ON PERMUTATIONAL CENTRAL LIMIT THEOREMS FOR GENERAL MULTIVARIATE LINEAR RANK STATISTICS

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1 ON PERMUTATIONAL CENTRAL LIMIT THEOREMS FOR GENERAL MULTIVARIATE LINEAR RANK STATISTICS by Praab Kumar Se Departmet of Biostatistics Uiversity of North Carolia at Chapel Hill Istitute of Statistics Mimeo Series No Sepetember 1981

2 ON PERMLITATIONAL CENTRAL LIMIT TIIEORfMS FOR GENERAL MULTIVARIATE LINEAR RANK STATISTICS * By PRANAB KUMAR SEN Uiversity of North Carolia~Chapel Hill SUMMARY. For multivariate liear rak statistics permutatioal cetral limit theorems have bee proved mostly either icorrectly or uder uecessarily striget regularity coditios. These theorems are revisited here with some special emphasis o a ovel martigale approach. 1. INTRODUCTION I oparametric multivariate aalysis permutatioal cetral limit theorems (PCLT) playa vital role. I the cotext of multvariate multisample rak order tests Puri ad Se(1966Theorem 4.1) cosidered a PCLT ad this was later [Puri -e ad Se(1969Theorem 3.2)J exteded to geeral liear models. I either case the proof provided by these authors is ot correct (as will be explaied i Sectio 2); evertheless the theorems remai valid uder the striget regularity coditios stated there. These theorems are multivariate geeralizatios of the classical (uivariate) PCLT which i its most geeral form is due to Hajek(196l).He was able to icorporate a powerful (quadratic mea) equivalece result for liear rak statistics ad liear combiatios of idepedet radom variables (r.v.) which provide the desired result through the classical CLT. Puri ad Se(197l Theorem 5.4.1) adapted this techique i the multivariate case; but their result. o the PCLT does ot properly follow from the Hajek(196l) result though the coclusio o the ucoditioal ull distributio remais true. Coclusio o the PCLT i the multivariate case based o the momet-covergece property [cf. Wald ad Wolfowitz(1944)J isof coursevalid but demads comparatively striget regularity coditios. All these call for a re-examiatio of multivariate PCLT. * Work partially supported by the Natioal Heart Blood ad Lug Istitute Cotract NIH-NHLBI-7l-2243-L from the Natioal Istitutes of Health.

3 -2- The object of the preset ivestigatio is to provide a systematic accout of multivariate PCLT's alog with a ovel martigale approach. The mai results alog with the prelimiary otios are preseted i Sectio 2 ad their proofs are cosidered i Sectio 3. I this cotext a martigale approachdeveloped i the cotext of progressive cesorig by Chatterjee ad Se(1973) ad exteded further by Se(1979) is icorporated i a coveiet proof of a multivariate PCLT uder less striget regularity coditios. 2. THE MAIN THEOREMS Let ~i = (XiI' ' Xip)' i=l be idepedet ad idetically distributed (i.i.d.) r.v.'s with a cotiuous distributio fuctio (d.f.) F defied o the Euclidea space R P where p is a positive iteger.let ~i = (cil c iq )' i=l be a set of kow (regressio) vectors where q > 1. For each j (=l p) let R ij be the rak of X ij amog X 1j ""'X j for i=l (ties amog the observatios are eglected i probability as F is assumed to be cotiuous) ad let a j (l) a () be a set of scores. The a set j of multivariate liear rak statistics (LRS) L L = jk L. l(c ij - cj)a j(r ). c = Li=lcij 1= ij j = «L jk» may be defied by (2.1) for j=l p ad k=l q. I geeral because of the iter-depedece of the p variates L is ot geuiely distributio-free. However it is permu-... tatioally (coditioally) distributio-free uder the followig rak permutatio model due to Chatterjee ad Se(1964). Cosider the rak-collectio matrix m "" (of order px) specified by ~ = (~l'" '~) = (~::.:::::.~~:) (2.2) R l R P p where R _i = (Ril"'" R) ip for i=l Cosider ow a permutatio of the t colums of ~ so that the top row is i the atural order (viz.l )ad deote the resultig matrix (termed the reduced rak-collectio matrix) by OR * Note that the totality of (!)p rak collectio matrices may thus be partitioed ito (l)p-l subsets where each subset correspods to a particular reduced rak

4 -3- collectio matrix m* ad the subset SQR *) has cardiality!. The coditioal distributio of ~ over the appropriate S(tR *) is uiform irrespective of the _ (cotiuous) d.f. F. We deote this coditioal (permutatioal) probability measure by ~ The we have [ see Puri ad Se(1969)] where E 8 ~ = ~ ad V~ ~ = ~e ~ ~ = «Vjj'»jj'=l p has the elemets -1 - }{ V jj ' = (-l) Li=l{aj(Rij)-aj aj(r ij ) -1. a j = Li=l a j (1) j=l p (2.3) (2.4) (2.5) ad -e Our primary cocer is to study the asymptotic multiormality of Luder the permutatio model Sice the ~i e; this is termed the multivariate PCLT. are specified vectors without ay loss of geerality we may assume that the rak of C = «C kk'» is q for adequately large. More ~ ~ specifically we let C = D Q D where D = Diag( C 11' 'C ) ad assume qq that TI the smallest characteristic root of Q satisfies the followig: TI > Q > 0 for every > (> q) Further as i Majumdar ad Se(1978) we let ~ = max {( -)' -1 ( -)} ~ l<i< c.-c C c i - c -1 - ad defie the (exteded) Noether coditio as (2.7) (2.8) ~ ~ 0 as ~ 00. (2.9) also the exteded Ha}ek(1968) coditio is defied by sup { ~ }< > -0 I the multi-sample case ~ ~* < 00. (2.10) treated by Puri ad Se(1966) (2.10) holds while i the liear model case Puri ad Se(1969) assumed that (2.10) holds; this may ot be really eeded. I both these papers sufficietly striget coditios o the scores were imposed which isure that V _ coverges i probability to a positive defiite (p.d.) matrix ~ Basically for the multivariate PCLT

5 -4- it suffices to show that for every oull A (of order pxq) Trace(AL ) is - /:) _ (1) (q) asymptotically ormal (uder cr). If we let L - (L L ) the Puri ad Se( ) cosidered arbitrary liear compouds of the form AlL(l)+ + A L(q) expressed the same i the form of E~lg ira l(ril) a (R. )]' q 1= p 1p ad the appealed to Theorem 7.1 of Hajek(196l) to show that uder E? the later (with suitable gi) is asymptotically multiormal. There appears to be some flaws i these steps. First oe eeds to cosider a arbitrary liear compoud of all the pq elemets of L (ot simply a liear combiatio of its q colums) i order that the Cram{r-Wold characterizatio theorem applies.for such a geeral ~ their simplified form for Trace(~~) may ot be obtaiable. Secod eve otherwise Theorem 7.1 of Hajek(196l) does ot apply here. I this case we have vectors Rl R whose joit distributio uder E( (beig - / differet from their ucoditioal d.f.) does ot coform to the model of Hajek (196l 0 where (Rl R ) assumes all possible permutatios of (l ) with the equal probability (l)-l ad a (i)i=l were p-vectors. This explais the iadequacy of the proofs of the multivariate PCLT's i Puri ad Se( ). Puri ad Se(1971 Theorem 5.4.1) have sketched a differet approach. Let F[j] be the jth margial d.f. for F for j=l p ad let ~i = (Uil U ip )' with U ij = F[j](X ij ) j=l p i=l. Also for eacj j(=l p) let Fially let aoj(u) = a j(i) for (i-l)/<u<i/ i=l. (2.11) - L~ = E~=l[ a~l(uil) a~p(uip)]' The by usig the coordiatewise proof of - (c i - c ). - Hajek(196l) " (V ~ C ) -~ II L - L O ll... 0 i probability as (2.12) Puri ad Se(197l) showed that (2.13) so that the asymptotic ormality (uder e ) of L O would esure the same for L However the classical CLT may ot apply to the coditioal (permutatioal) distributio of L O ad hece the proof (uder e ) remais icomplete; though the asymptotic ucoditioal multiormality of L O ad hece of L would follow. '

6 -5- From the above discussio it seems desirable to formulate multivariate PCLT's for liear rak statistics i a uambiguous maer ad to provide valid proofs for them. Towards this we defie (i) a = a = i=l (i) a - a ) (2.14) (2.15) (2.16) where for the time beigwe assume that V is of full rak (otherwise uder ~ L will have a degeerate d.f.). Note that the a(i) ad V are stochastic O' i ature ad hece ulike the case of p=l Y is i geeral a r.v.the we have the followig Theorem 1. If o (i probability) ~ + 00 ~ uder e L is asymptotically (i probability) ormal with mea ~ dispersio matrix -e V C It may be oted that uder (2.10) all we eed for the above theorem to hold is -1 that y + 0 (i probability) as + 00 ad this ca be established uder coditios much weaker tha the oes i Puri ad Se( l). I the. ext theorem we do ot wat to impose (2.10) ad desire to icorporate (2.9). For this we suppose that there exist score fuctios ~j(u)o<u<l o such that for the aj(u) defied by (2.11) l~;~p{ f~ {a~j(u) - ~j(u)}2 du} + 0 as + 00 j=l p (2.17) where for each j(=l p) ~j(u) = ~jl(u) - ~j2(u) O<u<l where ~jk(u) is odecreasig absolutely cotiuous ad square itegrable iside (01) for k=12. (2.18) These two coditios are less striget tha the oes i Puri ad Se( ). Theorem 2. 1 (2.9) (2.17) ad (2.18) hold the uder ~' ~ is as~ptotically (i probability) ormal with mea 0 ad dispersio matrix V ~C ~ ~ -0 Proofs of these theorems alog with other commets are preseted i the ext sectio.

7 -6- -e ad let 3. PROOFS OF THE THEOREMS Let us first cosider the proof of Theorem 1. For a arbitrary oull matrix A (of order pxq) we like to show that uder e Tr(AL ) is asymptotically ormal. - If we defie the a(i) ad a as i (2.14)-(2.15) we have the Tr(~~~ ) = L~.lL~=lL~=l(cik - c k ) A jk [ aj(r ij ) - a ] (3.1) = L~=l ~~[~~i) - ~ J; d ij = L~=lAjk(cik-ck)' l<j<pl~i~ ad ~~ = (dil d ip )' for i=l.note that by (2.3) EeTr(~~~) = 0 ad 2 _ P P q q _ 2 E~(Tr(~~» - Lj=lLj'=lLk=lLk'=lAjkAj'kVjjCkk' - L say. (3.2) * * * Express the reduced rak collectio matrixm as ( Rl R ) so that the first ~ - elemet of ~i* is equal to i for i=l. Defie the Sl 8 by lettig R~l=RS 1 = i for i=l. (3.3) i Further let b (i ' tr*) = a(8i ) - a f i 1 or = (3.4) Y = (Y l""'y ) be a radom vector which takes o each permutatio of (l ) with the equal probability (!) The the permutatio distributio of Tr(AL ) agrees with the distributio of... * Z = Li=l... dib (Y i; lr ) 1 give tr_ * J. (3.5) Sice uder 1 m.* is... held fixed while the vector Y has the discrete uiform... distributio lover the set of permutatios of (l )J we may ow virtually repeat the proof of Theorem 3 of Hoeffdig(195l) ad obtai the asymptotic ormality of Z (give m * ) uder the sole coditio that y ~ ~ 0 as Sice I- is a r.v~ wheever Y~ ~ 0 i probability as ~ 00 the method of momet proof of Hoeffdig holds for all m * exceptig a subset with probability tedig to 0 as ~ 00 ad hece the aforesaid ormality holds i probability. This completes the proof of Theorem 1. To prove Theorem 2 let F[jj'J be the bivariate margial d.. for the (jj')th variates for the d.f. F for j~ j'=l p. Let the ~ = «V jj» be defied by V jj = l:l:lpj(fu](x»lpj(fu](y»df[jj'j(xy) lpjlp j " (3.6) for jj'=l p where

8 -7- </>j = f~ </>j (u)du Note that expressig (2.4) for j= l p. ad (2.5) i the itegral form ivolvig the empirical d.f.'s ad the usig (2.17)-(2.18) alog with the usual Gliveko-Catelli lemma type result we obtai that uder (2.17) ad (2.18). v -+ V i probability as (3.8) The proof of (3.8) is essetially similar to that of Theorem 3.1 of Puri ad Se(1969). ad hece. the details are omitted. Now. without ay essetial loss of geerality. we assume that V is positive defiite (otherwise. the limitig permutatio distributio of L will be sigular i probability). Let us also defie the Si as i (3.3) ad for every k:l~k~ let S k = (Sl Sk)' ad. let S = o. Let ow e be the sigma-field geerated by S (uder )..O.k.k for k=l ad let ~.O be the trivial sigma field. Let the L k = Ee (L I ~ k ) for k = O.l (3.9). ~e At this stage we appeal to Chatterjee ad Se(1973 Sectio 4) for the case of p = 1 ad Se(1979 Sectio 2) for geeral p ~ 1. ad obtai that where k * *- L = ~i=li.al(rs.l)-al(k) a (R S )-a (k)j'(c S -c) (3.10).k 1 p i P p - i (3.11) ad. covetioally we let aj() * = O. for j=l p. By (3.9) we coclude that uder E? for every. {L k' ~ k: 0 < k < } _.. -- is a zero mea martigale. (3.12) Thus to prove Theorem 2 it suffices to cosider for a arbitrary oull A (of order pxq) the partial sequece { Tr(~~ k)jo<k<} ad verify the coditios for the martigale cetral limit theorem [ viz. Dvoretzky (1972)J.Note that by (3.12) uder 6?' {Tr(~~~.k)' O<k<} also forms a martigale sequece. Hece if we defie Yk Tr(~~k) - Tr(~~k_l)' k=l 2 the. it suffices to show that for T' defied by (3.2) as (3.13)

9 -8- ad Now { y 2 p ~. 1 Eel ~ i-l] }/T~ -+ 1 ~= i (3.14) { ~~=l p 2 E IY.1 )J}/T~ \:I e [Y.l( > T -+ 0 > O. (3.15) ~ ~ (3.14) ad (3.15) follow (as a direct vector-extesio) precisely o the same lie as i (4.32) through (4.40) of Se(1979) ad hece the proof is complete. We coclude this sectio with the remark that this martigale approach adapted from Chatterjee ad Se(1973) ad Se(1979) besides providig a valid proof of the PCLT i the multivariate case avoids the computatioal complicatios ad the extra regularity coditio (2.10) of the momet-covergece approach i Theorem 1. REF IRE N C E S e CHATTERJEE S.K. ad SEN P.K.(1964). Noparametric tests for the bivariate two-sample locatio problem. Calcutta Statist.Assa.Bull. 1l ad --- (1973). Noparametric testig uder progressive cesorig. Calcutta Statist.Asso.Bull. ~ DVORETZKY A.(1972). Cetral limit theorems for depedet radom variables. Proc. 6th Berkeley SymP. Math.Statist.Prob.Uiv.Califoria Press l HAJEK J.(196l). Some extesios of the Wald-Wolfowitz-Noether theorem. A. Math.Statist (1968). Asymptotic ormality of simple liear rak statistics uder alteratives. A. Math. Statist HOEFFDlNG W.(195l). A combiatorial cetral limit theorem. A. Math. Statist. ~ MAJUMDAR H. ad SEN P.K.(1978). Noparametric tests for multiple regressio uder progressive cesorig. Jour. Multivar. Aal. ~ PURl M.L. ad SEN P.K.(1966). O a class of multivariate multisample rak order tests. Sakhya Ser.A ad --- (1969). A class of rak order tests for a geeral liear hypothesis. A. Math. Statist ad --- (1971). Noparametric Methods i Multivariate Aalysis. Joh Wiley New York. SEN P.K.(1979). Rak aalysis of covariace uder progressive cesorig. Sakhya Ser.A WALD A. ad WDLFOWITZJ.(1944). Statistical tests based o permutatios of the observatios. A. Math. Statist

THE LIM;I,TING BEHAVIOUR OF THE EMPIRICAL KERNEL DISTRIBDTI'ON FUNCTION. Pranab Kumar Sen

-e ON THE LIM;I,TING BEHAVIOUR OF THE EMPIRICAL KERNEL DISTRIBDTI'ON FUNCTION By Praab Kumar Se Departmet of Biostatistics Uiversity of North Carolia at Chapel Hill Istitute of Statistic~ Mimeo Series