A New Approach to the BHEP Tests for Multivariate Normality

Transkript

1 joural of multivariate aalysis 6, 13 (1997) article o. MV A New Approach to the BHEP Tests for Multivariate Normality Norbert Heze ad Thorste Wager Uiversita t Karlsruhe, D-7618 Karlsruhe, Germay ad Ruhr-Uiversita t Bochum, D Bochum, Germay Let X 1,..., X be i.i.d. radom d-vectors, d1, with sample mea X ad sample covariace matrix S. For testig the hypothesis H d that the law of X 1 is some odegeerate ormal distributio, there is a whole class of practicable affie ivariat ad uiversally cosistet tests. These procedures are based o weighted itegrals of the squared modulus of the differece betwee the empirical characteristic fuctio of the scaled residuals Y j =S &1 (X j &X ) ad its almost sure poitwise limit exp(&&t& ) uder H d. The test statistics have a alterative iterpretatio i terms of L -distaces betwee a oparametric kerel desity estimator ad the parametric desity estimator uder H d, applied to Y 1,..., Y. By workig i the Fre chet space of cotiuous fuctios o R d, we obtai a ew represetatio of the limitig ull distributios of the test statistics ad show that the tests have asymptotic power agaist sequeces of cotiguous alteratives covergig to H d at the rate &1, idepedet of d Academic Press 1. INTRODUCTION There is a cotiued iterest i the problem of testig for multivariate ormality, as evideced by the recet papers of Ah [1], Bowma ad Foster [5], Heze [1], Horswell ad Looey [14], Kariya ad George [16], Koziol [17], Mudholkar, McDermott, ad Srivastava [0], Mudholkar, Srivastava, ad Li [1], Naito [], Ozturk ad Romeu [3], Rayer, Best, ad Matthews [4], Romeu ad Ozturk [5], Sigh [6], Versluis [7], ad Zhu, Wog, ad Fag [9] o this subject. The purpose of this paper is ot to review the huge literature o tests for multivariate ormality (for short; MVN tests), but to preset a ew approach to a whole class of MVN tests studied by Barighaus ad Heze Received October 9, 1996; revised April 9, AMS 1991 subject classificatio umbers: primary 6G10; secodary 6H15. Key words ad phrases: test for multivariate ormality, empirical characteristic fuctio, Gaussia process, cotiguous alteratives X Copyright 1997 by Academic Press All rights of reproductio i ay form reserved.

2 HENZE AND WAGNER [3] ad Heze ad Zirkler [11]. These tests have the desirable properties of v affie ivariace, v cosistecy agaist each fixed oormal alterative distributio, v asymptotic power agaist cotiguous alteratives of order &1, v feasibility for ay dimesio ad ay sample size, which o other MVN test shares, at least to our kowledge. To state the testig problem ad the tests uder discussio, let X 1, X,... be a sequece of idepedet copies of a radom d-dimesioal colum vector X, where d1 is a fixed iteger. The distributio of X will be deoted by P X. The problem is to test, o the basis of X 1,..., X, the hypothesis H d : P X # N d, where N d is the class of all odegeerate d-variate ormal distributios. Sice N d is closed with respect to full rak affie trasformatios ad the alteratives to H d are rarely kow or give i practice, we are iterested i (affie) ivariat ad cosistet tests. Such a test may be based o the test statistic T, ; := (4 1[S is sigular]+w, ; 1[S is osigular]). (1.1) Here, ;>0 is a parameter (the role of which is discussed later), 1[ } ] stads for the idicator fuctio ad S := 1 : (X j &X )(X j &X )$ is the empirical covariace matrix of X 1,..., X, where X := &1 X j ad the prime deotes traspose. W, ; is the weighted L -distace W, ; := R d } betwee the empirical characteristic fuctio 9 (t)&exp. \&&t& +} ; (t) dt (1.) 9 (t) := 1 : k=1 exp(it$y k ) (i =&1) of the scaled residuals Y k :=S &1 (X k &X ) (k=1,..., )

3 NEW APPROACH TO THE BHEP TESTS 3 ad the characteristic fuctio exp(&&t& ) of the stadard d-variate ormal distributio. S &1 is the symmetric positive defiite square root of S &1. Note that Y k ad, hece, W, ; are oly defied if S is osigular. Accordig to (1.1), W, ; is replaced by its maximum possible value 4 i the case whe S is ot ivertible (this suggestio is due to Cso rgo [7]). The weight fuctio. ; figurig i (1.) is. ; (t) :=(?; ) &d exp \ &&t& ; +. The extremely appealig feature of this choice is that W, ; takes the simple form W, ; = 1 : j, k=1 + exp \&; &Y j&y k & exp ; &Y j & \& (1+; )+ +(1+; ) &d. &(1+; ) &d 1 : This represetatio shows that T, ; is ivariat; i.e., we have T, ; (AX 1 +b,..., AX +b)=t, ; (X 1,..., X ) for each osigular A # R d_d ad each b # R d. Moreover, sice the computatio of &Y j &Y k & ad &Y j & ivolves oly S &1, ot eve the square root S &1 of S &1 is eeded. The statistic W, ; was proposed by Epps ad Pulley [9] i the special case d=1. Barighaus ad Heze [3] exteded the approach of Epps ad Pulley to the case d>1 ad obtaied the limit distributio of } W, ; uder H d for the case ;=1. S. Cso rgo [7] coied the term BHEP test with referece to these four authors. Usig a theorem of de Wet ad Radles [8], Heze ad Zirkler [11] proved that the limitig distributio of T, ; uder H d is that of j1 $ j (;) N, where N j 1, N,... are i.i.d. stadard ormal radom variables ad ($ j (;)) j1 is the sequece of eigevalues associated with a itegral operator give i Theorem 3.1 of Heze ad Zirkler [11]. If the distributio of X is ot i N d we have lim if &1 T, ; C(P X, ;)>0, almost surely for some costat C(P X, ;) (Cso rgo [7]). This shows that rejectig H d for large values of T, ; yields a affie ivariat ad uiversally cosistet MVN test (cf. the first two bullets).

4 4 HENZE AND WAGNER That the role of ; figurig i the weight fuctio. ; is that of a smoothig parameter may be see from the represetatio W, ; =(?) d ; &d R d_ g, ;(x)&(?{ ) &d exp dx, (1.3) \&&x& { +& where { =(; +1)(; ), ad g, ; (x)= 1 h d : k=1 (?) &d exp k& \&&x&y h + is a oparametric kerel desity estimator, applied to Y 1,..., Y, with a stadard Gaussia kerel ad badwidth h=1(; - ) (see Heze ad Zirkler [8, p. 3600]). I the spirit of desity estimatio, Bowma ad Foster [4, p. 1535] proposed to base a MVN test o BF :=; d (?)&d W, ;, where ; :=(h - ) &1 ad h := [4((d+))] 1(d+4). We cojecture that, i cotrast to the case of fixed ; (see Sectio 3), the MVN test based o BF is ot able to discrimiate betwee H 0 ad alteratives which are &1 -apart. Iterestigly, the class of MVN tests based o T, ; is ``closed at the boudaries ; 0 ad ; '' which correspod to ``ifiite ad zero smoothig,'' respectively. More precisely, we have (Heze [13]) where lim ; &6 W, ; = 1 } b 6 1, d+ 1 } b 4 1, d, (1.4) ; 0 b 1, d = 1 : j, k=1 (Y$ j Y k ) 3 is Mardia's time-hoored measure of multivariate sample skewess (Mardia [18]) ad b 1, d= 1 : j, k=1 Y$ j Y k &Y j & &Y k & is the sample versio of the populatio skewess measure ; 1, d= &E[&X& X]& (this otatio assumes X to be stadardized), itroduced by Mo ri, Rohatgi, ad Sze kely [19]. O the other had, we have lim ; d ; \ W, ;& 1 + =&d & : exp j& \&&Y (1.5) +

5 NEW APPROACH TO THE BHEP TESTS 5 which shows that, i the limit ;, rejectio of H d is equivalet to rejectig H d for small values of &1 exp(& 1 &Y j& ). Note that this statistic is similar to Mardia's measure &1 &Y j& 4 of multivariate kurtosis (Mardia [18]) i the sese that it ivestigates a aspect of the ``radial part'' of the stadardized uderlyig distributio. I view of the four properties featured at the begiig of this sectio, we stress that stadard MVN tests based o multivariate skewess ad kurtosis i the sese of Mardia lack the property of uiversal cosistecy (see Barighaus ad Heze [4] ad Heze [1]). The purpose of this paper is to provide a ew approach to the class of BHEP tests. The reasoig utilizes the theory of weak covergece i the Fre chet space C(R d ) of cotiuous fuctios o R d ad is preseted i Sectio. By meas of this approach, we obtai v a ew represetatio of the limitig ull distributio of T, ; i terms of a Gaussia process i C(R d ) v the joit limitig ull distributio of T, ; for several values of ; v the asymptotic power of the test based o T, ; agaist cotiguous alteratives. Iterestigly, the MVN test based o T, ; is able to detect cotiguous alteratives which coverge to the ormal distributio at the rate &1 (see Sectio 3). The fial sectio presets some empirical results ad cocludig discussio.. THE LIMIT DISTRIBUTION OF T, ; UNDER H d Throughout this sectio, we assume that the distributio of X is a cetered d-variate ormal with uit covariace matrix I d, for short, XtN d (0, I d ). Sice T, ; is affie ivariat, this assumptio meas o loss of geerality whe studyig the distributio of T, ; uder H d. Our startig poit is the observatio that for d+1, T, ; = R d Z (t). ;(t) dt, (.1) where Z (t)= 1 - _ : cos(t$y j)+si(t$y j )&exp \&&t& +&, (.)

6 6 HENZE AND WAGNER t # R d. Note that Z is a radom elemet i the Fre chet space C(R d )of cotiuous fuctios o R d, edowed with the metric where \(x, y)= : k=1 \ &k k (x, y) } 1+\ k (x, y), \ k (x, y)= max x(t)&y(t). &t&k Theorem.1. Let X 1, X,...be a sequece of i.i.d. N d (0, I d ) distributed radom d-vectors, ad let Z be defied i (.). There exists a cetered Gaussia process Z i C(R d ) havig covariace kerel K(s, t)=exp \&&s&t& + & { 1+s$t+(s$t) = exp +&t& \ &&s& (.3) + (s, t # R d ) such that Z w D Z i C(R d ), where `` w D '' deotes covergece i distributio. Theorem.. Uder the coditios of Theorem.1, we have Z (t). ;(t) dt w D R d Z (t ). ; (t ) dt. R d Note that this result is ot trivial sice the fuctioal x [ [x] := x (t). ; (t) dt (.4) is ot cotiuous o C(R d ) (it is ot eve defied o C(R d ) but oly o the subset of square-itegrable fuctios with respect to. ; ). Here ad i what follows, a uspecified itegral deotes itegratio over the whole space R d. Remark. There seems to be a coectio betwee Z(t) ad a limitig process Z (t) obtaied by Cso rgo [6] i the cotext of MVN testig via the empirical characteristic fuctio (see Theorem. of [6]). Both Z(t) ad Z (t) have the property that their values are idepedet at orthogoal vectors. Whereas the process Z is a simple trasform of a complex-valued Gaussia process Y (see formula (.5) of [6]), it is ot clear whether the

7 NEW APPROACH TO THE BHEP TESTS 7 process Z is a trasform of that same Y. If the uderlyig distributio is N d (0, I d ), the covariace kerel *(s, t) ofygive i (.) of [6] is related to the kerel K(s, t) by the formula Proof of Theorem.1. K(s, t)=*(s,&t)&(s$t) exp \ &&s& +&t& +. Let Z*(t) := 1 - _ : cos(t$x j)+si(t$x j )&exp \&&t& + ad + {1 (t$x j) & &t& &t$x j= exp \ &&t& +& (.5) Z (t) := 1 - _ : cos(t$x j)+si(t$x j )&exp \ &&t& + +[cos(t$x j )&si(t$x j )]t$ j&, (.6) where j =(S &1 &I d ) X j &S &1 X. (.7) The mai steps of the proof are to show that Z * w D Z i C(R d ), (.8) ad \(Z, Z ) w P 0, (.9) \(Z, Z*) w P 0. (.10) To prove (.8), use elemetary trigoometric idetities ad the formulae E cos(t$x)=exp(&&t& ), E si(t$x)=0, E(s$Xt$X)=s$t, E(t$X si(s$x))=s$t exp(&&s& ), E((s$X) (t$x) )=&s& &t& +(s$t), E((s$X) cos(t$x))=(&s& &(s$t) ) exp(&&t& )

8 8 HENZE AND WAGNER to obtai EZ*(t)=0 ad E(Z *(s) Z *(t))=k(s, t), where K(s, t) is give i (.3). By the multivariate cetral limit theorem, the fiite dimesioal distributios of Z * coverge to cetered multivariate ormal distributios with covariaces determied by the kerel K. To prove that the sequece (Z*) 1 is tight, it suffices to show that for each k1 the sequece Z *, restricted to R d =[t # k Rd : &t&k], is tight i the Baach space C(R d ) k (adapt the reasoig give i Karatzas ad Shreve [15, p. 6f.], to the preset case). To this ed, let N(R d k,!) deote the smallest umber m such that R d k ca be covered with m spheres of radius!. Sice d N(R d,!) k+1 k 1+ } \! + is a (crude) boud for N(R d k,!), the metric etropy coditio 1 0 [log N(R d k,!)]1 d!< holds. Moreover, lettig g(x, t) :=cos(t$x)+si(t$x)&exp \&&t& + + {1 (t$x) & &t& &t$x = exp \ &&t& +, (.11) straightforward calculatios yield the estimate g(x, s)&g(x, t) M } &s&t& (&s&, &t&k), where M=(k+k 3 )(1+&X& )+(3+k ) &X&. Sice M0 ad EM <, (.8) ow follows from Corollary 7.17 of Araujo ad Gie []. To prove (.9), ote that Y j =X j + j with j give i (.7). Usig trigoometric formulae, we have cos(t$y j )=cos(t$x j )&t$ j si(t$x j )+=, j (t) si(t$y j )=si(t$x j )+t$ j cos(t$x j )+', j (t), where =, j (t) &t& & j &, ',j (t) &t& & j &. Sice Z (t)&z (t)= 1 - : [=, j (t)+', j (t)],

9 NEW APPROACH TO THE BHEP TESTS 9 it follows that max Z (t)&z (t) k &t&k - } : & j &. (.1) From ad - (S &1 - (S &1 &I d )=& 1 - : &I d )=-(S &1 - (S &1 &I d )=& 1 - : (X j X$ j &I d )+O P ( &1 ) &I d )(S &1 +I d ) we obtai Writig tr( } ) for trace, a simple calculatio shows that & j & =X$ j (S &1 ad, thus, 1 - : & j & =- tr &I d ) X j &X$ j S &1 (S &1 &X $ S &1 (X j X$ j &I d )+O P ( &1 ). (.13) &1 &I \(S d ) 1 : X j X$ j+ &I d ) X +X $ S &1 X - (S &1 &I d ) X +- X $ S &1 X. I view of X =O P ( &1 ), &1 X jx$ j =I d +O P ( &1 ), (.1), (.13) ad the defiitio of the metric \, (.9) follows. To verify (.10), ote that max Z (t)&z *(t) &t&k where = max &t&k} U (t)& 1 - : \ (t$x j) U (t)= 1 - : t$ j (cos(t$x j )&si(t$x j )) & &t& &t$x j+ exp \ &&t& +}, (.14) 1 =t$ -(S &1 &I d )(A (t)&x B (t))&t$ - : X j B (t)

10 10 HENZE AND WAGNER ad A (t)= 1 : B (t)= 1 : X j (cos(t$x j )&si(t$x j )), (.15) (cos(t$x j )&si(t$x j )). (.16) From the compactess of [t: &t&k], the cotiuity of A ( } ) ad B (}) ad the strog law of large umbers, it is straightforward (although a little tedious) to show that ad max &t&k" A (t)+t exp \&&t& +" ww 0 almost surely max &t&k} B (t)&exp \&&t& +} ww 0 almost surely. O combiig (.15), (.16) with (.13) ad pluggig ito (.14), we obtai max Z (t)&z *(t) w P 0. &t&k This proves (.10) ad cocludes the proof of Theorem.1. K Proof of Theorem.. Use K(t,t). ; (t)dt< ad Toellis theorem to coclude that Z (t). ; (t)dt is fiite almost surely. The mai steps of the proof are to show that ad [Z *] w D [Z] (.17) [Z &Z *] w P 0 (.18) (recall the otatio [x] from (.4)). Note that, from (.17) ad the cotiuous mappig theorem (CMT), we have [Z *] w D [Z]], which, i view of the triagle iequality [Z *] & [Z ] [Z *&Z ] ad (.18), implies [Z ] w D [Z] ad, thus, [Z ] w D [Z]. (.19)

11 NEW APPROACH TO THE BHEP TESTS 11 From the proof of Theorem.1 (cf. (.1)) we obtai Z (t)&z (t) &t& - : & j & =&t& } o P (1) which, i tur, implies [Z &Z ] w P 0. The assertio of Theorem. ow follows from (.19), the CMT ad the iequality [Z ] & [Z ] [Z &Z ]. To prove (.17), recall that Z * ad Z have the same covariace kerel K defied i (.3). For fixed =>0, we may thus choose a compact set C such that where E(V,1 )=E(V 1 )=, (.0) V,1 = R d "CZ *(t). ; (t) dt, 1, V 1 = R d "C Z (t). ; (t) dt. Put V, = C Z *(t). ; (t) dt, V = C Z (t). ; (t) dt, ad write F ad F for the distributio fuctios of [Z *] (=V,1 +V, ) ad [Z] (=V 1 +V ), respectively. Usig [V,1 +V, t][v, t], [V t][v 1 +V t+=] _[V 1 =], [V,1 +V, t]_[v,1 =]$[V, t&=], [V t&=]$[v 1 +V t&=], together with V, w D V, (.0), ad the cotiuity of the distributio fuctio of V, we have

12 1 HENZE AND WAGNER F(t&=)&=P(V t&=)&= = lim P(V, t&=)&= lim if lim sup F (t) F (t) lim P(V, t) =P(V t) F(t+=)+= ad, thus, (.17) by lettig = ted to zero. To prove (.18), we deduce from the proof of Theorem.1 (cf. the reasoig followig (.14)) that (Z (t)&z *(t)) _&t& " &1 : +&t&&a (t)&o P (1) (X j X$ j &I d ) ""A (t)+t exp \&&t& +" +&t& - &X & }B (t)&exp \&&t& +} +&t& &-(S &1 &I d )&&X & B (t) &, (.1) where A (t) ad B (t) are give i (.15) ad (.16), respectively. Lettig W := \ B (t)&exp \ &&t& &t& ++. ; (t) dt, Toellis theorem ad some algebra give E(W )= 1 (1&exp(&&t& )) &t&. ; (t) dt ad, thus, W =o P (1). (.)

13 NEW APPROACH TO THE BHEP TESTS 13 I the same way, straightforward calculatios yield ad, thus, where Sice by (.1), we have t 1 E(W )= (1&&t& exp(&&t& )) &t&. ; (t) dt t W =op (1), (.3) t W := " A (t)+t exp &t& \&&t& +". ; (t) dt. [Z &Z *] O P (1) } " &t& A (t)+t exp \ &&t&. +" ; (t) dt +o P (1) } &t& &A (t)&. ; (t) dt +O P (1) } &t& +o P (1) } &t& B (t). ; (t) dt, \ B (t)&exp \ &&t&. ++ ; (t) dt the proof ow follows from (.), (.3), ad the fact that both the secod ad the last itegral form a tight sequece (we have covergece of expectatios). K It is well kow that the distributio of T ; (d) := Z (t). ; (t)dt is that of 7 j1 * j (;) N j, where N 1, N,... is a sequece of idepedet uit ormal radom variables, ad (* j (;)) j1 is the sequece of ozero eigevalues of the itegral operator A defied by Aq(s)= K(s, t) q(t). ;(t) dt. Although K give i (.3) looks much simpler tha the kerel h* ; figurig i Theorem 3.1 of Heze ad Zirkler [11], we did ot succeed i solvig the equatio Aq(s)=*q(s) ad thus gettig a explicit form for * j (;).

14 14 HENZE AND WAGNER However, some valuable iformatio o the distributio of T ; (d) may be obtaied from the relatios E(T ; (d))= K(t, t). ;(t) dt, Var(T ; (d))= K (s, t). ; (s). ; (t) ds dt, ad E(T ; (d)&et ; (d)) 3 =8 K(s, t) K(t, u) K(u, s). ;(s). ; (t). ; (u) ds dt du. By tedious maipulatios of itegrals we obtai the followig result (for ET ; (d) ad Var(T ; (d)); see also Theorem 3. of Heze ad Zirkler [11]): Theorem.3. We have E(T ; (d))=1&# &d _ 1+d; # Var(T ; (d))=(1+4; ) &d +# &d ;4 +d(d+) # &, ;8 _ 1+d;4 +3d(d+) # 4# & 4 E(T ; (d)&et ; (d)) 3 &4$ &d =8(1+3; ) &d &1(# ) &d _ 1+3d;4 $ ;8 +d(d+) $ &, _ +d;4 # +d;6 # +d;8 # +d(d+) ;1 # & +6(#$) _ &d ;8 3d(d+) ;1 4+4d;4 $ +4d;6 +d(d+) + #$ # $ # $ & &# &3d _ 8+1d;4 + 8d;6 ;8 d(d+)(d+8) ;1 +6d(d+) + # 3 # # 4 # &, 6

15 NEW APPROACH TO THE BHEP TESTS 15 where #=#(;)=1+;, $=$(;)=1+4; +3; 4, = (;)=1+4; +; CONTIGUOUS ALTERNATIVES I this sectio, we cosider a triagular array X 1,..., X, d+1, of rowwise idepedet ad idetically distributed radom vectors havig Lebesgue desity, f (x)=.(x)}(1+ &1 h(x)), where.=. 1 is the desity of N d (0, I d ) ad h is a bouded measurable fuctio such that h(x).(x)dx=0. To guaratee that f is oegative, we tacitly assume to be large eough. I what follows, we retai the otatio adopted i the previous sectios; i.e., we write X = &1 X j, S = &1 (X j&x )(X j &X )$, Y j :=S &1 (X j &X ), etc. Theorem 3.1. Uder the triagular array X 1,..., X ad the stadig assumptios, we have Z w D Z + c i C(R d ), where Z is defied i (.), Z is the Gaussia process figurig i Theorem.1 ad the shift fuctio c is give by c(t)= g(x, t) h(x).(x) dx, where g(x, t) is defied i (.11). Theorem 3.. Uder the coditios of Theorem 3.1, we have Z (t). ;(t) dt w D ( Z( t )+c(t)). ; (t) dt. From Theorem 3.1 ad Theorem 3., we coclude that the BHEP tests are able to detect alteratives which coverge to the ormal distributio at the rate &1, irrespective of the uderlyig dimesio d.

16 16 HENZE AND WAGNER Proof of Theorem 3.1. Cosider the probability measures P () := } (.* d ), Q () := } ( f * d ) o the measurable space (X, B ):=} (Rd, B d ), where * d is Lebesgue measure o the Borel sets B d of R d. Puttig L :=dq () dp (), we have log L (X 1,..., X )= : log(1+ &1 h(x j )) = : \ &1 h(x j )& h (X j ) + +o ()(1) P ad thus, by the LidebergFeller theorem ad the law of large umbers, log L ww D N \ &_, _+ uder P (), where _ :=h (x).(x)dx<. By LeCam's first lemma (see, e.g., Wittig ad Mu llerfuk [8, p. 311]) the sequece Q () is cotiguous to P (). Notig that, uder P (), lim Cov(Z*(t), log L )=c(t), where Z * is the auxiliary process itroduced i (.5), it is straightforward to show that, for fixed k ad t 1,..., t k # R d, the joit limitig distributio of Z *(t 1 ),..., Z *(t k ) ad log L uder P () is the (k+1)-variate ormal distributio N k+1 0 b 0 &_ &, \7 c$ c + &, _ where 7=(K(t l, t m )) 1l, mk ad c=(c(t 1 ),..., c(t k ))$ (recall K(s, t) from (.3)). Ivokig LeCam's third lemma (see, e.g., Wittig ad Mu llerfuk [8, p. 39]), we thus obtai that, uder Q (), the fiite dimesioal distributios of Z * coverge to the fiite dimesioal distributios of the shifted Gaussia process Z+c. Sice tightess of Z * uder P () ad the cotiguity of Q () to P () imply tightess of Z * uder Q (), we have Z * w D Z + c uder Q ().

17 NEW APPROACH TO THE BHEP TESTS 17 Sice, by (.9) ad (.10), \(Z, Z *) teds to zero stochastically uder P () ad thus also uder Q () (because of cotiguity), the assertio of Theorem 3.1 follows. K Proof of Theorem 3.. Sice the proof follows the reasoig give i the proof of Theorem., it will oly be sketched. First, from the boudedess of h ad the fact that g(x, t) 3+&t& &x& +&t& +&t&&x&, we have c(t) #(1+&t& ) for some costat # which esures that (Z(t)+ c(t)). ; (t) dt is fiite almost surely. To prove ote that, uder Q (), we have [Z *] w D [Z+c] uder Q (), (3.1) E Z *(t). ; (t) dt= \ K(t, t)+&1 c (t) +. ;(t) dt+o(1). This shows that, for fixed =>0, there is a compact set C such that, uder Q (), E(V 1 )= ad E(V,1 )=, 1, where V,1, V 1 are give i the proof of Theorem. with the oly exceptio that Z is replaced by the shifted process Z+c. The rest of the argumet for provig (3.1) rus alog the lies of the proof of (.17). Sice [Z ] & [Z *] coverges to zero stochastically uder P () (see the argumet followig (.18)) ad, thus, because of cotiguity, also uder Q (), the assertio of Theorem 3. is a cosequece of (3.1) ad the CMT. K 4. EMPIRICAL RESULTS AND DISCUSSION We stress that, for each fixed ;>0, rejectig the hypothesis H d of multivariate ormality for large values of T, ; yields a affie ivariat ad uiversally cosistet MVN test which may be carried out easily for ay umber of dimesios (cf. the four bullets at the begiig of Sectio 1). To determie approximate upper quatiles of the ull distributio of T, ;, a Mote Carlo experimet was performed. The results are give i Tables IIV for several values of, d, ad ; ad the sigificace levels :=0.1 ad :=0.05. Each tabulated value is based o Mote Carlo replicatios. A etry like & stads for 1.17_10 &5, ad meas The results clearly show that the test is practically sample size idepedet; i.e., the critical values seem to coverge rapidly to their correspodig asymptotic values. This observatio (which cotrasts with other oparametric settigs where a ormal limit arises) is typical for situatios where the statistic uder discussio is a degeerate U- or

18 18 HENZE AND WAGNER TABLE I Empirical Percetage Poits of T, ; (d=) ;=0.1 ;=0.5 ;=1.0 ;=3.0 1&: =0 & =50 & =100 & q ;, d (:) & q + ;, d (:) & & & & & & TABLE II Empirical Percetage Poits of T, ; (d=3) ;=0.1 ;=0.5 ;=1.0 ;=3.0 1&: =0 &5.14 =50 &5.55 =100 &5.67 q ;, d (:) &5.7 q + ;, d (:) &5.5 & & & & & TABLE III Empirical Percetage Poits of T, ; (d=5) ;=0.1 ;=0.5 ;=1.0 ;=3.0 1&: =0 & & =50 &5 6.4 & =100 & & q ;, d (:) & & q + ;, d (:) &5 6.7 &

19 NEW APPROACH TO THE BHEP TESTS 19 TABLE IV Empirical Percetage Poits of T, ; (d=10) ;=0.1 ;=0.5 ;=1.0 ;=3.0 1&: =50 &4.64 & =100 &4.79 & =00 &4.87 & q ;, d (:) &4.91 & q + ;, d (:) &4.91 & V-statistic (see Go tze [10]). That W, ; defied i (1.) is a degeerate V-statistic with estimated parameters has bee exploited by Heze ad Zirkler [11]. I each table, the row deoted by q ;, d (:) is the (1&:)-quatile of a logormal distributio havig expectatio + ;, d :=E(T ; (d )) ad variace _ ;, d =Var(T ;(d)) give i Theorem.3, i.e., &1 q ;, d (:)=+ ;, d\ 1+_ ;, d + ;, d+ exp (1&:) ;, d log \8&1 \1+_ + ;, d++, where 8 &1 ( } ) deotes the iverse of the stadard ormal distributio fuctio. Likewise, the row deoted by q + ;, d (:) is the (1&:)-quatile of a Fig Power of the MVN tests based o T, ; (d=5) for a ormal mea mixture.

20 0 HENZE AND WAGNER Fig. 4.. Power of the MVN tests based o T, ; (d=5) for a uiform distributio over the uit 5-cube [0, 1] 5. three-parameter logormal distributio havig the first three momets as give i Theorem.3; i.e., q + ;, d(:)=+ ;, d & where _ ;, d -a+1a&\ 1&exp(8&1 (1&:) - log(a+1a&1)) -a+1a&1 +, a= \1+ m ;, d _ 3 ;, d m ;, d =E(T ; (d)&et ; (d)) m ;, d + m4 13 ;, d, _ 3 ;, d _ 6 ;, d+ With the exceptio of the cases d=, ;=0.1 ad d=10, ;=3, both q ;, d (:) ad q + ;, d (:) show remarkably good agreemet with the empirical quatiles. We therefore suggest to use q ;, d (:) orq + ;,d(:) as a approximate critical value for a omial level : test based o T, ; if the sample size is ot too small. To illustrate the depedece of power of the MVN test based o T, ; o the parameter ;, Figs exhibit plots of the empirical power (based o Mote Carlo replicatios) of the T, ; test for the case d=5 ad the sample sizes =0, =50, ad =100 as a fuctio of ;. I each case, the omial level is 0.1. The alteratives to ormality chose are a equal mixture of the stadard ormal distributio N d (0, I d ) ad the ormal distributio N d (a, I d ), where a=(3, 3,..., 3) (Fig. 4.1), the uiform distributio

21 NEW APPROACH TO THE BHEP TESTS 1 Fig Power of the MVN tests based o T, ; (d=5) for a symmetric Pearso type II distributio give i (4.1). over the uit 5-cube (Fig. 4.), the spherically symmetric Pearso Type II distributio with desity 1(9)? 5 (1&&x& )}1[&x& <1] (4.1) (Fig. 4.3) ad the spherically symmetric Pearso Type VII distributio havig desity 3? 3 (1+&x& ) &5 (4.) It is strikig to see that, at least for the values of uder study ad a certai rage of values for ;, power does ot icrease with the sample size. We have o theoretical explaatio for this weird behavior which certaily costitutes a field of future research. Takig a large value of ; (which effectively amouts i rejectig H d for large values of the right had side of (1.5)) seems to be a powerful procedure agaist short-tailed symmetric alteratives. O the other had, takig a small value of ; which essetially results i computig a covex combiatio of two skewess measures (see (1.4)) is a good safeguard agaist symmetric heavy-tailed distributios. Of course, much more work eeds to be doe to uderstad the depedece of power o ; i order to obtai some kid of a adaptive test for multivariate ormality.

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