Permutation Tests for Reflected Symmetry

Transkript

1 joural of multivariate aalysis 67, (1998) article o. MV Permutatio Tests for Reflected Symmetry Georg Neuhaus ad Li-Xig Zhu* Uiversity of Hamburg, Hamburg, Germay ad Chiese Academy of Scieces, Beijig, People's Republic of Chia Received May 16, 1996; revised February 16, 1997 The paper presets a permutatio procedure for testig reflected (or diagoal) symmetry of the distributio of a multivariate variable. The test statistics are based i empirical characteristic fuctios. The resultig permutatio tests are strictly distributio free uder the ull hypothesis that the uderlyig variables are symmetrically distributed about a ceter. Furthermore, the permutatio tests are strictly valid if the symmetric ceter is kow ad are asymptotic valid if the ceter is a ukow poit. The equivalece, i the large sample sese, betwee the tests ad their permutatio couterparts are established. The power behavior of the tests ad their permutatio couterparts uder local alterative are ivestigated. Some simulatios with small sample sizes ( 20) are coducted to demostrate how the permutatio tests works Academic Press AMS 1991 subject classificatios: primary 62G10; secodary 62G20, 62G09. Key words ad phrases: empirical characteristic fuctio, empirical process, permutatio tests, reflected symmetry, validity of test. 1. INTRODUCTION Testig for symmetry of a radom variable has received cosiderable attetio i the literature. I uivariate cases, may statistics have bee proposed. For example, Butler (1969), Rothma ad Woodroofe (1972), Doksum, Festad, ad Aaberge (1977), Atille, Kerstig ad Zucchii (1982), Shorack ad Weller (1986, Sectio 22), Aki (1987), Cso rgo ad Heathcote (1987), ad Schuster ad Barker (1987). There are two differet but related issues i multivariate cases. Oe is to test for spherical (elliptical) symmetry. For example, Kariya ad Eato (1977), Bera (1979), Blough (1989), Barighaus (1991), Barighaus ad Heze (1991), Fag, Zhu ad Betler (1993), ad Zhu, Fag, ad Zhag (1994). Aother oe is to test for reflected symmetry (or diagoal symmetry), which will be ivolved i the preset paper, such as Aki (1993). Ghosh ad Ruymgaart (1992) exteded the statistic proposed by Feuerverger ad Mureika (1972) * Supported by the NSF of Chia ad a fellowship of the MaxPlack Gesellschaft zur Fo rderig der Wisseschafte of Germay, while o leave from Istitute of Applied Mathematics, Chiese Academy of Scieces, at Uiversity of Hamburg X Copyright 1998 by Academic Press All rights of reproductio i ay form reserved.

2 130 NEUHAUS AND ZHU to the multivariate case, which rests upo a itegrated empirical characteristic fuctio. Heathcote, Rachev, ad Cheg (1995) used, amog others, a maximized empirical characteristic fuctio to ivestigate this problem. Sice little is kow about the samplig ad limitig ull distributios of the test statistics (Ghosh ad Ruymgaart, 1992, p. 439; Heathcote, Rachev, ad Cheg, 1995, p. 99), some approximatio procedures icludig Bootstrap were suggested for practical use of the tests. It is ot clear, however, whether the approximatios have good performace. The purpose of the preset paper is to develop a permutatio procedure for testig the reflected symmetry of multivariate radom variables. We will show that the permutatio tests are, respectively, strictly (asymptotically) valid umber reflected symmetry about a kow (ukow) ceter. It will tur out that the tests ad their associated permutatio oes will be asymptotically equivalet. The permutatio tests are coditioally distributio-free uder the ull hypothesis. Sectio 2 will cotai a review of the tests proposed by Ghosh ad Ruymgaart (1992) ad Heathcote, Rachev, ad Cheg (1995). The permutatio tests will be defied i Sectio 3 ad the validity ad asymptotic validity of the tests will be preseted i the same sectio. A power study uder local alteratives will be made i Sectio 4. Sectio 5 will cotai some simulatio experimets. Sectio 6 will preset some cocludig remarks. All proofs are postpoed to the Appedix. 2. REVIEW OF TESTS As metioed i the previous sectio, a d-variate radom variable x is said to be reflectedly symmetric about a ceter + if (x&+) ad &(x&+) have the same distributio, (2.1) or equivaletly, if the imagiary part of the characteristic fuctio of x&+ equals zero, i.e., E[si(t$(x&+))]=0 for t # R d, (2.2) where t$ stads for the traspose of t. Letx 1,..., x be i.i.d. copies of x ad P ( } ) the correspodig empirical measure. Based o (2.2), Ghosh ad Ruymgaart (1992) ad Heathcote, Rachev, ad Cheg (1995) costructed the tests, respectively, Br [P (si(t$(x&+)))] 2 dw(t) (2.3)

3 PERMUTATION TESTS FOR SYMMETRY 131 ad sup A - P [si(t$(x&+^ ))], (2.4) where both B r (a sphere with the radius r) ad A (a geeral regio) are workig regios, w( } ) is a distributio fuctio o R d, ad P ( f(x)) stads for (1) j=1 f(x j). We ow use other otatios to represet them. Defie a empirical process, [V (X, t)=- P [si(t$(x&+))]: ], (2.5) ad a estimated empirical process, [V 1 (X, +^, t)=- P [si(t$(x&+^ ))]: ], (2.6) where X =(x 1,..., x ), +^ is a estimate of + whe + is ukow, ad A is a workig regio as Heathcote, Rachev, ad Cheg (1995) metioed. The test statistics are defied as Q 1 (X )= A (V (X, t)) 2 dw(t), (2.7) M 1 (X )=sup A V (X, t), (2.8) Q 2 (X, +^ )= A [V 1 (X, +^, t)] 2 dw(t), (2.9) ad M 2 (X, +^ )=sup A V 1 (X, +^, t), (2.10) where Q 1 ad M 2 are the oes i (2.3) ad (2.4). Whe + is kow Ghosh ad Ruymgaart (1992) derived some asymptotic properties of the test based o (2.3), while Heathcote, Rachev, ad Cheg (1995) ivestigated the limitig behavior of the test related to (2.4) for the case where the ceter + is ukow. 3. PERMUTATION TESTS Let a } b mea that every compoet of the vector b is multiplied by a commo uivariate variable a. Here P is the empirical measure of (e i, x i ), i=1,...,, where e 1,..., e are i.i.d. uivariate variables, e i =\1, i=1,...,, with probability values oe half; defie E =(e 1,..., e ). We use a geeric

4 132 NEUHAUS AND ZHU otatio where P stads for a probability measure which may rest upo differet sets of variables for each appearace. For the kow ceter case defie a empirical permutatio process, give X,by [V (E, X, t)=- P [si(t$e }(x&+))]: ]. (3.1) Comparig this process with that defied i (2.5), the versios are the same excepts that the iserted permutatio variables appear i the permutatio process. We will derive, as stated i Theorem 3.3, the (asymptotic) equivalece of the processes, that is, almost surely, both processes coverge i distributio to the same limit process. For the ukow ceter case, the situatio is ot so simple. I order to esure the equivalece betwee the empirical permutatio process, which will be defied below, ad its ucoditioal couterpart i (2.6), both versios caot be the same. The defiitio of our permutatio process is motivated by the followig fact which will be proved i the Appedix, Proof of Theorem 3.4: For a ukow ceter +, a estimate +^ is eeded replacig + i (2.6); here +^ =x, the sample mea, is applied. It ca be proved that, uiformly o, - P (si(t$(x&x )))=- P (si(t$(x&+)) cos(t$p (x&+))) &- (P ((t$(x&+)) si(t$p (x&+)))) =- P (si(t$(x&+)) &- si(t$p (x&+)))(p ((t$(x&+))))+o p (1). Accordigly, we defie a estimated empirical permutatio process [V 1 (E, X, x, t): ] give X by V 1 (E, X, x, t)=- P (si(t$e }(x&x ))) &- si(t$p (e }(x&x ))) P (cos(t$e }(x&x ))). (3.2) The resultig permutatio test statistics give X ceter +, are, for a kow Q 1 (E, X )= A (V (E, X, t)) 2 dw(t) (3.3) ad M 1 (E, X )=sup A V (E, X, t). (3.4)

5 PERMUTATION TESTS FOR SYMMETRY 133 The ukow ceter couterparts are defied i (3.10). Whe the workig regio A is a cube [&a, a] d ad the weight fuctio w( } ) is the uiform distributio o this cube, Q 1 (E, X ) will have a specific form which will be easy to compute. I fact, Q 1 (E, X )= (V (E, X, t)) 2 dw(t) [&a, a] d = 1 : i=1 : j=1 e i e j I(i, j), (3.5) where I(i, j)= 2\ 1 d ` k=1 si(a(x i &x j ) k ) d & ` a(x i &x j ) k k=1 si(a(x i +x j &2+) k ) a(x i +x j &2+) k +, ad (x) k meas the kth compoet of x. We will prove this formula i the Appedix. The followig states the validity of the tests i (3.3) ad (3.4). Theorem 3.1. Assume that x 1,..., x are i.i.d. d-variate variables which are reflectedly symmetric about a kow ceter. Let E (1) (m),..., E be idepedet copies of E. The for ay 0<:<1 ad T=Q 1 or M 1 P (i), m (:)=P[T(X )>m&[m:] of T(E ( j), X )$ s] [m:]+1 m+1, (3.6) where [z] stads for the largest iteger part of z. Remark 3.2. Iequality (3.6) is strict oly if T fails to resolve certai X, which ca happe because of discreteess of X or because of E (j) (e.g., T(X )=T(E (k), X ), if E (k) =(1,..., 1). But if m is reasoably large, ad x satisfies some regularity coditios like cotiuity, (3.6) will be close. I fact uder some coditios o the distributio of x, lim lim P (i), m (:)=: (3.7) m for ay 0<:<1, which is a cosequece of Theorem 3.3 below. Theorem 3.3. Assume, i additio to the coditios of Theorem 3.1, that the distributio of x is cotiuous with third absolute momet. The the empirical permutatio process [V (E, X, t): ] give X i (3.1) coverges weakly to a Gaussia process [V(t):] for almost all sequeces [x 1,..., x,...], which is likewise the limit of the empirical process

6 134 NEUHAUS AND ZHU [V (X, t): ] i (2.5). The Q 1 (E, X ) ad M 1 (E, X ) give X (i (3.3) ad (3.4)) ad Q 1 (X ) ad M 1 (X )(i (2.7) ad (2.8)) have almost surely the same limit, say Q 1 = (V(t)) 2 dw(t) ad M 1 =sup A V(t). Now, the covergece of the associated quatiles ca be established. Deote by * (:), * (:, X ), ad *(:) the1&: quatiles of the distributios of Q 1 (X ), Q 1 (E, X ), give X ad Q 1, respectively. Corollary 3.4. Uder the coditios i Theorem 3.3 for almost all sequeces [x 1,..., x,...], * (:, X ) *(:) i Probab. (3.8) * (:) *(:) i Probab. (3.9) as. A similar result holds for M 1 (X ) ad M 1 (E, X ) give X. For the symmetry testig about a ukow ceter, defie permutatio test statistics as Q 2 (E, X, x )= A (V 1 (E, X, x, t)) 2 dw(t), (3.10) M 2 (E, X, x )=sup V 1 (X, x, t), (3.11) where V 1 ( } ) is defied i (3.2). The followig theorem states the asymptotic validity of the permutatio test based o Q 2 ad M 2. Theorem 3.5. Assume that x 1,..., x,... are i.i.d. uivariate variables which are reflectedly symmetric about a ukow ceter +. Let E (1),..., E (m),... be idepedet copies of E. The for ay 0<:<1, lim P[Q 2 (X, x )>m&[m:] of Q 2 (E ( j), X, x )$ s] = lim P[Q 2 (E 0, X, +)+O p (1- )> m&[m:] of (Q 2 (E ( j), X, +)+O p (1- ))$ s] [m:]+1 m+1, (3.12) ad similarly for M 2, where Q 2 (X, x ) ad M 2 (X, x ) are defied i (2.9) ad (2.10). I additio, assume that the x's have a commo distributio with third absolute momet. The the assertios of Theorem 3.3 ad of Corollary 3.4

7 PERMUTATION TESTS FOR SYMMETRY 135 cotiue to hold for processes i (2.6), (3.2), ad a certai Gaussia process [V 1 (t)=]. Remark 3.6. For performig the above tests, oe has to choose a workig regio A ad a weight fuctio w( } ) (for Q 1 ad M 1 ). This issue was discussed by Heathcote, Rachev, ad Cheg (1995). I our simulatios i Sectio 5 below A=[&1, 1] d ad w( } ) is uiform distributio o A. The Gaussia processes [V(t): ] i Theorem 3.3 ad [V 1 (t): ] i Theorem 3.5 are just the oes i Ghosh ad Ruymgaart (1992) ad Heathcote, Rachev, ad Cheg (1995). 4. POWER STUDY Heathcote, Rachev, ad Cheg (1995, Theorem 3.2) show that the test defied i (2.4) is cosistet agaist ay fixed alterative. We here ivestigate the behavior of the tests ad the permutatio tests for local alteratives. For coveiece, let si (i) (t$x) be the ith derivative of si( } ) at the poit t$x. Suppose that i.i.d. d-variate variables have the represetatio x i + y i :, i=1,...,, for some :>0. This meas that the distributio of x is the covolutio of a symmetric distributio ad a distributio covergig to the degeerate oe. The followig theorem reveals the power behavior of the tests for such local alteratives. Theorem 4.1. Assume that the followig coditios hold: (1) Both distributios of x ad of y are cotiuous ad, i additio, x is reflectedly symmetric about a kow ceter +. The (2) Let l deote the smallest iteger, such that sup B l (t) :=sup E((t$( y&ey) l si (l) (t$(x&ex)))) {0, E(&y& 2l )<, ad E(&y& 2(l&1) &x& 2 )<. (4.1) [- P [si(t$(x+ y 1(2l) &Ex&Ey 1(2l) ))]: ] =[- P [si(t$(x&ex))+(1l!) B l (t)]: ]+o p (1). (4.2)

8 136 NEUHAUS AND ZHU This leads to covergece i distributio ( O ) [- P (si(t$(x+ y 1(2l) &Ex&Ey 1(2l) )))] 2 dw(t) A O (V(t)+(1l!) B l (t)) 2 dx(t), (4.3) A sup - P (si(t$(x+ y 1(2l) &Ex&Ey 1(2l) ))) O sup V(t)+(1l!) B l (t), (4.4) where V(t):] is a Gaussia process defied i Theorem 3.3. Remark 4.2. This coclusio meas that the tests ca detect local alteratives covergig to the ull hypothesis at 1(2l) -rate or slower (the test statistics will coverge i distributio to ifiity uder the local alterative with slower covergece rate). I some cases, this rate ca reach a parametric rate, that is, l=1. For example, if x has a uiform distributio o [&- 3, - 3] d ad if y=(x 2&1,..., 1 x2 d &1), we ca see easily that, via a little elemetary calculatio, sup t #[&1,1] d E(t$y cos(t$x)) {0. Hece, l will be oe. O the other had, whe x ad y are idepedet of each other, l is at least three, ad the tests ca detect, at most, alteratives covergig to the ull hypothesis at 16 -rate. I fact, it is clear that for l=1, 2 sup t #[&1,1] d E((t$y) l si (l) (t$x)) =0. We also ote that our tests are omibus because of the absolute ad square values i the test statistics, ad that therefore the tests are asymptotically ubiased for all shapes of the fuctios B l. Remark 4.3. Let us discuss the meaig of coditio (4.1). It should be easier to uderstad the implicatio of this coditio i the case where x ad y are idepedet. Clearly, ay d-variate variable, w say, ca be decomposed ito two compoets x ad y say, where x is symmetric. Assume further that the momet geeratig fuctio of w exists. I the case where these two compoets are idepedet, w is symmetric if ad oly if y is symmetric which, i tur, is equivalet to the fact that all odd cetered momets of t$y equal zero for ay t. I other words, the larger l is, the more symmetric the variable y is i a certai sese ad the the harder the alterative is detected. I view of sup B l (t), we ca see that sup B l (t){0 is equivalet to sup E(t$( y&e y )) l {0 uder the idepedece of x ad y ad the symmetry of x. Cosequetly, sup B l (t) is a idex measurig the symmetry extet of the variable. The case where

9 PERMUTATION TESTS FOR SYMMETRY 137 both compoets are depedet is more complicated, but the implicatio is similar. We ow study the permutatio procedures for symmetry about a kow ceter +. Of course, we hope that the tests are sesitive to alteratives. I cotrast, it is hoped that the permutatio procedure is ot sesitive to the uderlyig distributio. The reaso is as follows. As described i Sectio 3, the permutatio procedures are applied merely to determie the critical values. Therefore, it is importat that the permutatioal distributio, servig for computig the critical value, is ot affected by a asymmetry of the uderlyig distributio. The followig theorem idicates that the critical values determied by the permutatio tests, uder local alteratives, equals i fact approximately the oes uder the ull hypothesis. Hece the critical values remai uaffected, i the sese of large sample, by the uderlyig distributio of the sample with small perturbatio for symmetry. Theorem 4.4. Assume E &y& 2 <. The for ay %>0 the empirical permutatio processes [- P [si(t$e }(x+ y % &Ex&Ey % ))]: ], give the (x i, y i )'s, ad [- P [si(t$e }(x&ex))]: ], give the x$ i 's have, almost surely, the same limitig Gaussia process [V(t): ] as the empirical process [- P [si(t$(x&ex))]: ]. Hece the quadratic or maximum fuctioals of these processes have, almost surely, the same limitig distributio as the radom variables A (V(t)) 2 dw(t) or sup V(t). For the ukow ceter case, there also exists, similar to the kow ceter case, a oradom shift fuctio t (1l!) B l (t) i the limitig process of [- P [si(t$(x+ y 1(2l) &x & y 1(2l) ))]: ] uder local alteratives. The followig result describes this. Theorem 4.5. Assume the same coditios as i Theorem 4.1. The [- P [si(t$(x+ y 1(2l) &x & y 1(2l) ))]: ] =[- P (si(t$(x&ex)))&- si(t$p (x&ex) E(cos(t$(Ex)))) +(1l!) B l (t): ]+o p (1) (4.5) ad cosequetly, A [- P (si(t$(x+ y 1(2l) &x & y 1(2l) )))] 2 dw(t) O A (V 1 (t)+1l!) B l (t)) 2 dw(t) (4.6)

10 138 NEUHAUS AND ZHU ad sup - P (si(t$(x+ y 1(2l) &x & y 1(2l) ))) O sup V 1 (t)+(1l!) B l (t), (4.7) where [V 1 (t): ] is a Gaussia process metioed i Theorem 3.5. For the estimated empirical permutatio process, there is a parallel coclusio to Theorem 4.4. Theorem 4.6. Uder the coditios of Theorem 4.4, for ay %>0, the estimated empirical permutatio process, give (x i, y i )'s, [- P (si(t$e }(x+ y % &x & y % ))) &- P (cos(t$e }(x+ y % &x & y % ))) _si(t$p (e }(x+ y % &x & y % ))): ] ad the empirical permutatio process, give x i 's, [- P (si(t$e }(x&x ))) +- P (cos(t$e }(x&x )))(si(t$p (e }(x&x )))): ], as well as the ucoditioal process [- P (si(t$e }(x&x ))): ], have, almost surely, a commo limitig Gaussia process [V 1 (t): ] metioed i Theorem 3.5. The the quadratic or maximum fuctioals of these processes coverge weakly to A (V 1 (t)) 2 dw(t) or sup V 1 (t) (almost surely for the permutatio oes). 5. SIMULATIONS I order to demostrate the performace of the permutatio tests, some small-sample simulatio experimets have bee performed. I the simulatio results reported i the tables below the sample sizes are =10 ad =20, the dimesios of radom variable are xdi=2, 4, ad 6 ad the followig distributios of the variable have bee take:

11 PERMUTATION TESTS FOR SYMMETRY 139 Nx has stadard multivariate ormal distributio N(0, I d ); N+/ 2 x has stadard ormal distributio N(0, I d ) ad y=[x 2 &1,..., 1 x2 d &1]; the resultig radom variable is w=x+ y; N+N s x has stadard multivariate ormal distributio N(0, I d ) ad the idepedet y has multivariate ormal distributio N(1-, I d ); the resultig radom variable is w=x+ y. I order to get a critical value for fixed [(x 1, y 1 ),..., (x, y )], 2000 pseudo-radom umbers E =(e 1,..., e ) of size =10 ad =20 are geerated by the Mote Carlo method. The basic experimet was replicated 3000 times for each combiatio of sample size, dimesio of radom variable, ad the uderlyig distributio of the variable. The omial level is The proportio of times that the values of the statistics exceeded the critical values are recorded as the estimated power of the tests. I order to judge the performace of the permutatio tests, we compute the ``accurate'' power of the test usig the Mote Carlo method. Here ``accurate'' power meas that the ull distributio of the variable is assumed to be the stadard ormal N(0, I d ), a completely kow distributio. So we ca get by the Mote Carlo method the exact ull distributio of the test statistics ad the critical values (ot approximated oes) as log as the umber of replicatios is large eough. Based o these critical values, the obtaied power of the tests is called ``accurate'' power here. The replicatio umber of each sigle experimet is Let us explai why we cosider the alterative N+N s i the kow ceter case. Clearly w&1- is symmetric, but we regard N(1-, I d ) asaukow small perturbatio of the ull distributio N(0, I d ). Hece, we do ot ceter the variable w i the simulatios preseted i the Tables I ad II. TABLE I Estimated Power i the Kow Ceter Case, =10 N N N+/ 2 N+/ 2 N+N s N+N s Q 1 M 1 Q 1 M 1 Q 1 M 1 Accurate = Permutatio di= Accurate = Permutatio di= Accurate = Permutatio di=

12 140 NEUHAUS AND ZHU TABLE II Estimated Power i the Kow Ceter Case, =20 N N N+/ 2 N+/ 2 N+N s N+N s Q 1 M 1 Q 1 M 1 Q 1 M 1 Accurate = Permutatio di= Accurate = Permutatio di= Accurate = Permutatio di= Let us first look at Tables I ad II. We see that, uder the ull hypothesis, the simulated level of the permutatio tests Q 1 ad M 1 is close to the omial oe i most cases, eve if the sample size is quite small, such as =10. However, if the dimesio is large ad the sample size is too small as compared with the dimesio, M 1 does ot seem to be as good as we expected. This is the case whe the dimesio is 6 ad the sample size is 10. With icreasig sample size, the situatio becomes better. Uder the alterative cosidered here, Q 1 still has quite good performace. The results i Table II show that the power is very close to the ``accurate'' oe. This meas that the permutatio test Q 1 behaves as a test beig based o a kow ull distributio. Hece Q 1 is distributio-free, ot oly i theory but also i practice. O the other had, the performace of M 1 is discouragig, although theoretically it is also a coditioal distributio-free test. The estimated power uder alteratives is cosiderably lower tha the ``accurate'' oe. Hece M 1 is presumably applicable i large sample cases, sice, comparig Tables I ad II the power of M 1 is icreasig with icreasig the sample size. For the N+N s case, the tests caot detect such a alterative. However, this is reasoable sice the theory i Sectio 4 has told us that the tests hardly detect this kid of alterative, sice the mea of y is 1-, a too small shift. From Tables I ad II, we ca see that, eve if the critical value is based o the samplig ull distributio (the ``accurate'' oe), N+N s is also hard to detect. This meas that oe may eed to defie a more efficiet test for detectig such kids of local alteratives. We will discuss this problem further i the ext sectio. I the cases where the symmetric ceter is ukow, the permutatio test Q 2 ca still hold the level, but M 2 caot, especially i high-dimesioal cases. Uder alteratives Q 2 is still applicable especially i lower

13 PERMUTATION TESTS FOR SYMMETRY 141 TABLE III Estimated Power i Ukow Ceter Case, =10 N N N+/ 2 N+/ 2 Q 2 M 2 Q 2 M 2 Accurate = Permutatio di= Accurate = Permutatio di= Accurate = Permutatio di= dimesio cases. M 2 does ot seem to be recommedable i small sample cases. However, comparig Tables III ad IV, there is iformatio supportig M 2. That is, with icreasig the sample size, the power of M 2 is icreasig. Hece, M 2 may be applicable i large sample cases. Summarizig, the permutatio tests have good performace i the case where the ceter of symmetry is kow but are worse i the situatio where it is ukow. This is easy to explai. As show i the theorems ad i Remark 3.6, the tests are asymptotically valid at - -rate i the sese of Theorem 3.5. Whe the sample poits are too sparse i the high-dimesioal space, a O(1- ) perturbatio for the test statistic caot be igored i small sample cases. Hece, they caot be expected to have a satisfyig performace. TABLE IV Estimated Power i Ukow Ceter Case, =20 N N N+/ 2 N+/ 2 Q 2 M 2 Q 2 M 2 Accurate = Permutatio di= Accurate = Permutatio di= Accurate = Permutatio di=

14 142 NEUHAUS AND ZHU 6. CONCLUDING REMARKS We have developed a permutatio procedure for testig reflected symmetry of a multivariate variable, which is based o the empirical characteristic fuctio. Uder some regularity coditios o the distributio of the variable, we have ivestigated the validity of the permutatio tests ad the power behavior of the tests ad their permutatio couterparts uder local alteratives. Remark 6.1. I priciple, other test statistics may be foud for the above testig problems. Whe the symmetry ceter is assumed to be kow, a permutatio test would be based o its ucoditioal couterpart without ay modificatio followig our approach i Sectio 3 i (3.3) or (3.4). It is worthwhile to metio, that whe the symmetry ceter has to be estimated, the permutatio test will geerally ot have the same form as its ucoditioal couterpart as the oe i (3.5) or (3.6) which is based o (3.2). The modificatio will guaratee a equivalece, i the large sample sese, betwee the test ad the associated permutatio oe. O the other had, if we wat to have a test whose permutatio couterpart is strictly valid for the ukow ceter case, it seems to us that such a test has to be locatioivariat. Remark 6.2. Although the choice of the workig regio ad of the weight fuctio have bee uder cosideratio i the preset paper, it is ecessary to explore how these choices affect the performace of the tests. O the other had, i some cases, the choice of workig regios is ot very importat. We ow show a example i which the fact that the imagiary part of the characteristic fuctio equals zero i a compact subset of R d such as [&1, 1] d is equivalet to reflected symmetry of the variable. Suppose that the momet geeratig fuctio of a multivariate variable x, say, exists i a cube [&a, a] d, a>0. The the momet geeratig fuctio of #$x, the liear projector of x o R 1, exists i a iterval [&a 1, a 1 ] for ay # beig o the uit sphere i R d, where a 1 does ot deped o #. If the imagiary part of the characteristic fuctio of x equals zero i a cube [&a 2, a 2 ] d, so does the oe of #$x i a iterval [&a 3, a 3 ]. It is easy to see tha the all momets of #$x with odd orders equal zero. This meas that the characteristic fuctio of #$x is real; hece, #$x is symmetric about the origi for ay #. This coclusio implies, i tur, that x is reflectedly symmetric. Cosequetly, the choice of the workig regio is ot very importat i such a case. For example, we could choose > d [&a i=1 i, a i ]as a workig regio, where a i is the variace of the ith compoet of x. Remark 6.3. I the simulatio of Sectio 5, the permutatio test, for the kow ceter case, is a distributio-free test. The power is close to the

15 PERMUTATION TESTS FOR SYMMETRY 143 ``accurate'' oe obtaied by assumig the ull distributio to be kow. Oe may use the same priciple, as metioed i Remark 6.1, to defie further tests. Hece, it remais to compare the differet proposals. Remark 6.4. For the ukow ceter case, the permutatio tests defied i the preset paper do ot have a good performace i the small-samplehigh-dimesio case. Oe of the mai reasos is that the test is oly asymptotically valid. It would be iterestig to costruct a test beig strict validity eve i the ukow ceter case. We defer this to future research. Remark 6.5. I this paper we use the sample mea as a estimate of the ukow ceter. If the uderlyig distributio is heavy-tailed, its use may be questioable due to its lack of robustess, ad furthermore, accordig to the discussio i Heathcote, Rachev, ad Cheg (1995) ad Remark 6.4, it could ot be expected that the tests have good performace i small sample cases we coducted. The sample size may be cosiderably larger for havig good performace of the tests. We will study this questio i coectio with robustess cosideratios. Remark 6.6. We ame the coditioal test procedure as the permutatio test, although it is ot exactly like the classical permutatio procedure. Actually, it is a radom symmetrizatio procedure. The same idea could be applied to some other settig i the statistical iferece. APPENDIX: PROOFS OF THEOREMS We first prove the formulae i (3.5). Note that si(x) } si( y)= 12(cos(x& y)&cos(x+y). The Q 1 (E, X ) =(2a) &d [&a, a] d{ 1- : i=1 =1 : i=1 =1 : i=1 si(t$e i }(x i &+)) = 2 dt : j=1{ (2a)&d si(t$(x i &+)) si(t$(x i &+)) dt = e ie j [&a, a] d : j=1{ (2)&d&1 cos(t$a }(x i &x j )) [&1, 1] d &cos(t$a }(x i +x j &2+)) dt = e ie j :=1 : : i=1 j=1 e i e j I(i, j).

16 144 NEUHAUS AND ZHU Sice the uiform distributio o [&1, 1] d is symmetric, we have for u havig uiform distributio o [&1, 1] d, I(i, j)=2 &d&1 E(cos(u$a }(x i &x j ))&cos(u$a }(x i +x j &2+))) =2 &d&1 (Re E(e (u$a }(x i &x j )) )&Re E(e (u$a }(x i +x j &2+)) )) =2 &d&1 =12 `d \Re Re `d \ k=1 k=1 The proof is completed. d E(e (u k a(x i &x j ) k ) )&Re ` E(e (uka(xi+xj&2+)k ) ) + k=1 si(a(x i &x j ) k ) a(x i &x j ) d & ` k=1 si(a(x i +x j &2+) k ) a(x i +x j &2+) +. We ow start to prove theorems. For coveiece of the otatios, let a b b mea the vector with every compoet of a is multiplied by the correspodig compoet of b, ad c deotes a geeric costat which may chage its meaig, eve i the same formula. Proof of Theorem 3.1. Recall the defiitio of a } b at the begiig of Sectio 3. First, we prove that x is reflectedly symmetric if ad oly if x=e } x*, where e=\1 with probability oe-half ad is idepedet of x* ad, furthermore, x* has the same distributio as x. Sufficiecy is clear sice P[e } x*t]= 1 2 P[x*t]+ 1 2 P[&x*t] = 1 2 P[xt]+ 1 2 P[&xt]=P[xt]. Similarly P[&e } x*t]=p[&xt]; hece P[xt]=P[&xt]. For ecessity, let x* :=e } x, where e=\1 with probability oe half ad is idepedet of x. The x ad x* have the same distributio, x=e } x*. Furthermore, x* ade are idepedet sice for ay s # R d, P[e=\1, e } xs]= 1 2P[e } x*s e=\1] = 1 2 P[\xs] =P[e=\1]P[e} xs]. Let X*=(x 1 *,..., x *). The set [Q i (X )>m&[m:] of Q i (E ( j), X )$ s] equals exactly the set [Q i (E, X *)>m&[m:] of Q i (E b E ( j), X*), j=1,..., m]. It is easy to check that [E, E b E ( j), j=1,..., m] are i.i.d.

17 PERMUTATION TESTS FOR SYMMETRY 145 -dimesioal variables. Ideed, for ay t i the set cosistig of all -dimesioal variables of the form (\1,..., \1), say [t 1,..., t 2 ], P[E b E (1) =t, E b E (2) =s] = 1 2 : 2 j=1 = 1 2 : 2 j=1 P[E (1) b t j =t, E (2) b t j =s] =P[E b E (1) =t] P[E b E (2) =s]. The idepedece betwee E ad E b E ( j) ca be checked i the same way. Hece, give X*, the m+1 variables Q i (E, X*) ad Q i (E b E ( j), X *), are i.i.d., which implies that P[Q i (E, X*)>m&[m:] ofq i (E b E ( j), X *)$ s X *] [m:]+1 m+1. The proof is coclude by itegratig X *. Proof of Theorem 3.3. I the followig, we first show that the process [V (E, X, t): ], give X, coverges almost surely to the process [V(t): ] which is the limitig process of [V (X, t): ]. The coclusio of Theorem 3.3 will the hold. Defie sets D 1 =[lim 1 &x j=1 j&+& 2 =E &x&+& 2 ], D t, s = { lim 1 : si(t$(x j &+)) si(s$(x j &+)) j=1 =E(si(t$(x&+)) si(s$(x&+))) =, ad D=D 1 & [ t, s # A0 D t, s ], where A 0 is ay coutable dese set of A. D is a subset of the sample space havig probability measure oe. The by the Lipschitz cotiuity of the sie fuctio, it is clear that D=D 1 & [ t, s # A D t, s ]. We describe the covergece of the process i a lemma. Lemma 7.1. Uder the coditios of Theorem 3.3, give ay sequeces [x 1,..., x,...] # D, the process [V (E, X, t): ] coverges weakly to a cetered Gaussia process [V(t): ] with the covariace kerel E(si(t$(s j &+)) si(s$(x&+))) for t, s # A.

18 146 NEUHAUS AND ZHU Proof. I the followig, we always assume without further metioig that the give [x 1,..., x,...] belogs to D. We eed to prove the fidis covergece ad the uiform tightess of the process. (1) Fidis covergece. This part of the proof is stadard, so we oly give a outlie. For ay iteger k, t 1,..., t k # A. Let We have to show that V (k) =(E(si(t$ i (x&+)) si(t$ l (x&+)))) 1i, lk. V (k) =[V (E, X, t i ): i=1,..., k] O N(0, V (k) ). (7.1) It suffices to show that for ay uit k-dimesioal vector # Note that the variace of the LHS i (7.2) is #$V (k) O N(0, #$V (k) #) (7.2) #$(P (si(t$ i (x&+)) si(t$ l (x&+)))) 1i, lk # #$V (k) #. (7.3) Hece, if #$V (k) #=0, (7.2) is trivial. Assume #$V (k) #>0. Ivokig the Lyapuov coditio, #$V (k) - #$V (k) # N(0, 1) The fidis covergece holds via combiig with (7.3). (2) Uiform tightess. All we eed to do is to show that for ay '>0 ad =>0, there exists a $>0 such that lim sup P[sup [$] V (E, X, t)&v (E, X, s) ' X ]<=, (7.4) where [$]=[(t, s): &t&s&$]. Sice the limitig property is ivestigated for, is always cosidered to be large eough below which simplifies some argumets of the proof. Note that V (E, X, t)=- P (si(t$e }(x&+)))=- P (e si(t$(x&+))).

19 PERMUTATION TESTS FOR SYMMETRY 147 Write P% for the siged measure that places mass e i at (x i &+). We the write the LHS of (7.4) i aother form lim sup P[sup - P% (si(t$(x&+)))&(si(s$(x&+)) X, s) >' X ]. [$] (7.5) Note that whe &t&s&$ for large P (si(t$(x&+))&si(s$(x&+))) 2 &t&s& 1 : j=1 &x j &+& 2 2 &t&s& E &x&+& 2 =c &t&s&. (7.6) The applyig the Hoeffdig iequality for ay t, s # A, P[- (P% (si(t$(x&+))&si(s$(x&+))))>'c &t&s& X, E ] 2 exp(&' 2 32). (7.7) I order to apply the chaiig lemma (e.g., Pollard, 1984, p. 144), we eed to check that the coverig itegral J 2 ($, &}&, A)= $ [2 log [(N 2 (u, &}&, A)) 2 u]] 12 du (7.8) 0 is fiite for small $>0, where &}& is the Euclidea orm i R d ad the coverig umber N 2 (u, &}&, A) is the smallest m for which there exist m poits t 1,..., t m with mi 1im &t&t i &u for every. It is clear that Cosequetly, for small $>0 N 2 (uc, &}&, A)cu &d. (7.9) J 2 ($, &}&, A)c $ (log(1u)) 12 duc$ 12. (7.10) 0 Applyig ow the chaiig lemma, there exists a coutable dese subset [$]* of [$] such that P[sup - (P% (si(t$e }(x&+))&si(s$&+))) >26cJ 2 ($, &}&, A) X ] [$]* 2c$. (7.11) The coutable dese subset [$]* ca be replaced by [$] itself because - P% [si(t$(x&+))&si(s$(x&+))] is a cotiuous fuctio with respect

20 148 NEUHAUS AND ZHU to t ad s for each fixed X. Hece, choosig $ smaller tha 1e 2 i (7.10), =(8C), ad ('(26c)) 2 i (7.11) Eq. (7.4) is proved. The proof of the lemma is completed. O the other had, the weak covergece of the process [- P (si (t$(x&+)): ] to the process [V(t): ] ca follow from our results. The sequece e 1 } x 1, e 2 } x 2,... has the same distributio as the sequece x 1, x 2,... Hece, the limit process is the same for the [e i } x i ] as for the [x i ]. The above covergece of coditioal process implies immediately the covergece of ucoditioal process, which has bee derived by Ghosh ad Ruymgaart (1992) ad Heathcote, Rachev, ad Cheg (1995). The proof of the Theorem 3.3 is cocluded from oticig that Q i are cotiuous fuctioals of the process [V (E, X, t): ]. Proof of Theorem 3.5. Note that sie ad cosie fuctios are, respectively, odd ad eve fuctios, the ad - P (si(t$e }(x&x )))=- P (si(t$e }(x&+)) cos(t$p (x&+))) &- (P (e } cos(t$(x&+)) si(t$p (x&+)))) =: I 1 (t)&i 2 (t) (7.12) - si(t$p (e }(x&x )))=- si(t$p e }(x&+)) cos(t$e } P (x&+)) &- cos(t$p e }(x&+)) si(t$e } P (x&+)) =: I 3 (t)&i 4 (t), (7.13) where e =(1) j=1 e j. By the cetral limit theorem we have - P (x&+) =O p (1). It is the easy to see that I 1 (t)=- P (si(t$e }(x&+)))+o p (1- ), I 2 (t)=o p (1- ), I 3 (t)=- si(t$p (e }(x&+)))+o p ((1- ) 2 ), I 4 (t)=o p (1- ), (7.14) uiformly over, as log as we otice that [- P (si(t$e }(x&+))): ] ad [- P (e } cos(t$(x&+))): ] both coverge weakly to Gaussia processes. Cosequetly, for almost all sequeces [x 1,..., x,...]

21 PERMUTATION TESTS FOR SYMMETRY 149 V 1 (E, X, x, t) =- P (si(t$e }(x&+))) &- P (cos(t$e }(x&+)) si(t$p e }(x&+)))+o p (1- ) =V 1 (E, X, +, t)+o p (1- ), (7.15) uiformly over. The proof of (3.12) ca be based o (7.15). Moreover, followig the argumet used i the proof of Theorem 3.3 above, we see that the process [V 1 (E, X, x, t): ], give X, i (3.2) coverges weakly to a Gaussia process for almost all sequeces [x 1,..., x,...], which is likewise the limit of the process [V 1 (X, x, t): ] defied i (2.6); see Theorem 3.1 i Heathcote, Rachev, ad Cheg (1995). Hece, the coclusios i Theorem 3.3 ad Corollary 3.4 hold. The proof of Theorem 3.5 is complete. Proof of Theorem 4.1. Without loss of geerality, assume Ex=Ey=0. It is kow that max 1 j &y j & 1(2l) 0, a.s. Hece, by the Taylor expasio of sie fuctio for ay, - P [si(t$(x+ y 1(2l) ))] l&1 =- P [si(t$x)]+ : i=1 +(1l!) P [(t$y) l si (l) (t$x)] +(1l!) &1 : j=1 (1i!) &i(2l) - P [(t$y) i si (i) (t$x)] [(t$y j ) l si (l) (t$(x j +(t$y j )* 1(2l) ))&si l (t$x j )], (7.16) where (t$y j )* is a value betwee 0 ad t$y j. We have to show that the secod ad fourth summads i the RHS of the equality ted to zero i probability for ad that the third summad coverges i probability to E[(t$y) l ) si (l) (t$, x)]. First, cosider the secod summad. It is eough to show that for each 1il&1, [- P ((t$y) i si (i) (t$x)): ] coverges weakly to a cetered Gaussia process. By Theorem VII.21 ad the equicotiuity lemma (Pollard, 1984, p. 157, p. 150), all we eed to do is to check that for ay '>0 ad =>0, there exists a $>0 for which lim sup P[J 2 ($, P, 0 i )>']<=, (7.17)

22 150 NEUHAUS AND ZHU where 0 i =[(t$y) i si (i) (t$x)): ]. The coverig itegral J 2 ($, P, 0) is similar to that i (7.8) ad the semiorm i L 2 (P )is- P ( f &g) 2. Note that - P [(t$y) i si (i) (t$x)&(s$y) i si (i) (s$x)] 2 c &t&s& - P &y& 2i +P (&y& 2i &x& 2 )=: c &t&s& C 1, (7.18) where C 1 c=- E &y& 2i +E &y& 2i &x& 2, a.s. For the case C 1 <2c, we ca boud J 2 ($, P, 0 i )byc$ 12, similar to (7.10). Hece, P[J 2 ($, P, 0 i )>']<P[C 1 2c] +P[J 2 ($, P, 0 i )>'] 0 (7.19) as. The covergece of the third summad i (7.16) ca be derived by applyig Theorem II. 24 (Pollard, 1984, p. 26). The fourth summad teds to zero, sice for some costat c>0, sup (t$y j ) l (si (l) (t$(x j +(t$y j )* 1(2l) ))&si l (t$ j )) c &y j & l (max j &y j & 1(2l) ), max 1 j &y j & 1(2l) a.s., ad E &y& l <; Eq. (4.2) i Theorem 4.1 is proved. Both (4.3) ad (4.4) are cosequeces of (4.2). The proof is completed. Proof of Theorem 4.4. Sice Without loss of geerality, assume Ex=Ey=0. - P (si(t$e }(x+ y % ))) =- P (si(t$e } x)(cos(t$y % )))&cos(t$x) si(t$e } y % ). Hece, all we eed to do is to show that for almost all sequeces [(x 1, y 1 ),...] ad sup - P% (si(t$x)(1&cos(t$y % ))) 0 i Probab. (7.20) sup - P% (cos(t$x) si(t$y % )) 0 i Probab., (7.21) where P% is a siged measure that places mass e i at (x i, y i ). We show (7.20). A similar way ca be applied to show (7.21). Let Y =(y 1,..., y ).

23 PERMUTATION TESTS FOR SYMMETRY 151 If P is ay probability, the semiorm i L 1 (P) is P f &g ad 0 = [si(t$ } )(1&cos(t$} % ): ]. The it is easy to show that, by the Lipschitz cotiuity of the fuctio si(t$ } )(1&cos(t$} % )), the coverig umber N 1 (u, P, 0 ) ca be bouded by Bu &W for some B ad W uiformly over P. Precisely, we have for 0<u<1 Furthermore, ote that sup N 1 (u, P, 0 )Bu &W. (7.22) P sup P (si(t$e } x)(1&cos(t$e } y % ))) 2 (X, Y ) <cp &y& 2 2% c &2% (7.23) for some c>0. Applyig the formula (31) of Pollard (1984, p. 31), we have P[sup - P% (si(t$x)(1&cos(t$y % ))) >= (X, Y )] 2B \ = W exp(= (2c &2% ))0 (7.24) for ; (7.20) is proved. The proof of (7.21) is similar. This fiishes the proof of Theorem 4.4. Proof of Theorem 4.5. Note that sup P (cos(t$(x&ex)+(y&ey) 1(2l) ))&cos(t$(x&ex)) <cp &y&ey& 1(2l) =O( &1(2l) ) a.s., sup 1&cos(t$P ((x&ex)+(y&ey) 1(2l) )) <c(&p x&ex& 2 +&P y&ey& 2 1l )=O p ( &1 ), ad sup - (si(t$p (x&ex)+p ( y&ey) 1(2l) ))&si(t$p (x&ex)) <c - &P y&ey& 1(2l) =O p ( &1(2l) ).

24 152 NEUHAUS AND ZHU Based o these iequalities, it is easy to see that - P (si(t$(x+ y 1(2l) &(x + y 1(2l) )))) =- P (si(t$((x&ex)+(y&ey) 1(2l) )) _cos(t$p ((x&ex)+(y&ey) 1(2l) ))) &- P (cos(t$((x&ex)+(y&ey) 1(2l) )) _si(t$p ((x&ex)+(y&ey) 1(2l) ))) =- P (si(t$(x+ y 1(2l) &(Ex+Ey 1(2l) )))) &- P (cos(t$(x&ex)) si(t$p (x&ex))+o p ( &1(2l) )) =- P (si(t$(x&ex)))+(1l!) E[(t$( y&ey)) l si (l) (t$(x&ex))] &- si(t$p (x&ex)) E(cos(t$(x&Ex))+O p ( &1(2l) )). This is just the formula i (4.5), completig the proof. Proof of Theorem 4.6. By argumets similar to those i Theorems 4.5 ad 4.4, the coclusio ca be derived, so we omit the details. ACKNOWLEDGMENTS The authors are grateful to two referees for the commets which improved the represetatio of this paper. REFERENCES Aki, S. (1987). O o-parametric tests for symmetry. A. Ist. Statist. Math Aki, S. (1993). O o-parametric tests for symmetry i R m. A. Ist. Statist. Math Atille, L., Kerstig, G., ad Zucchii, W. (1982). Testig symmetry, I. Amer. Statist. Assoc Barighaus, L. (1991). Testig for spherical symmetry of a multivariate distributio. A. Statist Barighaus, L., ad Heze, N. (1991). Limit distributios for measures of skewess ad kurtosis based o projectios. J. Multivariate Aal Bera, R. (1979). Testig for spherical symmetry of a multivariate desity. A. Statist Blough, D. K. (1989). Multivariate symmetry via projectio pursuit. A. Ist. Statist. Math Cso rgo, S., ad Heathcote, C. R. (1987). Testig for symmetry. Biometrika Doksum, K. A., Festad, G., ad Aaberge, R. (1977). Plots ad tests for symmetry. Biometrika

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