M., HLUBINKA D., HUDECOV ´A S. et al. (1989) for VARMA time series models construct tests based on com-ponentwise ranks which, however, fail to be distribution-free. Building uponan ingenious multivariate extension of the L definition of quantiles, Oja(1999, 2010) defines the so-called spatial ranks ; the resulting tests are neitherdistribution-free nor efficient. Tests based on the ranks of various concepts ofstatistical depth also have been proposed (Liu (1992), Liu and Singh (1993),He and Wang (1997), Zuo and He (2006)). While distribution-free, the ranksof statistical depth, however, are failing to exploit any directional informa-tion, and hence typically do not allow for any type of asymptotic efficiency. As for the tests based on the
Mahalanobis ranks and signs proposed by Hallinand Paindaveine (2002a,b, 2004, 2005), they do achieve, within the class oflinear models and linear time series with elliptical densities, parametric orsemiparametric efficiency at correctly specified elliptical reference densities;their distribution-freeness, hence their validity, unfortunately, is limited tothe class of elliptical distributions.Inspired by measure transportation ideas, a new concept of ranks andsigns for multivariate observations has been introduced recently under thename of
Monge-Kantorovich ranks and signs in Chernozhukov et al. (2017),under the name of center-outward ranks and signs in Hallin (2017) and Hallinet al. (2020a), along with the related concepts of center-outward distributionand quantile functions . Contrary to earlier concepts, those ranks and signsare extending to dimension d > essential maximal ancillarity prop-erty of univariate ranks that can be interpreted as a finite-sample form ofsemiparametric efficiency; the corresponding empirical center-outward dis-tribution functions, moreover, satisfy a Glivenko-Cantelli result.Concepts of center-outward ranks and signs have been successfully applied(Boeckel et al. (2018); Deb and Sen (2019); Ghosal and Sen (2019); Shi et al.(2019, 2020)) in the construction of distribution-free tests of independencebetween random vectors and multivariate goodness-of-fit; applications tothe study of tail behavior and extremes can be found in De Valk and Segers(2018); Beirlant et al. (2019) are using the related center-outward empir- To be precise, depth-based ranks are not essentially maximal ancillary, and not infor-mative enough for the consistent reconstruction, in locally asymptotically normal experi-ments , of central sequences . The difference in the terminology reflects the fact that Chernozhukov et al. (2017)are working under finite second-order moment and compact support assumptions whichare needed if a Monge-Kantorovich optimization approach is adopted, while Hallin (2017)and Hallin et al. (2020a) are based on the less demanding geometric approach by McCann(1995), which only requires absolute continuity. See Section 2.4 and Appendices D1 and D.2 of Hallin et al. (2020a) for a precisedefinition of this crucial property. imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020
ENTER-OUTWARD RANK TESTS ical quantiles in the analysis of multivariate risk; Hallin et al. (2019) areproposing center-outward R-estimators for VARMA time series models withunspecified innovation densities. The present paper goes one step further inthe direction of a toolkit of distribution-free tests for multiple-output multi-variate analysis, by deriving a H´ajek-type asymptotic representation resultfor linear center-outward rank statistics. Asymptotic normality follows as acorollary, from which center-outward rank tests are constructed for multiple-output regression models (including, as special cases, MANOVA and two-sample location models). Those tests are distribution-free, hence valid, overthe entire family of absolutely continuous distributions; for adequate choiceof the scores, parametric efficiency is attained at chosen densities. Outline of the paper
The paper is organized as follows. Section 2 brieflydescribes the main tools to be used: center-outward distribution and quantilefunctions (Section 2.1) and their empirical counterparts, the center-outwardranks and signs (Section 2.2). The main properties of these concepts aresummarized in Section 2.3 (Proposition 2.1); their invariance/equivarianceproperties are established in Proposition 2.2. Section 3 is entirely devoted tothe key results of this paper, which extend and generalize the classical ap-proach by H´ajek and ˇSid´ak (1967): a H´ajek type asymptotic representationfor multivariate center-outward linear rank statistics (Section 3.2) and theresulting asymptotic normality result (Section 3.3). Section 4.1 describes themultiple-output regression model to be considered throughout, which con-tains, as particular cases, the two-sample location and MANOVA models, ofobvious practical importance. Local asymptotic normality is established inSection 4.2 for this model under general error densities (Proposition 4.1) and,for the purpose of future comparisons, for the particular case of elliptical dis-tributions (Proposition 4.2). The center-outward tests we are proposing aredescribed in Section 5.2, along with (Corollary 5.2) their local asymptoticoptimality properties. Due to their importance in applications, the particularcases of the hypothesis of equal locations in the two-sample problem and thehypothesis of no treatment effect in MANOVA are considered in Section 5.3.Sections 6.1 and 6.2 propose some simple choices of score functions, extend-ing the classical median-test-score (based on center-outward signs only),Wilcoxon, and van der Waerden (normal-score) tests. Section 6.3 discussesaffine invariance issues. Section 7 is devoted to a numerical exploration ofthe finite-sample performance of our rank tests which appear to outperformtheir competitors in non-elliptical situations while performing equally wellunder ellipticity. Section 7.3 presents an application to a four-dimensionalreal dataset of (highly non-elliptical) archeological observations from three imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020
HALLIN M., HLUBINKA D., HUDECOV ´A S. excavation sites in present-day Israel. While traditional MANOVA methodscannot reject the hypothesis of no treatment effect, our fully distribution-freecenter-outward rank-based test rejects it quite significantly.
2. Center-outward distribution functions, ranks, and signs in R d . Center-outward distribution functions.
Throughout, denote by Z ( n ) a triangular array ( Z ( n )1 , . . . , Z ( n ) n ), n ∈ N of i.i.d. d -dimensional randomvectors with distribution P in the family P d of absolutely continuous distri-butions on R d . The notation spt(P) is used for the support of of P, spt(P)for its interior. The open (resp. closed) unit ball and the unit hyperspherein R d are denoted by S d (resp. S d ) and S d − , respectively; U d stands forthe spherical uniform distribution over S d , µ d for the Lebesgue measureover R d ; I d is the d × d unit matrix, A the indicator of the Borel set A .The definition of the center-outward distribution function of P is particu-larly simple for P in the so-called class P + d of distributions with nonvanishingdensities —namely, the class of all distributions with density f := dP / d µ d such that, for all D ∈ R + , there exist constants λ − D ;P and λ + D ;P satisfying(2.1) 0 < λ − D ;P ≤ f ( z ) ≤ λ + D ;P < ∞ for all z with (cid:107) z (cid:107) ≤ D (so that spt(P) = R d and P-a.s. is equivalent to µ d -a.e.). The main result in McCann (1995) then implies the existence ofan a.e. unique convex lower semi-continuous function ϕ : R d → R withgradient ∇ ϕ such that ∇ ϕ d . Call F ± := ∇ ϕ the center-outwarddistribution function of P. It follows from Figalli (2018) that F ± defines ahomeomorphism between the punctured unit ball S d \{ } and its image R d \ F − ± ( { } ):call Q ± : u (cid:55)→ Q ± ( u ) := F − ± ( u ), u (cid:54) = the center-outward quantile func-tion . Figalli also shows that, defining Q ± ( { } ) := F − ± ( { } ) yields a convexand compact subset with Lebesgue measure zero in R d , the center-outwardmedian set of P.All the intuition and all the properties of center-outward distribution andquantile functions hold for this special case (considered in Hallin (2017)), Namely, the spherical distribution with uniform (over [0 , S d − . For d = 1, it coincides with the Lebesgue uniform; for d ≥
2, ithas unbounded density at the origin. We borrow from measure transportation the convenient notation T T : R d → R d pushes P forward to T Z ∼ Pof T ( Z ). imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 ENTER-OUTWARD RANK TESTS and the reader may like to restrict to it. The general case P ∈ P d (consideredin Hallin et al. (2020a)), however, requires the slightly more technical defi-nitions below, due to the fact that P-a.s., in general, is no longer equivalentto µ d -a.e. and the possibility of a more or less regular boundary to spt(P) .For P in P d but not necessarily in P + d , McCann (1995) implies the exis-tence of an a.e. unique real-valued convex lower semi-continuous function ψ with domain S d such that ∇ ψ d = P. Without loss of generality, we fur-ther can extend ψ to R d by defining ψ ( u ) := lim inf S d (cid:51) v → u ψ ( v ) for (cid:107) u (cid:107) = 1and ψ ( u ) := ∞ for (cid:107) u (cid:107) >
1. The center-outward quantile function Q ± thenis defined as the gradient ∇ ψ , with domain S d , of that extended ψ . Consid-ering the Legendre transform ϕ ( z ) := sup u ∈ S d (cid:0) (cid:104) u , z (cid:105) − ψ ( u ) (cid:1) of ψ , definecenter-outward distribution function of P as F ± := ∇ ϕ , with domain R d and range in S d . See Section 2.3 for the main properties of F ± and Q ± andSection 2 of Hallin et al. (2020a) for further details.Some properties of the center-outward ranks, such as distribution-freenessor the independence between ranks and signs, hold for all P ∈ P d . Someothers—essentially, asymptotic results—require slightly more regular distri-butions. Following Hallin et al. (2020a), define P ± d := (cid:8) P = ∇ Υ d (cid:12)(cid:12) Υ is convex, ∇ Υ a homeomorphism over S d \{ } with ∇ Υ( { } ) compact, convex, and µ d (cid:0) ∇ Υ( { } ) (cid:1) = 0 (cid:9) . Obviously, P ± d ⊂ P d ; it follows from Proposition 2.3 in Hallin et al.(2020a) that P ± d also contains the class P conv d of all distributions P ∈ P d with(bounded or unbounded) convex support satisfying (2.1) for all z ∈ spt(P)with (cid:107) z (cid:107) ≤ D , which in turn contains P + d : hence, P + d ⊂ P conv d ⊂ P ± d ⊂ P d .2.2. Center-outward ranks and signs.
Except for a few particular casessuch as spherical distributions, the above definitions are not meant for theanalytical derivation of F ± and Q ± which typically involves Monge-Amp`erepartial differential equations; estimation is possible, though, via their em-pirical counterparts F ( n ) ± and Q ( n ) ± , based on center-outward ranks and signs,which we now describe.Associated with the n -tuple Z ( n )1 , . . . , Z ( n ) n , the empirical center-outwarddistribution function F ( n ) ± is mapping Z ( n )1 , . . . , Z ( n ) n to a “regular” grid G n of the unit ball S d . That grid G n is obtained as follows: (a) first factorize n into n = n R n S + n , with 0 ≤ n < min( n R , n S ); Regularity problems in the area are notoriously tricky: see, e.g., Caffarelli (1996). In particular, no closed forms of F ± and Q ± are known for non-spherical ellipticaldistributions. imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 HALLIN M., HLUBINKA D., HUDECOV ´A S. (b) next consider a “regular array” S n S := { s n S , . . . , s n S n S } of n S points onthe sphere S d − (see the comment below); (c) finally, the grid consists in the collection G n of the n R n S points g ofthe form (cid:0) r/ (cid:0) n R + 1 (cid:1)(cid:1) s n S s , r = 1 , . . . , n R , s = 1 , . . . , n S , along with ( n copies of) the origin in case n (cid:54) = 0: a total number n − ( n −
1) or n of distinct points, thus, according as n > n = 0.By “regular” we mean “as uniform as possible”, in the sense, for example, ofthe low-discrepancy sequences of the type considered in numerical integrationand Monte-Carlo methods (see, e.g., Niederreiter (1992), Judd (1998), Dickand Pillichshammer (2014), or Santner et al. (2003)). The only mathemat-ical requirement needed for Proposition 2.1 below is the weak convergence,as n S → ∞ , of the uniform discrete distribution over S n S to the uniformdistribution over S d − . A uniform i.i.d. sample of points over S d − satisfiessuch a requirement but fails to produce as neat a concept of ranks and signs;moreover, one easily can construct arrays that are “more regular” than ani.i.d. one. For instance, one could see that n S or n S − s n S s in S n are such that − s n S s also belongs to S n S , so that (cid:107) (cid:80) n S s =1 s n S s (cid:107) = 0or 1 according as n S is even or odd. One also could consider factorizationsof the form n = n R n S + n with n S even and 0 ≤ n < min(2 n R , n S ),then require S n to be symmetric with respect to the origin, automaticallyyielding (cid:80) n S s =1 s n S s = .The empirical counterpart F ( n ) ± of F ± is defined as the (bijective, once theorigin is given multiplicity n ) mapping from Z ( n )1 , . . . , Z ( n ) n to the grid G n that minimizes the sum of squared Euclidean distances (cid:80) ni =1 (cid:13)(cid:13) F ( n ) ± ( Z ( n ) i ) − Z ( n ) i (cid:13)(cid:13) . That mapping is unique with probability one; in practice, it is ob-tained via a simple optimal assignment (pairing) algorithm (a linear pro-gram; see Section 4 of Hallin (2017) for details).Call center-outward rank of Z ( n ) i the integer (in { , . . . , n R } or { , . . . , n R } according as n = 0 or not) R ( n ) i ; ± s := ( n R + 1) (cid:13)(cid:13) F ( n ) ± ( Z ( n ) i ) (cid:13)(cid:13) i = 1 , . . . , n and center-outward sign of Z ( n ) i the unit vector S ( n ) i ; ± := F ( n ) ± ( Z ( n ) i ) / (cid:13)(cid:13) F ( n ) ± ( Z ( n ) i ) (cid:13)(cid:13) for F ( n ) ± ( Z ( n ) i ) (cid:54) = ;for F ( n ) ± ( Z ( n ) i ) = , put S ( n ) i ; ± s = . imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 ENTER-OUTWARD RANK TESTS Some desirable finite-sample properties, such as strict independence be-tween the ranks and the signs, only hold for n = 0 or 1, due to the fact thatthe mapping from the sample to the grid is no longer injective for n ≥ n of tiedvalues involved is o ( n ) as n → ∞ ), is easily taken care of by the followingtie-breaking device: (i) randomly select n directions s , . . . , s n in S n S , then (ii) replace the n copies of the origin with the new gridpoints(2.2) [1 / n R + 1)] s , . . . , [1 / n R + 1)] s n . The resulting grid (for simplicity, the same notation G n is used) no longerhas multiple points, and the optimal pairing between the sample and the gridis bijective; the n smallest ranks, however, take the non-integer value 1 / Main properties.
This section summarizes some of the main prop-erties of the concepts defined in Sections 2.1 and 2.2; further properties anda proof for Proposition 2.1 can be found in Hallin et al. (2020a).
Proposition . Let F ± denote the center-outward distribution func-tion of P ∈ P d . Then,(i) F ± is a probability integral transformation of R d : namely, Z ∼ P iff F ± ( Z ) ∼ U d ; by construction, (cid:107) F ± ( Z ) (cid:107) is uniform over the in-terval [0 , , F ± ( Z ) / (cid:107) F ± ( Z ) (cid:107) uniform over the sphere S d − , and theyare mutually independent.Let Z ( n ) i , . . . , Z ( n ) i be i.i.d. with distribution P ∈ P d and center-outward dis-tribution function F ± . Then,(ii) (cid:0) F ( n ) ± ( Z ( n )1 ) , . . . , F ( n ) ± ( Z ( n ) n ) (cid:1) is uniformly distributed over the n ! /n ! permutations with repetitions of the gridpoints in G n with the origincounted as n indistinguishable points (resp. the n ! permutations of G n if either n ≤ or the tie-breaking device described in Section 2.2 isadopted);(iii) if either n = 0 or the tie-breaking device described in Section 2.2 isadopted, the n -tuple of center-outward ranks (cid:0) R ( n )1; ± , . . . , R ( n ) n ; ± (cid:1) and the n -tuple of center-outward signs (cid:0) S ( n )1; ± , . . . , S ( n ) n ; ± (cid:1) are mutually indepen-dent;(iv) if either n ≤ or the tie-breaking device described in Section 2.2 isadopted, imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 HALLIN M., HLUBINKA D., HUDECOV ´A S. the n -tuple (cid:0) F ( n ) ± ( Z ( n )1 ) , . . . , F ( n ) ± ( Z ( n ) n ) (cid:1) is strongly essentially maxi-mal ancillary . Assuming, moreover, that P ∈ P ± d ,(v) (Glivenko-Cantelli) max ≤ i ≤ n (cid:13)(cid:13)(cid:13) F ( n ) ± ( Z ( n ) i ) − F ± ( Z ( n ) i ) (cid:13)(cid:13)(cid:13) → as n → ∞ . Center-outward distribution functions, ranks, and signs also inherit, fromthe invariance features of Euclidean distances, elementary but quite re-markable invariance and equivariance properties under orthogonal trans-formations. Denote by F Z ± the center-outward distribution function of Z and by F Z ;( n ) ± the empirical distribution function of a sample Z , . . . , Z n associated with a grid G n . Proposition . Let µ ∈ R d and denote by O a d × d orthogonalmatrix. Then,(i) F µ + OZ ± ( µ + Oz ) = OF Z ± ( z ) , z ∈ R d ;(ii) denoting by F µ + OZ ;( n ) ± the empirical distribution function of the sample µ + OZ , . . . , µ + OZ n associated with the grid O G n (hence, by F Z ;( n ) ± the empirical distribu-tion function of the sample Z , . . . , Z n associated with the grid G n ), (2.3) F µ + OZ ;( n ) ± ( µ + OZ i ) = OF Z ;( n ) ± ( Z i ) , i = 1 , . . . , n ; (iii) the center-outward ranks R ( n ) i ; ± and the cosines S ( n ) (cid:48) i ; ± S ( n ) j ; ± computed fromthe sample Z , . . . , Z n and the grid G n are the same as those computedfrom thesample µ + OZ , . . . , µ + OZ n and the grid O G n . Proof.
Starting with (ii) , note that, for any ( z , . . . , z n ) ∈ R nd , ( u , . . . , u n ) ∈ R nd , and µ ∈ R d , denoting by π a permutation of { , . . . , n } , n (cid:88) i =1 (cid:107) µ + z i − u π ( i ) (cid:107) − n (cid:88) i =1 (cid:107) z i − u π ( i ) (cid:107) = n µ (cid:48) µ + 2 µ (cid:48) n (cid:88) i =1 z i − µ (cid:48) n (cid:88) i =1 u i does not depend on π ; the optimal pairing between the µ + z i ’s and the u i ’sthus does not depend on µ , so that F µ + Z ;( n ) ± ( µ + Z i ) = F Z ;( n ) ± ( Z i ) for See Section 2.4 and Appendices D1 and D.2 of Hallin et al. (2020a) for a precisedefinition of this crucial property and a proof. imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020
ENTER-OUTWARD RANK TESTS all i (with F µ + Z ;( n ) ± and F Z ;( n ) ± constructed from the same grid G n ). Asfor F OZ ;( n ) ± ( OZ i ) computed from O G n and OF Z ;( n ) ± ( Z i ) computed from G n ,they obviously coincide since the Euclidean distances on which they arebased coincide. Part (iii) of the proposition is an immediate consequence.Turning to (i) , note that F ± , as the gradient of a convex function, enjoys(see, e.g., Rockafellar (1966)) cyclical monotonicity : for any finite collectionof points z , . . . , z k ∈ R nd , it holds that (cid:104) F ± ( z ) , z − z (cid:105) + (cid:104) F ± ( z ) , z − z (cid:105) + . . . + (cid:104) F ± ( z k ) , z − z k (cid:105) ≤ . Equivalently, considering the grid G k := (cid:8) F ± ( z ) , . . . , F ± ( z k ) (cid:9) , any k -tuple of theform ( z i , F ± ( z i )), i = 1 , . . . , k constitutes an optimal coupling minimizing S ( k ) z := k (cid:88) i =1 (cid:107) F ± ( z i ) − z π ( i ) (cid:107) over the k ! permutations π of { , . . . , k } : denoting by F z ;( k ) ± the minimizerof S ( k ) z , thus,(2.4) F z ;( k ) ± ( z i ) = F ± ( z i ) , i = 1 , . . . , k for any k .Now, for fixed k , (ii) applies, so that, similar to (2.3),(2.5) F µ + Oz ;( k ) ± ( µ + Oz i ) = OF z ;( k ) ± ( z i ) . In view of (2.4) (for F ± = F Z ± ), however,(2.6) F µ + Oz ;( k ) ± ( µ + Oz i ) = F µ + OZ ± ( µ + Oz i ) and F z ;( k ) ± ( z i ) = F Z ± ( z i ) . The result follows from piecing together (2.5) and (2.6). (cid:3)
These orthogonal equivariance and invariance properties, however, do notextend to non-orthogonal affine transformations.
3. Linear center-outward rank statistics: H´ajek representationand asymptotic normality.
Linear center-outward rank statistics.
Linear rank statistics in thiscontext depend on a score function J : S d → R d and are indexed by tri-angular arrays { c ( n )1 , . . . , c ( n ) n } of real numbers (regression constants). Onthose score functions and regression constants we are making the followingassumptions. imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 HALLIN M., HLUBINKA D., HUDECOV ´A S.
Assumption . (i) J : S d → R d is continuous over S d ; (ii) for any sequence s ( n ) = { s ( n )1 , . . . , s ( n ) n } of n -tuples in S d such thatthe uniform discrete distribution over s ( n ) converges weakly to U d as n → ∞ ,(3.7) lim n →∞ n − tr n (cid:88) r =1 J ( s ( n ) r ) J (cid:48) ( s ( n ) r ) = tr (cid:90) S d J ( u ) J (cid:48) ( u ) dU d where (cid:82) S d J ( u ) J (cid:48) ( u ) dU d < ∞ has full rank.As we shall see, a special role is played, in relation with spherical distri-butions, by score functions of the form(3.8) J ( u ) := J ( (cid:107) u (cid:107) ) u (cid:107) u (cid:107) [ (cid:107) u (cid:107)(cid:54) =0] u ∈ S d for some function J : [0 , → R . Assumption 3.1 then holds if (i) J iscontinuous and (ii) (3.9) 0 < lim n →∞ n − n (cid:88) r =1 J (cid:0) r/ ( n + 1) (cid:1) = (cid:90) J ( u ) d u < ∞ (a sufficient condition for (3.9) is the traditional assumption that J hasbounded variation, i.e. is the difference of two nondecreasing functions).As for the regression constants, we assume that the classical Noetherconditions hold.
Assumption . The c ( n ) i ’s are not all equal (for given n ) and satisfy(3.10) n (cid:88) i =1 ( c ( n ) i − ¯ c ( n ) ) / max ≤ i ≤ n ( c ( n ) i − ¯ c ( n ) ) −→ ∞ as n → ∞ where ¯ c ( n ) := n − (cid:80) ni =1 c ( n ) i .Associated with the score functions J , consider the d -dimensional statis-tics T ∼ ( n ) a = (cid:16) n (cid:88) i =1 ( c ( n ) i − ¯ c ( n ) ) (cid:17) − / n (cid:88) i =1 ( c ( n ) i − ¯ c ( n ) ) J ( F ( n ) ± ( Z ( n ) i )) , (3.11) T ∼ ( n ) e := (cid:16) n (cid:88) i =1 ( c ( n ) i − ¯ c ( n ) ) (cid:17) − / n (cid:88) i =1 ( c ( n ) i − ¯ c ( n ) )E (cid:20) J ( F ± ( Z ( n ) i )) (cid:12)(cid:12)(cid:12)(cid:12) F ( n ) ± ( Z ( n ) i ) (cid:21) , imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 ENTER-OUTWARD RANK TESTS and T ( n ) := (cid:16) n (cid:88) i =1 ( c ( n ) i − ¯ c ( n ) ) (cid:17) − / n (cid:88) i =1 ( c ( n ) i − ¯ c ( n ) ) J ( F ± ( Z ( n ) i )) . Adopting H´ajek’s terminology, call T ∼ ( n ) a an approximate-score linear rankstatistic and T ∼ ( n ) e an exact-score linear rank statistic. As we shall see, both T ∼ ( n ) a and T ∼ ( n ) e admit the same asymptotic representation T ( n ) , hence are asymp-totically equivalent. For score functions of the form (3.8), we have T ∼ ( n ) a = (cid:16) n (cid:88) i =1 ( c ( n ) i − ¯ c ( n ) ) (cid:17) − / n (cid:88) i =1 ( c ( n ) i − ¯ c ( n ) ) J (cid:16) R ( n ) i ; ± n R + 1 (cid:17) S ( n ) i ; ± , T ∼ ( n ) e = (cid:16) n (cid:88) i =1 ( c ( n ) i − ¯ c ( n ) ) (cid:17) − / × n (cid:88) i =1 ( c ( n ) i − ¯ c ( n ) )E (cid:34) J (cid:0)(cid:13)(cid:13) F ± ( Z ( n ) i ) (cid:13)(cid:13)(cid:1) F ± ( Z ( n ) i ) (cid:13)(cid:13) F ± ( Z ( n ) i ) (cid:13)(cid:13) (cid:12)(cid:12)(cid:12)(cid:12) F ( n ) ± ( Z ( n ) i ) (cid:35) , and T ( n ) = (cid:16) n (cid:88) i =1 ( c ( n ) i − ¯ c ( n ) ) (cid:17) − / n (cid:88) i =1 ( c ( n ) i − ¯ c ( n ) ) J ( (cid:13)(cid:13) F ± ( Z ( n ) i ) (cid:13)(cid:13) ) F ± ( Z ( n ) i ) (cid:13)(cid:13) F ± ( Z ( n ) i ) (cid:13)(cid:13) . Asymptotic representation.
The following proposition is a center-outward multivariate counterpart of the asymptotic results in Section V.1.6of H´ajek and ˇSid´ak (1967). Throughout this section, we assume that theempirical distribution function F ( n ) ± is computed from a triangular array( Z ( n )1 , . . . , Z ( n ) n ), n ∈ N of i.i.d. d -dimensional random vectors with distribu-tion P ∈ P ± d and center-outward distribution function F ± . Proposition . Let Assumptions 3.1 and 3.2holdand Z ( n )1 , . . . , Z ( n ) n be i.i.d. with distribution P ∈ P ± d . Then, ( i ) T ∼ ( n ) a − T ( n ) = o q.m. (1) , ( ii ) T ∼ ( n ) e − T ( n ) = o q.m. (1) , and ( iii ) T ∼ ( n ) a − T ∼ ( n ) e = o q.m. (1) as n → ∞ in such a way that n R → ∞ and n S → ∞ . The notation o q.m. (1) stands for a sequence of random vectors tending to zero inquadratic mean (hence also in probability). imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 HALLIN M., HLUBINKA D., HUDECOV ´A S.
Proof.
We throughout write Z i for Z ( n ) i . First consider Part (i) of theproposition. We have T ∼ ( n ) a − T ( n ) = (cid:16) n (cid:88) i =1 ( c ( n ) i − ¯ c ( n ) ) (cid:17) − / n (cid:88) i =1 ( c ( n ) i − ¯ c ( n ) )[ J ( F ( n ) ± ( Z i )) − J ( F ± ( Z i ))] . Let a ( n ) i := a ( F ( n ) ± ( Z i ) , F ± ( Z i )) := J ( F ( n ) ± ( Z i )) − J ( F ± ( Z i )). Then, (cid:13)(cid:13) T ∼ ( n ) a − T ( n ) (cid:13)(cid:13) = (cid:16) n (cid:88) i =1 ( c ( n ) i − ¯ c ( n ) ) (cid:17) − (cid:34) n (cid:88) i =1 ( c ( n ) i − ¯ c ( n ) ) (cid:13)(cid:13) a ( n ) i (cid:13)(cid:13) + (cid:88) i (cid:54) = j ( c ( n ) i − ¯ c ( n ) )( c ( n ) j − ¯ c ( n ) ) a ( n ) (cid:48) i a ( n ) j . Since E (cid:107) a ( n ) i (cid:107) = E (cid:107) a ( n )1 (cid:107) and E a ( n ) (cid:48) i a ( n ) j = E a ( n ) (cid:48) a ( n )2 , we getE (cid:13)(cid:13) T ∼ ( n ) a − T ( n ) (cid:13)(cid:13) = E (cid:13)(cid:13) a ( n )1 (cid:13)(cid:13) − E a ( n ) (cid:48) a ( n )2 ≤ (cid:13)(cid:13) a ( n )1 (cid:13)(cid:13) . Hence, it only remains to show thatE (cid:13)(cid:13) a ( n )1 (cid:13)(cid:13) = E (cid:13)(cid:13)(cid:13) J ( F ( n ) ± ( Z )) − J ( F ± ( Z )) (cid:13)(cid:13)(cid:13) = E (cid:13)(cid:13) ζ n − ζ (cid:13)(cid:13) → ζ n := J ( F ( n ) ± ( Z )) and ζ := J ( F ± ( Z )). It follows from the Glivenko-Cantelli theorem in Hallin et al. (2020a), the continuity of J , and the continu-ity over R \{ } of x (cid:55)→ x / (cid:13)(cid:13) x (cid:13)(cid:13) that ζ n → ζ a.s. Furthermore, Assumption 3.1implies thatE (cid:13)(cid:13) ζ n (cid:13)(cid:13) = tr E J ( F ( n ) ± ( Z )) J (cid:48) ( F ( n ) ± ( Z )) → tr E J ( F ± ( Z )) J (cid:48) ( F ± ( Z )) = E (cid:13)(cid:13) ζ (cid:13)(cid:13) . It follows (see, for instance, part (iv) of Theorem 5.7 in (Shorack, 2000,Chapter 3)) that E (cid:13)(cid:13) ζ n − ζ (cid:13)(cid:13) →
0. This concludes the proof for Part (i) ofthe proposition.Turning to Part (ii) , put b ( n ) i := J ( F ± ( Z i )) − E (cid:20) J ( F ± ( Z i )) (cid:12)(cid:12)(cid:12)(cid:12) F ( n ) ± ( Z i ) (cid:21) , and let us show that E (cid:13)(cid:13) b ( n )1 (cid:13)(cid:13) = o (1). Since (cid:107) ζ n − ζ (cid:107) tends a.s. to zero, itfollows from the Egorov theorem (see, e.g., Theorem 7.5.1 in Dudley (1989))that, for any ε >
0, there is a set A ⊂ Ω such thatP( A ) > − ε and sup ω ∈ A (cid:107) ζ n ( ω ) − ζ ( ω ) (cid:107) → . imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 ENTER-OUTWARD RANK TESTS Denoting by A c the complement of A in Ω, we haveE (cid:13)(cid:13) b ( n )1 (cid:13)(cid:13) = E (cid:13)(cid:13) ζ − E[ ζ | F ( n ) ± ( Z )] (cid:13)(cid:13) = E (cid:13)(cid:13) ζ A + ζ A c − E[ ζ A + ζ A c | F ( n ) ± ( Z )] (cid:13)(cid:13) ≤ (cid:13)(cid:13) ζ A − E[ ζ A | F ( n ) ± ( Z )] (cid:13)(cid:13) + 3E (cid:13)(cid:13) ζ A c (cid:13)(cid:13) + 3E (cid:13)(cid:13)(cid:13) E[ ζ A c | F ( n ) ± ( Z )] (cid:13)(cid:13)(cid:13) =: 3 (cid:16) I ( n )1 + I + I ( n )3 (cid:17) , say . In view of the square-integrability of ζ , I can be made arbitrarily smallas ε →
0. As for I ( n )3 , we have I ( n )3 = E (cid:13)(cid:13)(cid:13) E[ ζ ¯ A (cid:12)(cid:12) F ( n ) ± ( Z )] (cid:13)(cid:13)(cid:13) ≤ E (cid:16) E (cid:2) (cid:107) ζ ¯ A (cid:107) | F ( n ) ± ( Z ) (cid:3)(cid:17) = E (cid:13)(cid:13) ζ ¯ A (cid:13)(cid:13) where P( ¯ A ) ≤ ε , so that I ( n )3 also is arbitrarily small as ε → I ( n )1 → n → ∞ . Recall that F ( n ) ± ( Z ), withprobability (1 − n /n ) tending to one, is a point of the regular grid G n of n − ( n − n > points in the unit ball used in the construction of F ( n ) ± .Moreover, for any g ∈ G n \{ } , we have P[ F ( n ) ± ( Z ) = g ] = 1 /n . Define B ( n ) g := { ω : F ( n ) ± ( Z )( ω ) = g } . Clearly, (cid:8) B ( n ) g , g ∈ G n (cid:9) constitutes a disjoint partition of Ω and P( B ( n ) g ) → g ∈ G n as n → ∞ . Then, I ( n )1 = E (cid:13)(cid:13)(cid:13) (cid:88) g ∈ G n (cid:0) ζ A B ( n ) g − E[ ζ A | F ( n ) ± ( Z )] B ( n ) g (cid:1)(cid:13)(cid:13)(cid:13) = E (cid:88) g ∈ G n (cid:13)(cid:13)(cid:13) ζ A B ( n ) g − E[ ζ A | F ( n ) ± ( Z )] B ( n ) g (cid:13)(cid:13)(cid:13) where the latter equality follows form the fact that B ( n ) g B ( n ) h = ( g = h ) .Since B ( n ) g is an atom of σ (cid:0) F ( n ) ± ( Z ) (cid:1) , the latter conditional expectation isa constant on B ( n ) g , namelyE (cid:2) ζ A (cid:12)(cid:12) F ( n ) ± ( Z ) (cid:3) B ( n ) g = B ( n ) g P( B ( n ) g ) (cid:90) η ∈ B ( n ) g ζ ( η ) A ( η )dP( η ) . imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 HALLIN M., HLUBINKA D., HUDECOV ´A S.
Hence, I ( n )1 = (cid:88) g ∈ G n (cid:90) Ω (cid:13)(cid:13)(cid:13)(cid:13) B ( n ) g ( ω ) (cid:90) η ∈ B ( n ) g [ ζ ( ω ) A ( ω ) − ζ ( η ) A ( η )] dP( η )P( B ( n ) g ) (cid:13)(cid:13)(cid:13)(cid:13) dP( ω )= (cid:88) g ∈ G n (cid:90) Ω (cid:13)(cid:13)(cid:13)(cid:13) B ( n ) g ( ω ) (cid:90) η ∈ B ( n ) g (cid:2)(cid:0) ζ ( ω ) − ζ n ( ω ) (cid:1) A ( ω )+ (cid:0) ζ n ( η ) − ζ ( η ) (cid:1) A ( η ) (cid:3) dP( η )P( B ( n ) g ) (cid:13)(cid:13)(cid:13)(cid:13) dP( ω )since ζ n ( ω ) A ( ω ) B ( n ) g ( ω ) = J ( g ) = ζ n ( η ) A ( η ) B ( n ) g ( η ) on A ∩ B ( n ) g . Now,we are almost done. Since, for ω ∈ A , we have the uniform convergenceof (cid:107) ζ n ( ω ) − ζ ( ω ) (cid:107) to zero, we may bound the integrand uniformly. Moreprecisely, for any ˜ ε > n ˜ ε such that (cid:107) ζ ( ω ) − ζ n ( ω ) (cid:107) < ˜ ε forall n ≥ n ˜ ε and all ω ∈ A , so that, from Jensen’s inequality, I ( n )1 ≤ (cid:88) g ∈ G n (cid:90) Ω B ( n ) g ( ω ) (cid:90) B ( n ) g (cid:104) (cid:13)(cid:13) ζ ( ω ) − ζ n ( ω ) (cid:13)(cid:13) A ( ω )+ 2 (cid:13)(cid:13) ζ ( η ) − ζ n ( η ) (cid:13)(cid:13) A ( η ) (cid:105) dP( η )P( B ( n ) g ) dP( ω ) ≤ (cid:88) g ∈ G n (cid:90) Ω B ( n ) g ( ω ) 4˜ ε P( B ( n ) g )P( B ( n ) g ) dP( ω ) = 4˜ ε E (cid:88) g ∈ G n B ( n ) g = 4˜ ε . Part (ii) of the proposition follows. Part (iii) is an immediate consequenceof Parts (i) and (ii) . (cid:3) Asymptotic normality.
The asymptotic normality of T ∼ ( n ) a and T ∼ ( n ) e follows from Proposition 3.1 and the asymptotic normality of T ( n ) , alongwith the distribution-freeness of T ∼ ( n ) a and T ∼ ( n ) e . Proposition . Let Assumptions 3.1 and 3.2 hold and Z ( n )1 , . . . , Z ( n ) n be i.i.d. with distribution P ∈ P d . Then, T ∼ ( n ) a , T ∼ ( n ) e , and T ( n ) are asymptot-ically normal as n → ∞ (in such a way that n R → ∞ and n S → ∞ ), withmean and covariance (cid:82) S d J ( u ) J (cid:48) ( u ) dU d reducing, for J of the form (3.8) ,to d − (cid:82) J ( u ) d u I d . Proof.
First assume that P ∈ P ± d , with center-outward distributionfunction F ± . In view of Proposition 3.1, establishing the result for T ( n ) imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 ENTER-OUTWARD RANK TESTS is sufficient. Put V := F ± ( Z ( n )1 ). Then V D = U W , where U and W aremutually independent, U is uniform over [0 , W is uniform over theunit sphere S d − . Clearly,E T ( n ) = and Var ( T ( n ) ) = Var J ( V ) = (cid:90) S d J ( u ) J (cid:48) ( u ) dU d so that, for J of the form (3.8),Var ( T ( n ) ) = E J ( U )Var W = 1 d (cid:90) J ( u ) d u I d since Var W = d I d (see, e.g. page 34 of Fang et al. (2017)). Now, T ( n ) is asum of independent variables, and the Noether condition (3.10) ensures thatthe Feller-Lindenberg condition holds. The desired asymptotic normality re-sult (for T ∼ ( n ) a and T ∼ ( n ) e , under P ∈ P ± d ) thus follows from the central limittheorem. Finally, consider the general case P ∈ P d . Distribution-freenessimplies that the finite- n distributions of T ∼ ( n ) a and T ∼ ( n ) e are the same un-der P ∈ P d as under P (cid:48) ∈ P ± d . Hence, their asymptotic distributions un-der P ∈ P d and P (cid:48) ∈ P ± d also coincide. This completes the proof. (cid:3)
4. Multiple-output linear models.
The objective of this section isto construct, based on the center-outward ranks and signs of Section 3,rank tests for the slopes of multiple-output linear models, extending to amultivariate setting the methods developed, e.g. in Puri and Sen (1985) forthe single-output case.4.1.
The model.
Consider the multiple-output linear (or multiple-outputregression) model under which an observed Y ( n ) satisfies(4.1) Y ( n ) = n β (cid:48) + C ( n ) β + ε ( n ) , where n := (1 , . . . , (cid:48) , Y ( n ) = Y ( n )11 Y ( n )12 . . . Y ( n )1 d ... ... ... Y ( n ) n Y ( n ) n . . . Y ( n ) nd = Y ( n ) (cid:48) ... Y ( n ) (cid:48) n is an n × d matrix of n observed d -dimensional outputs, C ( n ) = c ( n )11 c ( n )12 . . . c ( n )1 m ... ... ... c ( n ) n c ( n ) n . . . c ( n ) nm = c ( n ) (cid:48) ... c ( n ) (cid:48) n imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 HALLIN M., HLUBINKA D., HUDECOV ´A S. an n × m matrix of (specified) deterministic covariates, β (cid:48) = ( β , . . . , β d ) and β = β β . . . β d ... ... ... β m β m . . . β md = β (cid:48) ... β (cid:48) m a d -dimensional intercept and an m × d matrix of regression coefficients, and ε ( n ) = ε ( n )11 ε ( n )12 . . . ε ( n )1 d ... ... ... ε ( n ) n ε ( n ) n . . . ε ( n ) nd = ε ( n ) (cid:48) ... ε ( n ) (cid:48) n an n × d matrix of nonobserved i.i.d. d -dimensional errors ε ( n ) i , i = 1 , . . . , n with density f ε . If β is to be identified, a location constraint has to beimposed on f ε . One could think of the classical constraint E ε ( n ) i = (requiring the existence of a finite mean): β + β (cid:48) c ( n ) i then is to be interpretedas the expected value of Y ( n ) i for covariate values c ( n ) i . In the context of thispaper, however, a more natural location constraint (which moreover doesnot require any integrability condition) is F ε ± ( ) = , where F ε ± stands forthe center-outward distribution function of the ε ( n ) i ’s: and β + β (cid:48) c ( n ) i thenare center-outward medians for ε and Y ( n ) i , respectively.In most applications, however, one is interested mainly in the impact ofthe input covariates c ( n ) i on the output Y ( n ) i : the matrix β is the parameter ofinterest, and β is a nuisance. There is no need, then, for identifying β norqualifying β + β (cid:48) c ( n ) i as a mean or a center-outward median for Y ( n ) i : β is tobe interpreted as a matrix of treatment effects governing the shift δ (cid:48) β in thedistribution of the d -dimensional output produced by a variation δ in the m -dimensional covariate. Center-outward ranks and signs being insensitive toshifts, there is even no need to specify, nor to estimate β .4.2. Local Asymptotic Normality (LAN).
The model (4.1) is easily seento be locally asymptotically normal (LAN) under the following two classicalassumptions.
Assumption . The square root z (cid:55)→ ( f ε ) / ( z ) of the error density is differentiable in quadratic mean, with quadratic mean gradient ∇ ( f ε ) / . It follows from a result by Lind and Roussas (1972) independently rediscovered byGarel and Hallin (1995) that quadratic mean differentiability is equivalent to partialquadratic mean derivability with respect to all variables. imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020
ENTER-OUTWARD RANK TESTS Letting ϕ f ε := − ∇ ( f ε ) / / ( f ε ) / , assume moreover that the information matrix I f ε := E (cid:104) ϕ f ε ( ε ) ϕ (cid:48) f ε ( ε ) (cid:105) hasfull rank d .On the regression constants C ( n ) , we borrow from Hallin and Paindaveine(2005) the following assumptions; note that Part (iii) requires that each ofthe m triangular arrays of constants c ( n ) ij , i ∈ N , j = 1 , . . . , m satisfiesAssumption 3.2. Assumption . Letting V ( n ) c := n − (cid:80) ni =1 (cid:0) c ( n ) i − ¯ c ( n ) (cid:1)(cid:0) c ( n ) i − ¯ c ( n ) (cid:1) (cid:48) with ¯ c ( n ) := n − (cid:80) ni =1 c ( n ) i , denote by D ( n ) c the diagonal matrix with ele-ments (cid:0) V ( n ) c (cid:1) jj , j = 1 , . . . , m ; (i) (cid:0) V ( n ) c (cid:1) jj > j = 1 , . . . , m ; (ii) defining R ( n ) c := D ( n ) − / c V ( n ) c D ( n ) − / c , the limit R c := lim n →∞ R ( n ) c exists, is positive definite, and factorizes into R c = (cid:0) K c K (cid:48) c (cid:1) − forsome full-rank m × m matrix K c ; (iii) the following Noether conditions hold: letting ¯ c ( n ) j := n − (cid:80) ni =1 c ( n ) ij ,lim n →∞ n (cid:88) i =1 (cid:0) c ( n ) ij − ¯ c ( n ) j (cid:1) / max ≤ i ≤ n (cid:0) c ( n ) ij − ¯ c ( n ) j (cid:1) = ∞ , j = 1 , . . . , m. Letting Z ( n ) i = Z ( n ) i ( β ) := Y ( n ) i − n β (cid:48) − β (cid:48) c ( n ) i , the following result readilyfollows from, e.g., (Lehmann and Romano, 2005, Theorem 12.2.3). In orderto simplify the notation, we throughout adopt the same contiguity rates asin Hallin and Paindaveine (2005). Namely, we consider local perturbationsof the parameter β of the form β + ν ( n ) τ where τ is an m × d matrixand ν ( n ) := n − / K ( n ) c , with K ( n ) c := (cid:0) D ( n ) c (cid:1) − / K c . This is a notationalconvenience and has no impact on the form of locally asymptotically optimaltest statistics. Proposition . Under Assumptions 4.1 and 4.2, the model (4.1) isLAN (with respect to β ), with central sequence ∆ ( n ) β ; f ε ( β ) := n / vec Λ ( n ) β ; f ε where (4.2) Λ ( n ) β ; f ε := 1 n n (cid:88) i =1 K ( n ) (cid:48) c (cid:0) c ( n ) i − ¯ c ( n ) (cid:1) ϕ (cid:48) f ε ( Z ( n ) i ) and Fisher information I f ε ⊗ I m imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 HALLIN M., HLUBINKA D., HUDECOV ´A S.
LAN for the same linear model (4.1) has been established (in the broadercontext of regression with VARMA errors in Hallin and Paindaveine (2005)under the assumption that the error density f ε is centered elliptical , thatis, has the form(4.3) f ε ( z ) = κ − d, f (cid:0) det Σ (cid:1) − / f (cid:0) ( z (cid:48) Σ − z ) / (cid:1) with κ d, f := (cid:0) π d/ / Γ( d/ (cid:1) (cid:82) ∞ r d − f ( r ) d r for some symmetric positive def-inite shape matrix Σ and some radial density f (over R +0 ) such that f ( z ) > R +0 and (cid:82) ∞ r d − f ( r ) d r < ∞ . When ε is elliptical withshape matrix Σ and radial density f , the modulus (cid:107) Σ − / ε (cid:107) has density f (cid:63)d ( r ) = ( µ d − f ) − r d − f ( r ) I [ r > , where µ d − f := (cid:82) ∞ r d − f ( r )d r , and distribution function F (cid:63)d ; f .Assumption 4.1 then is equivalent to the mean square differentiability,with quadratic mean derivative (cid:0) f / (cid:1) (cid:48) , of x (cid:55)→ f / ( x ), x ∈ R +0 (a scalar);letting ϕ f := − (cid:0) f / (cid:1) (cid:48) / f / , we automatically get I d ; f := (cid:90) (cid:16) ϕ f ◦ (cid:0) F (cid:63)d ; f (cid:1) − ( u ) (cid:17) d u < ∞ . Define the sphericized residuals Z ( n ) ell i := (cid:0) (cid:98) Σ ( n ) (cid:1) − / (cid:0) Y ( n ) i − β − β (cid:48) c ( n ) i (cid:1) = (cid:0) (cid:98) Σ ( n ) (cid:1) − / (cid:0) Z ( n ) i (cid:1) , i = 1 , . . . , n (4.4)where the matrix (cid:0) (cid:98) Σ ( n ) (cid:1) / is the symmetric root of a consistent estima-tor (cid:98) Σ ( n ) of some multiple a Σ of Σ ( a > Assumption . Under (4.1), (cid:98) Σ ( n ) − a Σ = O P ( n − / ) as n → ∞ , forsome a >
0; moreover, (cid:98) Σ ( n ) is invariant under permutations and reflections(with respect to the origin) of the residuals Z ( n ) i = ( Y ( n ) i − n β (cid:48) − β (cid:48) c ( n ) i )’s,and equivariant under their affine transformations.A traditional choice which, however, rules out heavy-tailed radial densi-ties with infinite second-order moments, is the empirical covariance matrix For simplicity, we henceforth are dropping the word “centered.” imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020
ENTER-OUTWARD RANK TESTS of the Z ( n ) i ’s. An alternative, which satisfies Assumption 4.3 without any mo-ment assumptions, is Tyler’s estimator of scatter, see Theorem 4.1 in Tyler(1987) for strong consistency, Theorem 4.2 for asymptotic normality.Under Assumption 4.3, which entails the affine invariance of Z ( n ) ell i , Propo-sition 4.1 takes the following form. Proposition . Under Assumptions 4.2 and 4.3, the model (4.1) witherror density f ε of the elliptical type (4.3) and quadratic mean differen-tiable f / is LAN (with respect to β ), with central sequence ∆ ( n ) ell (cid:98) Σ ( n ) , β ; f ( β ) := n / (cid:16)(cid:0) (cid:98) Σ ( n ) (cid:1) − / ⊗ I m (cid:17) vec Λ ( n ) ell (cid:98) Σ ( n ) , β ; f ( β )(4.5) = ∆ ( n ) ell Σ , β ; f ( β ) + o P (1) where Λ ( n ) ell (cid:98) Σ ( n ) , β ; f ( β ) := 1 n n (cid:88) i =1 ϕ f (cid:0)(cid:13)(cid:13) Z ( n ) ell i (cid:13)(cid:13)(cid:1) K ( n ) (cid:48) c (cid:0) c ( n ) i − ¯ c ( n ) (cid:1) (cid:32) Z ( n ) ell i (cid:13)(cid:13) Z ( n ) ell i (cid:13)(cid:13) (cid:33) (cid:48) (4.6) leading to a Fisher information matrix d I d ; f Σ − ⊗ I m . This LAN result, where the residuals are subjected to preliminary (empir-ical) sphericization via (cid:0) (cid:98) Σ ( n ) (cid:1) − / , highlights the fact that elliptical familieswith given f are parametrized spherical families (indexed by the shape pa-rameter Σ ). Actually, since ∆ ( n ) ell Σ , β ; f ( β ) = (cid:16) I d ⊗ Σ − / (cid:17) ∆ ( n ) ell I d , β ; f ( β ) , the limiting Gaussian shift experiments associated with elliptical and spher-ical errors coincide (with the perturbation vec( τ ) of vec( β ) in the ellipticalcase corresponding to a perturbation vec( ς ) = (cid:16) I d ⊗ Σ − / (cid:17) vec( τ ) in thespherical case). That invariance under linear sphericization of local limitingGaussian shifts, however, does not extend to the general case of Proposi-tion 4.1.
5. Rank tests for multiple-output linear models.
Elliptical or Mahalanobis rank tests .
Rank-based inference for el-liptical multiple-output linear models was developed in Hallin and Paindav-eine (2005). The ranks and the signs there are the elliptical or Mahalanobis imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 HALLIN M., HLUBINKA D., HUDECOV ´A S. ranks R ( n ) ell i and signs S ( n ) ell i —namely, the ranks of the moduli (cid:13)(cid:13) Z ( n ) ell i (cid:107) and the signs S ( n ) ell i := Z ( n ) ell i / (cid:107) Z ( n ) ell i (cid:107) , both computed, in agreement withthe above remark on the spherical nature of elliptical families, after theempirical sphericization (4.4).The validity of tests based on those elliptical ranks and signs, unfortu-nately, requires elliptical f ε . A welcome relaxation of stricter Gaussianityassumptions, ellipticity remains an extremely strong symmetry requirement;it is made, essentially, for lack of anything better but is unlikely to hold inpractice. If the assumption of ellipticity is to be waived, elliptical ranksand signs are losing their distribution-freeness for the benefit of the center-outward ranks and signs. And, since center-outward ranks and signs, in viewof Proposition 2.2, are invariant under location shift, center-outward ranktests can address the (more realistic) unspecified intercept case without anyadditional estimation step.5.2. Center-outward rank tests .
Denote by F ( n ) ± the empirical center-outward distribution associated with the n -tuple ( Z ( n )1 , . . . , Z ( n ) n ) where Z ( n ) i now is defined as Y ( n ) i − β (cid:48) c ( n ) i , by R ( n ) i ; ± and S ( n ) i ; ± , respectively, the cor-responding center-outward ranks and signs. In line with the form of thecentral sequence (4.2), consider(5.1) Λ ∼ ( n ) ± J := n − n (cid:88) i =1 K ( n ) (cid:48) c (cid:0) c ( n ) i − ¯ c ( n ) (cid:1) J (cid:48) (cid:32) R ( n ) i ; ± n R + 1 S ( n ) i ; ± (cid:33) . It follows from the asymptotic representation result of Proposition 3.1that, when the actual density is f ε , for the scores J = ϕ f ε ◦ F − ± , with ϕ f ε defined in Assumption 4.1(5.2) ∆ ∼ ( n ) β ; f ε ( β ) := n / vec Λ ∼ ( n ) ± J = ∆ ( n ) β ; f ε ( β ) + o P (1)and ∆ ∼ ( n ) β ; f ε ( β ) thus constitutes a version, based on the center-outward ranksand signs and hence distribution-free, of the central sequence ∆ ( n ) β ; f ε ( β )in (4.2). The following asymptotic normality result then holds. Proposition . Assume that Y ( n ) i satisfies (4.1) and let Assump-tions 3.1 and 4.2 hold. Then,(i) n / vec Λ ∼ ( n ) ± J is asymptotically normal, with mean and covariance I J ⊗ I m where I J := (cid:90) S d J ( u ) J (cid:48) ( u )dU d , imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 ENTER-OUTWARD RANK TESTS under the null hypothesis H ( n )0 ( β ) that β = β while the intercept β and the distribution P ∈ P d of the ε ’s remain unspecified;(ii) the test rejecting H ( n )0 ( β ) whenever the test statistic (5.3) Q ∼ ( n ) ± J := n (cid:16) vec Λ ∼ ( n ) ± J (cid:17) (cid:48) I − J ⊗ I m (cid:16) vec Λ ∼ ( n ) ± J (cid:17) exceeds the (1 − α ) quantile of a chi-square distribution with md degreesof freedom has asymptotic level α as n → ∞ ; (iii) for J = ϕ f ε ◦ F − ± where F ± denotes the center-outward distribu-tion function associated with f ε , the covariance I J ⊗ I m coincideswith I f ε ⊗ I m and the test based on Q ∼ ( n ) ± J f ε is, under error density f ε , locally asymptotically maximin, at asymptotic level α , for the nullhypothesis H ( n )0 ( β ) . Proof.
First assume that the error distribution P, with center-outwarddistribution function F ± , is in P ± d . Noting that, for column vectors a and b ,we have vec (cid:0) ab (cid:48) (cid:1) = b ⊗ a , n / vec Λ ∼ ( n ) ± J = n − / n (cid:88) i =1 vec (cid:104) K ( n ) (cid:48) c ( c ( n ) i − ¯ c ( n ) ) J (cid:48) (cid:16) R ( n ) i ; ± n R + 1 S ( n ) i ; ± (cid:17)(cid:105) = n − / n (cid:88) i =1 J (cid:16) R ( n ) i ; ± n R + 1 S ( n ) i ; ± (cid:17) ⊗ (cid:16) K ( n ) (cid:48) c ( c ( n ) i − ¯ c ( n ) ) (cid:17) . It follows from Proposition 3.1 that this latter statistic is asymptoticallyequivalent to T J = n − / n (cid:88) i =1 J (cid:16) F ± ( Z ( n ) i ) (cid:17) ⊗ (cid:16) K ( n ) (cid:48) c ( c ( n ) i − ¯ c ( n ) ) (cid:17) Since Q ∼ ( n ) ± J is distribution-free under the null hypothesis H ( n )0 ( β ), the finite- n sizeof this test is uniform over H ( n )0 ( β ), hence uniformly close to α for n large enough. This isin sharp contrast with daily practice pseudo-Gaussian tests, which remain asymptoticallyvalid under a broad range of distributions, albeit not uniformly so. imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 HALLIN M., HLUBINKA D., HUDECOV ´A S. which is a sum of independent variables such that E T J = 0 andVar T J = n − n (cid:88) i =1 Var (cid:104) J (cid:16) F ± ( Z ( n ) i ) (cid:17) ⊗ (cid:16) K ( n ) (cid:48) c ( c ( n ) i − ¯ c ( n ) ) (cid:17)(cid:105) = n − n (cid:88) i =1 E (cid:104) J (cid:16) F ± ( Z ( n ) i ) (cid:17) ⊗ (cid:16) K ( n ) (cid:48) c ( c ( n ) i − ¯ c ( n ) ) (cid:17) × J (cid:48) (cid:16) F ± ( Z ( n ) i ) (cid:17) ⊗ (cid:16) ( c ( n ) i − ¯ c ( n ) ) (cid:48) K ( n ) c (cid:105) = n − n (cid:88) i =1 E (cid:104) J (cid:16) F ± ( Z ( n ) i ) (cid:17) J (cid:48) (cid:16) F ± ( Z ( n ) i ) (cid:17) ⊗ (cid:16) K ( n ) (cid:48) c ( c ( n ) i − ¯ c ( n ) )( c ( n ) i − ¯ c ( n ) ) (cid:48) K ( n ) c (cid:17)(cid:105) = (cid:90) S d J ( u ) J (cid:48) ( u ) dU d ⊗ n − K ( n ) (cid:48) c n (cid:88) i =1 (cid:104) ( c ( n ) i − ¯ c ( n ) )( c ( n ) i − ¯ c ( n ) ) (cid:48) (cid:105) K ( n ) c which tends to I J ⊗ I m as n → ∞ . The Lindeberg condition is satisfied,so that T J , hence also n / vec Λ ∼ ( n ) ± J , has the announced asymptotic normaldistribution.Finally, consider the general case of an absolutely continuous P ∈ P d : asin the proof of Proposition 3.2, distribution-freeness implies that the asymp-totic distribution of n / vec Λ ∼ ( n ) ± J is the same under P ∈ P d as under P ∈ P ± d .This completes the proof of Part (i) . In view of (5.2); Parts (ii) and (iii) readily follow. (cid:3) Corollary . (i) In the particular case of a spherical score of theform (3.8) , the test statistic Q ∼ ( n ) ± J simplifies into (5.4) Q ∼ ( n ) ± J = nd (cid:82) J ( u )d u (cid:16) vec Λ ∼ ( n ) ± J (cid:17) (cid:48) (cid:16) vec Λ ∼ ( n ) ± J (cid:17) where Λ ∼ ( n ) ± J := n − n (cid:88) i =1 J (cid:32) R ( n ) i ; ± n R + 1 (cid:33) K ( n ) (cid:48) c (cid:0) c ( n ) i − ¯ c ( n ) (cid:1) S ( n ) (cid:48) i ; ± and n / vec Λ ∼ ( n ) ± J is asymptotically normal with mean and vari-ance d − (cid:82) J ( u )d u I md .(ii) The test statistic Q ∼ ( n ) ± J with spherical score (5.5) J f := ϕ f ◦ (cid:0) F (cid:63)d ; f (cid:1) − imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 ENTER-OUTWARD RANK TESTS yields locally asymptotically optimal tests under the spherical densitywith radial density f . Some particular cases.
In this section, we provide explicit forms ofthe test statistic for the two-sample and MANOVA problems. Because oftheir simplicity and practical value (see Section 6.1), we concentrate on thecase (5.4) of spherical scores, from which the general case (5.3) is easilydeduced (essentially, by substituting J (cid:16) R ( n ) i ; ± n R +1 S ( n ) i ; ± (cid:17) for J (cid:16) R ( n ) i ; ± n R +1 (cid:17) S ( n ) i ; ± ).5.3.1. Center-outward rank tests for two-sample location.
An importantparticular case is the two-sample location model, where n = n + n and (4.1)holds with covariates of the form C ( n ) = ( (cid:48) n , (cid:48) n ) (cid:48) (with n an n -dimen-sional column vector of ones, n an n -dimensional column vector of zeros);the parameter β = ( β , . . . , β d ) (cid:48) here is a d -dimensional row vector.The objective is to test the null hypothesis H : β = d under which thedistributions of Y ( n )1 , . . . , Y ( n ) n and Y ( n ) n +1 , . . . , Y ( n ) n coincide. Elementarycomputation yields¯ c ( n ) = n /n, V ( n ) c = n n /n , and K c = 1 . If the regular grid G n is chosen such that (cid:107) (cid:80) n S s =1 s n S s (cid:107) = (which is alwayspossible, see Section 2.2), (cid:80) ni =1 J (cid:16) R ( n ) i ; ± n R +1 (cid:17) S ( n ) i ; ± = and the test statistic (5.4)takes the simple form Q ∼ ( n ) ± J = (cid:18) nd/n n (cid:90) J ( u )d u (cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n (cid:88) i =1 J (cid:16) R ( n ) i ; ± n R + 1 (cid:17) S ( n ) i ; ± (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . (5.6)Assumption 4.2 (iii) yields lim n →∞ n min { n , n } / max { n , n } = ∞ , whichholds whenever(5.7) lim n →∞ min { n , n } = ∞ . Under Assumptions 3.1 and (5.7), with P ∈ P d , Q ∼ ( n ) ± J is, under H , asymp-totically χ with d degrees of freedom and the null hypothesis can be rejectedat asymptotic level α whenever Q ∼ ( n ) ± J exceeds the (1 − α ) quantile of a χ d distribution.5.3.2. Center-outward rank tests for MANOVA.
Another important spe-cial case of model (4.1) is the multivariate K -sample location or MANOVA imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 HALLIN M., HLUBINKA D., HUDECOV ´A S. model. The observation here decomposes into K samples, with respectivesizes n , . . . , n K and n = (cid:80) Kk =1 n k . Precisely, Y ( n ) =: (cid:0) Y ( n ;1) , . . . , Y ( n ; k ) , . . . , Y ( n ; K ) (cid:1) with Y ( n ; k ) = Y ( n ) k ;11 Y ( n ) k ;12 . . . Y ( n ) k ;1 d ... ... ... Y ( n ) k ; n k Y ( n ) k ; n k . . . Y ( n ) k ; n k d and (4.1) holds with the matrix of covariates C ( n ) = d ( n )11 d ( n )12 . . . d ( n )1 ,K − d ( n )21 d ( n )22 . . . d ( n )2 ,K − ... ... · · · ... d ( n ) K d ( n ) K . . . d ( n ) K,K − , where d ( n ) ij = n i I [ i = j ], i = 1 , . . . , K and j = 1 , . . . , K −
1. The null hypoth-esis is the hypothesis of no treatment effect H : β = ( K − × d .Letting v ( n ) := ( n /n, . . . , n K − /n ) (cid:48) , the matrix V c ( n ) in Assumption 4.2takes the form V c ( n ) = diag { v ( n ) } − v ( n ) v ( n ) (cid:48) , where diag { v ( n ) } stands forthe diagonal matrix with diagonal entries v ( n ) . If the regular grid S n ischosen such that (cid:107) (cid:80) n S s =1 s n S s (cid:107) = 0, the test statistic (5.4) simplifies into Q ∼ ( n ) ± J = d (cid:82) J ( u )d u K (cid:88) k =1 n k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n + ... + n k (cid:88) i = n + ... + n k − +1 J (cid:16) R ( n ) i ; ± n R + 1 (cid:17) S ( n ) i ; ± (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) where ( V ( n ) c ) − / is substituted for its limit K ( n ) c .Assumption 4.2 (iii) is satisfied as soon as lim n →∞ min { n , . . . , n K } → ∞ . Assuming moreover that lim inf n →∞ n k /n > n →∞ n k /n < ∞ )for 1 ≤ k ≤ K , the limit matrix R c is positive definite and Assump-tion 4.2 (ii) is satisfied as well. Then, under the null hypothesis of no treat-ment effect, Q ∼ ( n ) ± J is asymptotically chi-square with ( K − d degrees offreedom and the test rejecting H whenever Q ∼ ( n ) ± J exceeds the correspon-ding (1 − α ) quantile has asymptotic level α irrespective of the actual errordistribution P ∈ P d . This test is a multivariate generalization of the well-known univariate rank test for K -sample equality of location (the univariate This limit possibly can exist along subsequences, with asymptotic statements modi-fied accordingly. For the sake of simplicity, we do not include this in subsequent results. imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020
ENTER-OUTWARD RANK TESTS one-way ANOVA hypothesis of no treatment effect), see (H´ajek and ˇSid´ak,1967, p.170). Note that, for K = 2, Q ∼ ( n ) ± J coincides with the two-sample teststatistic obtained Section 5.3.1.
6. Choosing a score function.
Section 5 allows us to construct, basedon any score function ( J or J ) satisfying Assumption 3.1 (either with (3.7)or (3.9)), strictly distribution-free center-outward rank tests of the null hy-pothesis H ( n )0 ( β ) under which β = β while the intercept β and the errordistribution P ∈ P d remain unspecified. All these tests, however, depend ona score function to be selected by the practitioner. Some will favor simplescores of the spherical type (see Section 6.1); others may want to base theirchoice on efficiency considerations (see Section 6.2).6.1. Standard score functions.
Popular choices are the spherical signtest, Wilcoxon and van der Waerden scores. Let us describe them, in moredetails, in the particular case of the two-sample problem.The two-sample sign test is based on the degenerate score J sign ( r ) := 1for r ∈ [0 , (cid:80) ni =1 S ( n ) i ; ± = , one gets for (5.6), with thenotation of Section 5.3.1, the very simple test statistic Q ∼ ( n ) ± sign = ndn n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n (cid:88) i =1 S ( n ) i ; ± (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . The choice J Wilcoxon ( r ) := r similarly yields the Wilcoxon two-sample test:noting that (cid:80) ni =1 R ( n ) i ; ± S ( n ) i ; ± = holds if (cid:80) ni =1 S ( n ) i ; ± = and that (cid:82) r d u = 1 / Q ∼ ( n ) ± Wilcoxon = 3 ndn n ( n R + 1) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n (cid:88) i =1 R ( n ) i ; ± S ( n ) i ; ± (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . As for the two-sample van der Waerden test, it is based on the Gaussian orvan der Waerden scores J vdW ( r ) := (cid:0) Ψ − d ( r ) (cid:1) / , where Ψ d denotes the cumu-lative distribution function of a chi-square variable with d degrees of freedom.Clearly (cid:82) J vdW ( r )d r = (cid:82) ∞ x dΨ d ( x ) = d and, provided that (cid:80) ni =1 S ( n ) i ; ± = 0, n (cid:88) i =1 (cid:16) Ψ − d (cid:16) R ( n ) i ; ± n R + 1 (cid:17)(cid:17) / S ( n ) i ; ± = 0 . Hence, the van der Waerden center-outward rank test statistics takes the form Q ∼ ( n ) ± vdW = nn n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n (cid:88) i =1 (cid:16) Ψ − d (cid:16) R ( n ) i ; ± n R + 1 (cid:17)(cid:17) / S ( n ) i ; ± (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 HALLIN M., HLUBINKA D., HUDECOV ´A S.
Score functions and efficiency.
The tests statistics in Section 6.1offer the advantage of a structure paralleling the structure of the numeratorof the classical Gaussian F test—basically substituting, in the latter, S ( n ) i ; ± (sign test scores), R ( n ) i ; ± S ( n ) i ; ± (Wilcoxon scores), or (cid:16) Ψ − d (cid:16) R ( n ) i ; ± n R +1 (cid:17)(cid:17) / S ( n ) i ; ± (vander Waerden scores) for the sphericized residuals (4.4) and adopting theadequate standardization.The choice of a score function also can be guided by efficiency consid-erations, selecting J in relation to some reference distribution under whichefficiency is to be attained. This, in the univariate case, yields the normal(van der Waerden), Wilcoxon or sign test scores, achieving efficiency un-der Gaussian, logistic, or double exponential reference densities; as we shallsee, Q ∼ ( n ) ± sign and Q ∼ ( n ) ± vdW similarly achieve efficiency at spherical exponentialand Gaussian reference distributions. In the same spirit, one could contemplate the idea of achieving, basedon center-outward rank tests, efficiency at some selected reference distri-bution P ε in P d (with density f ε and center-outward distribution func-tion F ε ± satisfying the adequate regularity assumptions). Indeed, it followsfrom Proposition 5.1 that efficiency under P ε can be achieved by a test basedon the test statistic Q ∼ ( n ) ± J given in (5.3) with score J = ϕ f ε ◦ (cid:0) F ε ± (cid:1) − .This, however, raises two problems. First, in order for ϕ f ε to be ana-lytically computable, the distribution P ε has to be fully specified (up tolocation and a global scaling parameter), with closed-form density func-tion f ε . Second, the corresponding score J = ϕ f ε ◦ (cid:0) F ε ± (cid:1) − also involvesthe center-outward quantile function ( F ε ± ) − for which, except for a fewparticular cases (spherical distributions), no explicit form is available in theliterature.Once P ε is fully specified, in principle, it can be simulated, and an ar-bitrarily precise numerical evaluation of ( F ε ± ) − can be obtained, to beplugged into J . This may be computationally heavy, but increasingly effi-cient algorithms are available in the very active domain of numerical measuretransportation: see, e.g., M´erigot (2011) or Peyr´e and Cuturi (2019).Now, choosing a fully specified reference P ε may be embarrassing—this The computation of which, moreover, requires the specification of β or its consistentestimation—something center-outward ranks and signs do not need in view of their shift-invariance. Due to the fact that the density f (cid:63)d ; f of the modulus of a spherical logistic fails to belogistic for d > Q ∼ ( n ) ± Wilcoxon , however, does not enjoy efficiency under spherical logistic;this is also the case of the elliptical rank tests based on Wilcoxon scores in Hallin andPaindaveine (2002a,b, 2005). imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020
ENTER-OUTWARD RANK TESTS means, for instance, a skew- t distribution with specified degrees of freedom,shape matrix, and skewness parameter (without loss of generality, locationcan be taken as ), a multinormal or elliptical distribution with specified ra-dial density and specified (up to a positive global factor) covariance (again,the mean can be taken as ), ... Fortunately, a full specification of P ε canbe relaxed to the specification of a parametric family with parameter ϑ , say,such as the family P skew t of all skew- t distributions with location (parame-ters: a shape matrix and a d -tuple of skewness parameters) or the family P ell f of all elliptical distributions (4.3) with radial density f (parameter: a scat-ter matrix). The unspecified parameter ϑ of P ε indeed can be replaced,in the numerical evaluation of F ε ± , with consistent estimated values pro-vided that the estimator ˆ ϑ are measurable with respect to the order statis-tic of the residuals Z ( n ) i . Plugging these estimators into the score J —thisinclude the standardization factor and the numerical evaluation of F ε ± —yields data-driven (order-statistic-driven) scores J ( n ) . Conditionally on theorder statistic, the corresponding test statistic is still distribution-free and its(conditional) critical values remain unconditionally correct. However, thesecritical values involve the order statistic: the resulting tests therefore nolonger are ranks tests but permutation tests. The theoretical properties, feasibility, and finite-sample performance ofthis data-driven approach should be explored and numerically assessed—this is, however, beyond the scope of this paper and we leave it for futureresearch.In view of this, no obvious non-spherical candidate emerges, in dimen-sion d >
1, as a reference density. The center-outward test statistic achiev-ing optimality at the spherical distributions with radial density f is Q ∼ ( n ) ± J f with J f as in (5.5).6.3. Affine invariance and sphericization.
Affine invariance (testing) orequivariance (estimation), in “classical multivariate analysis,” is generallyconsidered an essential and inescapable property. Closer examination, how-ever, reveals that this particular role of affine transformations is intimatelyrelated to the affine invariance of Gaussian and elliptical families of distri- The order statistic of the n -tuple Z , . . . , Z n of d -dimensional ( d >
1) random vectorscan be defined as any reordering Z (1) , . . . , Z ( n ) generating the σ -field of permutation-invariant Borel sets of σ (cid:0) Z , . . . , Z n (cid:1) ; for instance, the one resulting from ordering theobservations Z i from smallest to largest first component. Similar data-driven score ideas have been proposed in the univariate context by Dodgeand Jureˇckov´a (2000). A permutation test is a test enjoying Neyman α -structure with respect to the sufficientand complete order statistic. imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 HALLIN M., HLUBINKA D., HUDECOV ´A S. butions. When Gaussian or elliptical assumptions are relaxed, affine trans-formations are losing this privileged role and the relevance of affine invari-ance/equivariance properties is much less obvious.When Y ( n ) A (cid:48) (where A is an arbitrary full-rank d × d matrix), is ob-served instead of Y ( n ) , (cid:98) Σ ( n ) is replaced with (cid:98) Σ ( n ) A = A (cid:98) Σ ( n ) A (cid:48) , yieldingsphericized residuals of the form Z ( n ) ell A ; i := (cid:0) A (cid:98) Σ ( n ) A (cid:48) (cid:1) − / AZ ( n ) i instead of Z ( n ) ell i . It follows from elementary calculation that Z ( n ) ell A ; i = PZ ( n ) ell i with P = ( A (cid:98) Σ ( n ) A (cid:48) ) − / A ( (cid:98) Σ ( n ) ) / orthogonal. Strictly speaking, sphericizedresiduals, thus, are not affine-invariant. This possible discrepancy betweensphericized residuals is due to the fact that square roots such as ( (cid:98) Σ ( n ) ) − / are only defined up to an orthogonal transformation; choosing the symmet-ric root is a convenient choice, but does not yield P = I d . However, themoduli (cid:107) Z ( n ) ell A ; i (cid:107) and (cid:107) Z ( n ) ell i (cid:107) coincide, irrespective of P , and so do thecosines (cid:104) Z ( n ) ell A ; i , Z ( n ) ell A ; j (cid:105)(cid:107) Z ( n ) ell A ; i (cid:107)(cid:107) Z ( n ) ell A ; j (cid:107) and (cid:104) Z ( n ) ell i , Z ( n ) ell j (cid:105)(cid:107) Z ( n ) ell i (cid:107)(cid:107) Z ( n ) ell j (cid:107) , i, j = 1 , . . . , n. The affine-invariance of typical elliptical-rank-based test statistics, whichare quadratic forms involving those moduli and cosines, follows.Being measurable with respect to the ranks of the moduli (cid:107) Z ( n ) ell i (cid:107) and thescalar products (cid:104) Z ( n ) ell i , Z ( n ) ell j (cid:105) / (cid:107) Z ( n ) ell i (cid:107)(cid:107) Z ( n ) ell j (cid:107) , the elliptical rank statis-tics developed in Hallin and Paindaveine (2005) are affine-invariant; this isin full agreement with our previous remark that the limiting local Gaussianshifts in elliptical experiments are unaffected under affine transformations.The center-outward distribution functions, ranks and signs cannot be ex-pected to enjoy similar affine-invariance properties—actually, it has beenproved (Proposition 3.14 in Cuesta-Albertos et al. (1993)) that they do not.If, however, affine invariance is considered an indispensable property, it iseasily restored: choosing your favorite (consistent under ellipticity) estimatorof scatter (cid:98) Σ ( n ) (which also requires, in case β is not specified, an estima-tor of location ˆ µ ( n ) ), just compute the sphericized residuals Z ( n ) ell i definedin (4.4) prior to computing the center-outward ranks and signs and per-forming the tests: in view of Proposition 2.2, the resulting center-outwardranks and signs enjoy the same affine-invariance properties (invariance ofthe ranks and the cosines of signs) as the elliptical ones.If the actual density f ε is elliptical, this linear sphericization does notmodify the local experiment, hence local asymptotic powers and, in case The Cholesky square root does: see Proposition 2 in Hallin et al. (2020b). imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020
ENTER-OUTWARD RANK TESTS the scores themselves are spherical, efficiency properties, are preserved. Ifthe actual density f ε is not elliptical, however, such linear sphericization has a nonlinear impact on f ε -based central sequences: the correspondingGaussian shift experiments are not preserved and, irrespective of the scoresthey are based on, the local asymptotic powers of center-outward rank testsare affected. Summing up, preliminary sphericization does restore affine-invariance of center-outward rank tests while preserving their local powersunder ellipticity, but distorts those local powers under non-elliptical errordensities. Figures 4 and 5 below provide examples where that distortionsignificantly deteriorates the power.Whether affine-invariance is desirable or not is open to discussion. In“classical multivariate analysis,” that is, under Gaussian or elliptical den-sities, linear sphericization preserves local experiments, making affine in-variance a natural requirement. When considering more general error dis-tributions P, linear transformations are losing their privileged status: theyno longer sphericize the distribution P of a typical Z ∼ P and no longerpreserve local experiments. Moreover, while all consistent-under-ellipticityestimators (cid:98) Σ ( n ) and ˆ µ ( n ) yield, under ellipticity, the same limiting locationand scatter values, distinct estimators, under non-elliptical densities, willconverge to distinct and sometimes hardly interpretable limits: ˆ µ ( n ) = X ( n ) (the arithmetic mean) and ˆ µ ( n ) = X ( n ) Oja (the
Oja median , Oja (1983)), whichasymptotically coincide under ellipticity, may yield completely distinct lo-cations; what is the relevance of Tyler’s scatter matrix in a distributionwhere sign curves are not straight lines and decorrelation of radii (throughwhich center ˆ µ ( n ) ?) makes little sense? etc. The distortion of local pow-ers under non-elliptical error densities thus depends on the choice of (cid:98) Σ ( n ) and ˆ µ ( n ) , which is hard to justify. While easily implementable, affine invari-ance/equivariance, in such a context, is thus a disputable requirement.
7. Some numerical results.
A small Monte Carlo simulation studyis conducted (Sections 7.1–7.2) in order to explore the finite-sample per-formance of our tests. Results are presented for two-sample location andMANOVA models, and limited to Wilcoxon score function J ( r ) = r ; otherchoices for J lead to very similar figures, which we therefore do not report.7.1. Two-sample location test.
Consider first a two-sample location prob-lem in dimension d = 2. Two independent random samples of size n = n = Actually, only a “second-order sphericization,” as the distribution of Z ( n ) ell i still failsto be spherical unless the error distribution itself was. The signs, indeed, now are sitting “inside” the function ϕ f ε . imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 HALLIN M., HLUBINKA D., HUDECOV ´A S. n/ Q ∼ ( n ) ell Wilcoxon and Q ∼ ( n ) ± Wilcoxon (see Sec-tion 5.3.1) were computed. The sample covariance matrix (cid:98) Σ was used for thecomputation of Λ ∼ ( n ) ell Wilcoxon and the elliptical or Mahalanobis ranks and signs.Rejection frequencies were computed for the following error densities: (a) a centered bivariate normal distribution with unit variances and cor-relation ρ = 1 / (b) a centered bivariate t -distribution with the same scaling matrix as in (a) and ν degree of freedom, ν = 1 (Cauchy) and ν = 3; (c) a mixture, with weights w = 1 / w = 3 /
4, of two bivariatenormal distributions with means µ = (3 / , (cid:48) and µ = ( − / , (cid:48) and covariance matrices Σ = (cid:18) / / (cid:19) and Σ = (cid:18) − / − / (cid:19) , respectively; (d) a mixture, with weights w = 1 / w = 3 /
4, of two bivariate t (Cauchy) distributions centered at µ = (3 / , (cid:48) and µ = ( − / , (cid:48) ,with the same scaling matrices Σ and Σ as in (c) ; (e) a “U-shaped” mixture, with weights w = 1 / w = 1 /
4, and w =1 /
4, of three bivariate normal distributions, N ( µ , Σ ), N ( µ , Σ ),and N ( µ , Σ ) where µ = (0 , (cid:48) , µ = ( − , (cid:48) , µ = (3 , (cid:48) , and Σ = (cid:18) / (cid:19) , Σ = (cid:18) / − / − / / (cid:19) , Σ = (cid:18) / / / / (cid:19) ; (f ) an “S-shaped” mixture, with equal weights w = 1 /
3, of three bivariatenormal distributions, N ( µ , Σ ), N ( µ , Σ ), and N ( µ , Σ ) where µ = ( − / , − / (cid:48) , µ = (0 , − / (cid:48) , µ = (9 / , (cid:48) , and Σ = (cid:18) / − (cid:112) / − (cid:112) / (cid:19) , Σ = (cid:18) / (cid:112) / (cid:112) / (cid:19) , Σ = (cid:18) / − (cid:112) / − (cid:112) / (cid:19) ; The bivariate t -distribution with m degrees of freedom and scaling matrix A (cid:48) A is theone defined in Example 2.5 of Fang et al. (1990) as the distribution of a random vector ξ := µ + A (cid:48) ζ √ m/ √ s where ζ ∼ N ( , I ) and s ∼ χ m , independent of ζ —not to beconfused with the elliptical distribution with Student radial density f . imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 ENTER-OUTWARD RANK TESTS l ll l lll lll ll lll lll l llll l ll ll l l lll lll ll l ll ll lllll ll ll ll ll lllll lll l lll lll ll ll ll lllll lll l l llll llll l ll llllll l ll l l l ll l llll ll ll llll lll ll l lll l ll ll ll ll lll llll ll lll l lll l lll lll llll l llll ll lll lll ll ll l lll l ll ll ll ll −3 −2 −1 0 1 2 − − − Mixture of two normals ll lll l ll lll l llll ll l ll lll lllll lll ll lll l l ll ll lllll ll lll l ll ll l l lllll llll ll llll ll llll lll l llll ll ll ll ll ll lll ll lll ll ll l ll ll ll ll llllll ll l l ll ll l ll ll lll lll l lll ll l lll l ll l l ll l ll lll ll lll ll ll ll ll l ll lll lll l ll ll l lll l −4 −2 0 2 4 − U−shaped mixture ll lll l ll lll ll lll ll llll ll ll llllll l l l ll l lll ll llll lll l ll l ll ll l lll lll llll ll ll l ll l lllll ll l llll ll llll ll l llll ll lll l lll l ll l l ll l llll l ll ll ll ll ll l ll ll ll llll l llll l l lll lll ll lll llll l ll lll ll ll ll ll ll l l ll lll l ll ll l lll l −5 0 5 − − − S−shaped mixture lll ll l ll ll lll lll ll lllll lll l lllll l lll lll l ll ll ll ll ll lll ll ll lll l ll lll lll l llll ll ll l lll lll l llll lll ll l lll l lll lll ll lll lll lll ll llll lll ll ll ll ll ll lll lllll ll llll ll l ll lll l ll ll ll lll ll lll llll l ll lll lllll l llll ll l lll l −1 0 1 2 3 4 5 − − − − − Skew t, 3 df
Fig 1 . Sample plots of observations drawn form the the Gaussian mixtures (c), (e),and (f) and from the skew t -distribution with degrees of freedom (g). (g) a skew t -distribution with ν degrees of freedom, ν = 1 and 3, withskewness parameter α = (5 , − (cid:48) , scaling matrix Σ = (cid:18) − / − / (cid:19) ,and location parameter ξ = . For the interpretation of the param-eters of the skewed t -distribution see (Azzalini and Capitanio, 2014,Chapter 6).Mixture error densities quite naturally appear in the context of hiddenheterogeneities due, for instance, to omitted covariates; as for asymmetries,they are likely to be the rule rather than the exception. Samples of size 200from the Gaussian mixtures (c) , (e) , and (f ) and from the skew t -distributionwith 3 degrees of freedom (g) are shown in Figure 1.The first sample was generated from one of the distributions (a)–(g) , andthe second sample was drawn from the same distribution shifted by the imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 HALLIN M., HLUBINKA D., HUDECOV ´A S. vector ( δ, δ ) (cid:48) for δ ∈ [0 . , . N = 1000times and the empirical size and power of the test were computed for α =0 .
05. The resulting rejection frequencies illustrate the dependence of thepower on the parameter δ ; they are provided in Figures 2, 3, 4, and 5.The power of the center-outward rank test statistic Q ∼ ( n ) ± is plotted asa solid line, the power of the elliptical rank test statistic Q ∼ ( n ) ell as a dashedline. For the sake of comparison, we also provide the power of the classicaltwo-sample Hotelling test, shown as a dotted line. Three different samplesizes n = n = 50 , n = 100 , G n for computation of the center-outward ranks and signs are constructed with n S = n R = 10 for n = 100, n S = n R = 20 for n = 400, and n S = n R = 30 for n = 900.Figure 2 displays the empirical power curves for the elliptical distributions (a) and (b) . The results for the normal distribution are very similar for thethree tests: rank-based tests (Wilcoxon scores), thus, are no less powerfulthan the optimal Hotelling test. As expected, the Hotelling test fails forthe t distribution, while the elliptical test based on the sample covariancematrix performs surprisingly well here (the robustness benefits of ranks).The tests based on Q ∼ ( n ) ell Wilcoxon and Q ∼ ( n ) ± Wilcoxon both outperform the Hotellingtest also for the t -distribution with 3 degrees of freedom. The conclusion isthat center-outward rank tests perform equally well as elliptical rank testsunder elliptical densities.The remaining distributions (c)–(g) are non-elliptical ones. Results forthe mixtures (c) and (d) are shown in Figure 3. For the mixture (c) oftwo normals, the results obtained for the three tests are still quite similar,but the center-outward rank test based on Q ∼ ( n ) ± Wilcoxon , in general, yields thelargest power. For the mixture (d) of two t (Cauchy) distributions, theHotelling test fails miserably and the center-outward rank test very clearlyoutperforms the elliptical rank test for all sample sizes.Figures 4 and 5 provide the results for the mixtures (e)–(f ) and the skew t -distribution (g) , respectively. The power curve for the test statistic Q ∼ ( n ) ± Wilcoxon computed from the linearly sphericized residuals (using the sample meanand the sample covariance matrix as estimators of location and scatter) isadded as a dot-dashed line. In all these plots, the center-outward rank teststatistic leads to the largest power. Note that the linear sphericization of theresiduals, which makes the test affine-invariant, may noticeably deterioratethe power (see the discussion in Section 6.3). imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020
ENTER-OUTWARD RANK TESTS One-way MANOVA.
The performance of center-outward rank testsis very briefly studied here for one-way MANOVA with K = 3 groups, stillfor d = 2. Two random samples were generated from the distribution (a) (Gaussian) or (e) (U-shaped mixture of three Gaussians), as described inSection 7.1, and the third sample was drawn from the same distributionshifted by the vector ( δ, δ ) (cid:48) for δ ∈ [0 . , . n = n = n = 75 (hence n = 225) and n = n = n = 300(hence n = 900) was considered. For n = 225, the grid G n is constructedwith n R = n S = 15; for n = 900, we set n R = n S = 30. As in Section 7.1, theresults are presented for the Wilcoxon scores J ( r ) = r only—other choiceslead to very similar conclusions.The empirical power curves are plotted in Figure 6 for the center-outwardrank test based on Q ∼ ( n ) ± Wilcoxon (solid line), the elliptical rank test statistic Q ∼ ( n ) ell Wilcoxon (dashed line), and the classical Pillai’s test based on an approximateF-distribution (dotted line). For the normal distribution, all the three testsperform very similarly. For the non-elliptical mixture distribution, however,the center-outward rank test provides a noticeably larger power comparedto the other two methods.7.3.
An empirical illustration.
The practical value of the center-outwardrank tests is illustrated by the following archeological application whereclassical methods fail to detect any treatment effect. The data consist of n = 126 measurements of MgO, P O , CoO, and Sb O in natron glassvessels excavated from three Syro-Palestinian sites in present-day Israel:Apollonia ( n = 54 observations), Bet Eli’ezer ( n = 17 observations), andEgypt ( n = 55 observations); a fourth site only has two observations, andtherefore was dropped from the analysis. This dataset is has been originallyanalyzed by Phelps et al. (2016) with the objective of detecting possibledifferences among the three sites. Bivariate plots of these four variables areshown in Figure 7, where one can observe that the marginal distributionsof CoO, and Sb O exhibit heavy tails and are very far from normal, whilethe joint distribution seems far from elliptical.First, all the two-dimensional data subsets corresponding to the bivari-ate plots in Figure 7 were analyzed (six bivariate MANOVA models, thus).Pillai’s classical test yields non-significant p -values for all combinations, seeTable 1. On the other hand, the center-outward tests are able to detect dif-ferences between the three groups whenever the variable CoO is included inthe analysis. The center-outward ranks and signs can be computed from agrid G n with either n S = 7 and n R = 18 or n S = 18 and n R = 7; both choiceshave been implemented in Table 1 below (c-o tests I and II, respectively).Table 1 reveals some differences in the p -values for c-o tests I and II which, imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 HALLIN M., HLUBINKA D., HUDECOV ´A S. however, yield the same conclusions at significance level α = 0 .
05. The testsbased on elliptical ranks (based on the sample covariance function) lead tohighly non-significant p -values for all couples of varables; consequently, thecorresponding results are not presented here. Pillai’s test c-o test I c-o test IIMgO P O O O CoO 0.1491 0.0000 0.0000P O Sb O O Table 1 p -values for the bivariate Pillai MANOVA and the Wilcoxon center-outward rank testsbased on n R = 7 , n S = 18 (c-o test I) and n R = 18 , n S = 7 (c-o test II), respectively. Next, the MANOVA comparison is conducted for the full 4-dimensionaldataset. Pillai’s test leads to a p -value 0 . α = 0 .
05: no difference gets detected among the three groups, thus,with this classical method. In sharp contrast, the Wilcoxon center-outwardrank test (with n R = 7 and n S = 18) yields a p -value 10 − , which is highlysignificant. The elliptical Wilcoxon rank test (based on sample covariancematrix), on the other hand, with p -value 0 . α = 0 .
8. Conclusion and perspectives.
Classical multivariate analysis meth-ods, which are daily practice in a number of applied domains, remain deeplymarked by Gaussian and elliptical assumptions. In particular, no distribution-free approach is available so far for hypothesis testing in multiple-output re-gression models, which include the fundamental two-sample and MANOVAmodels—except for the elliptical or Mahalanobis rank tests developed inHallin and Paindaveine (2004) which, unfortunately, require the strong as-sumption of elliptic symmetry, an assumption which is unlikely to hold inmost applications. Based on the recent concept of center-outward ranks andsigns, this paper proposes the first fully distribution-free tests of the hypoth-esis of no treatment effect in the context, thereby extending to the multi-variate case the classical H´ajek approach to univariate rank-based inference(H´ajek and ˇSid´ak, 1967). Simulations and an empirical example demonstratethe excellent performance of the method. This is only a first step into the di-rection of complete toolkit of distribution-free methods for multiple-outputanalysis of variance problems, but it lays the theoretical bases (asymptoticrepresentation and asymptotic normality results for linear center-outward imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020
ENTER-OUTWARD RANK TESTS rank statistics) and theoretical guidelines ( H´ajek projection of LAN centralsequences) for further developments.
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E-mail: [email protected]
ECARES and D´epartement de Math´ematiqueUniversit´e libre de BruxellesBrusselsBelgium E-mail: [email protected]ff.cuni.cz
Faculty of Mathematics and PhysicsCharles UniversityPragueCzech RepublicE-mail: [email protected]ff.cuni.cz
Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 HALLIN M., HLUBINKA D., HUDECOV ´A S. . . . . . . Normal distribution d P o w e r n=100n=400n=900 0.00 0.05 0.10 0.15 0.20 0.25 . . . . . . t−distribution, 1 df d P o w e r n=100n=400n=9000.00 0.05 0.10 0.15 0.20 0.25 . . . . . . t−distribution, 3 df d P o w e r n=100n=400n=900 Fig 2 . The empirical powers of two-sample location tests based on the Wilcoxon center-outward rank statistic (solid line), the elliptical rank statistic (dashed line), and the two-sample Hotelling test (dotted line) as functions of the shift δ under normal and Student t error densities; sample sizes n = n = 50 , , and . imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 ENTER-OUTWARD RANK TESTS . . . . . . Mixture of two normals d P o w e r n=100n=400n=900 0.00 0.05 0.10 0.15 0.20 0.25 . . . . . . Mixture of two t−distributions d P o w e r n=100n=400n=900 Fig 3 . The empirical powers of two-sample location tests based on the Wilcoxon center-outward rank statistic (solid line), the Wilcoxon elliptical rank test statistic (dashed line),and the two-sample Hotelling test (dotted line), as functions of the shift δ , for the mixturesof two normal (left panel) and two t error densities (right panel), respectively; samplesizes n = n = 50 , , and . . . . . . . U−shaped mixture d P o w e r n=100n=400n=900 0.00 0.05 0.10 0.15 0.20 0.25 . . . . . . S−shaped mixture d P o w e r n=100n=400n=900 Fig 4 . The empirical powers of two-sample location tests based on the Wilcoxon center-outward rank statistic (solid line), the Wilcoxon elliptical rank test statistic (dashed line),the two-sample Hotelling test (dotted line), and the center-outward rank statistic computedfrom linearly sphericized residuals (dot-dashed line), as functions of the shift δ , for the”U-shaped” (left panel) and the ”S-shaped” mixtures of three normal error densities (rightpanel), respectively; sample sizes n = n = 50 , , and . imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 HALLIN M., HLUBINKA D., HUDECOV ´A S. . . . . . . Skew t, 1 df d P o w e r n=100n=400n=900 0.00 0.05 0.10 0.15 0.20 0.25 . . . . . . Skew t, 3 df d P o w e r n=100n=400n=900 Fig 5 . The empirical powers of two-sample location tests based on the Wilcoxon center-outward rank statistic (solid line), the Wilcoxon elliptical rank test statistic (dashed line),the two-sample Hotelling test (dotted line), and the center-outward rank statistic computedfrom linearly sphericized residuals (dot-dashed line), as functions of the shift δ for skew t error densities with ν = 1 . (left panel) and ν = 3 (right panel), respectively; samplesizes n = n = 50 , , and . . . . . . . Normal distribution d P o w e r n=225n=900 0.00 0.05 0.10 0.15 0.20 0.25 . . . . . . U−shaped mixture d P o w e r n=225n=900 Fig 6 . The empirical powers of MANOVA tests based on the Wilcoxon center-outward rankstatistic (solid line), the Wilcoxon elliptical rank test statistic (dashed line), and Pillai’sclassical test (dotted line), as functions of the shift δ , for the normal distribution (left panel)and the U-shaped mixture of three normals (right panel); sample sizes n = n = n = 75 and . imsart-aos ver. 2014/10/16 file: HHH_arXiv_200730.tex date: July 31, 2020 ENTER-OUTWARD RANK TESTS MgO . . . . lllllllll llll lll llll lllllllll lll lllllllllll lllll lllll llllllllll llll lll llll lllllllll lll lllllllllll lllll lllll l lllllllll llll lll llll lllllllll lll lllllllllll lllll lllll l ll lll lll lllllllll lllllllllllll ll llll ll lll lllllll lllll l P2O5 ll lll lll lllllllll lllllllllllll ll llll ll lll lllllll lllll lll lll lll lllllllll lllllllllllll ll llll ll lll lllllll lllll l llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
CoO llllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . llllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllll llllll llllllllllllllllllllllllllllllllllllllllllllllll llllll Sb2O3
Fig 7 . The content of MgO, P O , CoO, and Sb O in natron glass vessels from Appolonia(circles), Bet Eli’ezer (triangles), and Egypt (squares).in natron glass vessels from Appolonia(circles), Bet Eli’ezer (triangles), and Egypt (squares).