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(1)Berry-Esseen Rates for Simple Linear Rank Statistics Kenici-Yoshihara* and Hiroshi Negishi (Received April 30, 1983). g1. Introduction. Let X;zi, ''', Xhn be independent random variables with continuous distribution functions Fni(x), ''', Fnn(x), respectively; all defined on the real line (-oo, oo).. Let u(x) be equal to 1 or O according T is;-;iO or<O and for every n(ll:1) let R.i =Ri be the rank of Xhi among .X}ti, ''･, Xh. for IS-i<..,n, i.e.. n (1.1) Rni == Ri= Z u(Xni -X}vd), 1=<i=< ne e'=1. Consider a simple linear rank statistic of the following form:. n (1.2) Sn ==Zcnian(Ri) i.=1. where cni,'",cnn are known regression constants and a.(i) for i'---1,･･･,n are scores.. Recently, Mason (1981) proved the convergence rate to normality of Sn to be O(n-"")(O<S*<S<1/2) when X}ti, ''', Xhn are independently and identically distributed, IJ'(u)l=<K(u(1-u))-3i2'6 and scores an(i) are defined by a.(i)= EJ(Uhi)(Uni being the i-th order statistic of n independent uniform (O, 1) random variables Uhi, '", Uhn).. Concerning the rate in the non-i.i.d. case, Jure6kova and Puri (1975), Bergstr6m and Puri (1977) and Hugkov6 (1977) obtained rates at which S. converges to normality under local regression alternatives when J is bounded. (See, also Serfling (1980)). But, to date rates for the case when J is unbounded are not known except the result of Ghosh and Sen (1972) which states that if IJ'(u)l -<-M(u(1--u))-', then it is possible to obtain a O(n-") rate for all O<6<1/4.. In this paper, using the projection technique introduced by Hajek (1968) and the method in Yoshihara (1978), we prove that the rate in the non-i.i.d. case is O(n-"')(O<6'<6<1/2) when IJ"'(u)ISM(u(1-u))-i'2-j'6(7'---O, 1, 2) and an(i)=:J(i/(n+1)) (Theorem 2.1). g2. Assumptions and notations. In the following sections, we always assume that Xhi, ''', Xhn are independent * Yokohama National University, Department of Mathematics, Faculty of Engineering..

(2) 2 , Berry-Esseen Rates '. ,. '. random variables with continuous distribution functions Fni, '･', Fnn, respectivity.. Let 6(>O) be any number. Let L6=={J} be the class of scores generating functions possessing the next properties:. (i) J is twice differentiable almost everywhere inside (O, 1) (ii) There exists a constant Mb such that. (2.1) I(d(`'/du(`'),J(u)iSMo{u(1-u)}-'/2-i"O (i---O, 1, 2). We shall consider the following conditions: Condition (!). The scores a.(1), "･, an(n) are given by the relation. (2.2) an(i)=J(.il) (1$i<=n)･. t' ). 1. Condition (ll). For the regression constants.c.i, ''', cnn there exist positive cons-. K (. tants b and M such that. is,. (2.3) max lcndlgM(logn)b for all n(n)-2). IEJ'Sn. In what follows, for brevity, we write ct, Fi, etc. instead of c.i, Fhi, etc.. Let n (2.4) H(x)==n-i:n(x) i=1. and for each i(1<.--.i-Sn) ,. (2･5) Yi=n-i ,,Z,... f{u(y-X))-F,(y)}J'(H(rv))dFi(rv) s'abi +{cJ<IHi(Xi)---EJ(HT(Xi))}, where J' denotes the derivative of J. Put. (2.6) Zij={u(Xl-"-Xli)--"Fj(Xi)}J'(H(Xi)) -f{u(y--Xi)-Fd(N)}J'(H(or))dF,(rv) (ISi, j' <.,, n, i#7'). Zi,i=o (1$i<..n). .. Let. ' (2.7) ei=E(RilXi)==1+ Z] Fd(Xi) (IEi<,.,,n). . ISd-Sn e'Si. '. and. R,-&. (2･8) rpi=.+1 (IS'i<==n)･ Put. '. '. / t/. n(2.9) ptn= ZciEJ(H(Xi)). i--1. If J E L6 for some 6(>O), then by Taylor's expansion we have the follow-. ing experession: (2.10) Sn-ptn-,Z. n1.,CtX. =,#,ci(J(nR+il)'-J(n6+'1)]+i",ci(J(ne+'1)-J(H(Xi))) --"-i,#,Ci,.Zi..f{u(or--X))-F,(or)}J'(H(pt)dFi(or). e'4i. ill. v.

(3) K. YosHiHARA and H. NEGism 3 = nil±1 tl.Ili tll.iCiZid+ ,2], copi( n6+'1 --"H(Xi))J"(oii). + ,#, cop2J"(02i)+ tl.II, ci(n//1 --H(Xi)) tJ'(o3t). "-"" n(i+1) tl-iCìs¥･s.{"(Xi-XD-Fd(Xl)}J'(H(Xi)). iti. where. (2･11) 0ii= .ttl +Tit H(Xle)- nttl' div. <l'i. (2.12) ･ 02i=.+1+T2ilrpiLand. '. IJ Ii. ;. (2･T3) 03i== .ttl +T3` H(X`)- n//1' with Tki(k==1, 2, 3;i--'1, '",n) being random variables such that ITidlSl.. Put. (2.14) sn2 =var (Z cii=1nYi). Condition (M). There exist positive constants M2 and M3 such that. (2.15) M2n<s.2<M3n.. Theorem 2.1. Let 6(O<6<1/2) and JEL6 be .fixed. Assume Conaitions (I)(Lll). Then the follozvings hold:. (i) For every 6'(O<6'<6). (2･16) An=suplP(Sn--yn<xsn)-￠(T)[=O(n""') (n.oo). x. (ii) The assertion remains true with sn r(iplace by var (Sn). (iii) Both assertions remaz'n true 2vith ptn replaced bov ESn.. Here. (2.17) ￠(T)=:JIX. v21. e-t2i2dt. Since if JGLi/2, then JEL6 for any 6(O<6<1/2), so we have the following theorem which is a generalization of Serfling's result (1980. Chap. 9. Theorem B (p.302)).. Theorem 2.2. Let JeLii2 be fixed. Assume Conditions (I)-(LII). Then the followings hold: ,g j-. (i) Foreverye(O<E<1/2). (2.18) a.=O(n-ii2"e) (n.oo).. (ii) The assertion remains ture zvith sn retzt)laced by var (S.). (iii) Both assertions remain ture zvith ptn replaced by ESn.. In what follows, we shall agree to donote by the letter M(with or without subscript) a quantity bounded in absolute value.. g3. Auxiliary lemmas. We use the next well-known inequality proved by S.N. Bernstein..

(4) 4 Berry-Esseen Rates Lemma A (Bernstein). Let Ci, ''', Cn be independent randbm variables satisL7?ying. P(ICi-ECilSM4)=1(IS-i<=n). Then, for t>O n (3.D P(IZ(C,-EC,)Ilnt) i=1. S2eXP("-2ivar(c7it+22/3A(Lnt] ("=i'2''")' O'---1. Firstly, we prove the following lemma. ･ Lemma 3.l. Let a(O<a<1) be fixed. 1[]C x is a number such that H(x);.)tl--n-". or H(x)S.n-", then for anor r(O<7<1--a) there exist constants Mk and Mb. t. 1,ts. such that for all n()-1). ' (3.2) P(n-il Z" {u(x-Xi)-Fi(x)}l).n-i/2(i-r)(i+a)) i=1 $Ms exp{-M6nr(i"a)}.. q. u.. Proof. If 'x is a number such that H(x)$1-n-a, then. nn. Ei Z {u(x-Xi)-`-'Fi(x)}l2= ￡ (1-Fi(x))Fi(x). i=1 i,=1. S Zn(1--Fi(x)) =n(1-H(x))Sn'-". Y=1. Simiarly, if x is a number such that H(x)S-n'cr, then. nn. El X {u(x--"Xi)--Fi(x)}l2.<., X Fi(x)==nH(T)=<ni-".. i--1 i=1 Hence, (3.2) follows from Lemma A.. Lemma 3.2. Let cr and r be the ones in Lemma 3.1. Then for each i(IS.i<..n). (3.3) P(lnytl4n'i/2(i-r'Ci'a', H(Xi)).1-n-cr) $Mk exp{-M6nr<i"a)}. and (3.4) P(lrpil l-}-.; n-i!2ì-r'("a', H(x,)s.n-a) $Ms exp{-M6n'('"cr)} Proof. (3.3) follows easily from Lemma 3,1 since 1' h'S'Of(3'3) =.1[,,,.,),-.-.) "P(lrp`l )' n-i12('-r'('i'"'IXi == T)dFi(T)･. The proof of (3.4) is similar, So, we have the lemma.. Next, for any a(O<a<1) let A :{u: O<uSn-a or 1--n-aSu<1}. For convenience, let ii 7k'''#ik denote that ii, ''n, ik are mutually different integers.. .,L. i [,t,x. {1. Since Xi, ''', X. are independent, so. nnk nnk. (3.s) Z ･･･ Z P( n {El(Xi,)eA})$ Z ･･･ Z R P(H(X,,)eA) it=1 ik=i o'--1 ii=1 ik=ld---1 il""'4ik =: ,nl, {,Z.,P(H(Xi,) E A)} " A.k, (n.4.,.,..} dH( x)]. k $ " (2nl'") :2icnte(i-a). i=1 Hence, by Lemma 3.2 we have the following lemma..

(5) K.YosmHARAandH.NEGisHi ' 5 Lemma 3.3. Let r be the one dofned in Lemma 3.2. Then, for aay P()-1) there exists a constant M> depending only on P for which the following inequalities hold:. Case (I) where Pm[P]>O:. n (3･6) EIZZ(H(Xi):A)13$Min([B]'i'(i-"); `".'. (3.7) E[ZlptIZ(H(X,):A)IB$ann-(Bl2)(i-r)(i+a)+([B]+i)(i-a); i".1. Y r}. [. `. (3.8) EIZ]rpi2X(H(Xi):A)IBSMin-B(i-r)(i"a)+([B]+i)(i-a). i=1. Case (II) where P =[P]. In this case, (3.6)-(3.8) hold zvith P instead of [P]+1.. Hizre, X(u: A) is the indicator ojC the set A and [P] denotes the integer part of P. Proof. We only prove Case (I). Firstly, as. nn. 1. h.s. of(3.6)SE[{ ￡ z(H(Xi): A)}[B] Z x(H(Xd): A)],. i=1 g' ---1. so (3.6) follows from (3.5).. Secondly, let k(ISk$[P]) be fixed. Let li(1;Si$k) be positive integers such that li+'''+lk=[P], Bic=={maxinidl<n-'i2(i-')(i'ev)} and let B-k be the comple1-S3'Sk. mentary set of Bte. As lrpilSl for each j'(IS7'<=n),. nn. E[{ Z 1ptIz(H(X,): A)][B]{ Z 1rp,,,,IB-'[B]X(H(Xi,.,): A)}]. ik+1=1. i=1. nnk n (3.g) ;sZ･･･Z 2] {n-B!2(i-r)(i+a)P(B,.,n[n{H(X,,)GA}]) nnn k+1 il=1 ik=lik+1=1 j'--･1. SE[{ Z ･･･l n Irp,,lt,X(H(X,,): A)}{ Z 1rp,,.,IB-"[B]Z(H(X,,,,): A)}] ik+1=1 il=1ik=1e'--1 ilab"'"ik. +P(BM,.,n[k+1 n {H(X,,)GA}])}. e'=1 = Ii+I2, (say), By (3.5). n Zk+1 hs.n-B/2(i-r)(i+a) Z n･･･ { " P(H(Xi,)EA)} il=1 iic+1=1 o'--･1 ;lil2k+1n'Bf2(1-r)(i+cx)+(ic+1)(1-ct).. 0n the other hand, by Lemma 3.2 eJ. I, S. nk+iP(B-,., nk+1 [ n {H(X,,)EA}]) o' --1 snk+i Z k+1 p(1rp,,1).n-i/2(i-r)(i+a), H(Xi,)eA) o'=1 $Mls(k+1)nte"exp{--Mbn'(i"a'}.. ]･. Hence, Lh.s.of(3.9)SMn-(B!2)(i-r)(i+a)'([B]+i)(i-ev) (k=1,･･･,[p]), which implies (3.7), since. nn. 1. h.s. of (3.7)SE[{ 2] ]mlZ(H(Xf): A)}[B] Z IrpklB-[B]Z(H(Xk): A)].. i=1. k=1. (3.8) can be proved analogously and so is omitted..

(6) 6 Berry-Esseen Rates g4. Moment inequalities of Sn'-ptn' Z ncdY)･ O' =1 In this section, we use the following notations, Let 6(O<6<1/2) and JEL6 be fixed. Let e(O<E<6) be arbitrary. Put cr=1-(e/8). Choose an in.t, eger d such. than ad-'<e/8. Define sets At(OSISd) as follows:. (4.1) Ao={u:O<uS-n-a or l-n-"Eu<1}. ' At={u:n'"i<uS.n-"t'iorl--n'aZ"'$u<1--n-at}. Put. (l=1, ･･･, d-1) A, =(O, 1)-d-1 U A,. I=1. il. ,-. ]. L. (4.2) Ke(u)==J(u)Z(u:Ai) (OEISd). Then, it'is obvious that KtEL6 (OSI$d),. d (4.3) J(u)=ZK,(u)X(u: A,), l= 1 and (4.4) l(d(i)/du(i))K,(u)ISMnai(-if2-i+a)X(u:A,) i-----O,1,2;l=O,1,･･･,d). Now, let S.(i', etc. be the ones obtained from (1.2), (2.5), (2.6) and (2.9) on replacing J by Ki, respectively. For brevity, we put. (4.5) K',l=Kit(0,d(t)),KL,1=K,M(0,d(t)),K,,d"=Kl,"(0,j(i)) (l=O, 1, ･･･, d; 7'=1, ''', n) Firstly, we provethe following theorem.. Theorem 4.1. Let JEL6(O<6<1/2) be fixed. Sumpose that Condition (I) holds and that ragression coeJOicients {cd} are unzformorl bounded. 77ien, for all. e(O<e<S) ･ (4･6) IES.-pt.1=<Mni/2-"'e. To prove Theorem 4.1, we need some lemmas.. Lemma 4.1. U>2der the assecnzptions of Theorem 4.1. (4･7) iES.(O)-dpt.(O'ISMni/2-6'e. Proof. Let cr be the one defined above. Putr==e/2(1+cr2). Using Taylor's expansion we have that for each i(IS-i<...n). KO( nttl)"KO( n//1)=rp2Ko'( n//1 +TzlV`l)Z( ntt'1 : Ao)X( n6+'1 : Ao). ･7･. L(. where Tt is a random variable with iTtl;;;;1. Since. ntl S ne+'1 +T`1'7ilS' n:Zl so. (4･8) Ko( nR+'1)-- Ko(n//1) $Mn3'2'61rpilX(ne+'1: Ae). Hence, by (3.5), (4.8), Lemma 3.3 and the fact lei/(n+1)-H(Xl)ISI/n. (4'9) tl.#,C`(EKo(.R+`1)'-EKo(.6+'1)). 1t... u.

(7) Berry-Esseen Rates 7 n S Mn3!2-6E{ Z 1tsIX(H(Xi): Ao U Ai)} i=1. SMn3/2-6n-1/2(1-r)(1+a2)+1(1-a2)=<Mnl12-6+e.. Moreover, since by (2.1). ' (4･10) Ke(.//1)SMni/2'6Z(.e+'1:Ao)5.Mnii2-"z(H(Xi):Ao), so by Lemma 3.3 (Case (ll)). (4･11) te.,ciEKo(.//1)S-Mnii2-6E{ ,Z(H(Xi):Ao)} f $Mnl12-6nlma2SMnl!2-6+e. ' b. On the other hand, we have. (4.12) 1pt.(O)1$te.,IcillEK,(H(X)))I$Mte.,flK,(H(T))IdF,(T). =MnflKo(H(T))ldH(T) $Mn(l(l'"-"+.Ill.-.]{u(i-u)}-i!2'6du. SMnl'ct(1/2"6)$Mnl12-6+e. Thus, (4.7) follows from (4.9), (4.11) and (4,12).. Lemma 4.2. U>zder the assumptions of Theorem 4,1. (4.13) IES.(`'-pt.(i'I$Mnii2-6'e (l=1,･･･,d-1). Proof. Let l(ISISd-1) be fixed. We note that as E{rpilXi}=O (ISi.gn), so by (2.11). E[rpi(Ki'(H(Xi))+(.//1-H(Xi))K"tit]]=O (ISiSn)･ Hence, putting r=e/4(1+at"2) and using (2.10), (4.4) and Lemma 3.3 (Case (E)) 1. h.s. of (4.13)$M[ t2.#, Erp2K"i2iZ(.//1 :A) +n-i t/i.;, EK'tiX(H(X): iAi)]. n e+1i=1 n l+1 e'---o. $M[nat(5i2'-"' Z E{rp,2x(HT(Xi): U Ad)}+n"`(3i2-6'n-i Z E{X(H(Xi): U AD}]. i=1. o' --o. $M[naC(5!2-6)n-(1-r)(1+at"Z)+(1-at+2)+nat(312-6)n-at"2] n. SMnll2-6+e. Thus, the proof is completed s. Lemma 4.3. U>zder the assumptions of Theorem 4.1. (4･14) IES.(d'-pt.(d'ISMni!2-6'e, Proof. We note first that by the difinition of d and (4.4). (4.15) sup{Kd'(u):uEA,}SMnad(3/2-"6)SMn3el8 and. (4･16) sup{IKd"(u)1:uEA,}$Mn"d(5/2-'6)sMn3ei8. As Erpi2SMn-i(IS-i<=n), so.

(8) 8 K. YosHiHARA and H. NEGiSHr Lhs. of (4.13)SM[ 2 E{rpi21K"d2il}+n-' ￡ EIKdi'l]. i=1. i=1 SM[n3ei8 Zn Ept2+ne!2]$Mne, i=1. which implies (4.14) and the proof is completed.. Proof of Theorem 4.1. The proof follows from Lemmas 4.1-4.3. Theorem 4.2. Su2eipose conditions of Theorem 4.1 hold. Then for aay P(2S-P< 2/(1-26)) and for anor e(O<e<6). n (4･17) EISn-pt.'ZciYilB$MnG(if2-6"e'. i=1 To prove Theorem 4.2 we need some lemmas. Lemma 4.4. U>zder the assumptions of Theorem 4.2 n (4.18) EISn(O)"pt.(O)-"' i¥, ci Yi(O)IBSMnB('l2-6"e).. c1) il. l 6. Proof. It is enough to prove the following inqualities:. (4'19) Etll.l,ci(Ko(nR+`1)-Ko(n8+'1)IBsMnB(ii2-6'e);. (4"20) 'EtlX.,ciKo(n//1)B.$MnB(i/2-6"e';. (4･21) lpt.Ò'l$Mni/2-6'e; n (4･22) Ell ciY,(O'IB$MnB('/2-6"e).. i=1 But, (4.21) was shown in Lemma 4.1. Now, put r= e/4(1+a2). By (4.8) and Lemma 3.3 (Case (I)) 1. h.s. of (4.19)SMnB(3/2-"'E{ Zn lrpilZ(H(Xi): AoUAi)}B. i=1. gMnB(312-6)n-Bf2(1-"r)(1+a2)+([B]+1)(1-a2)sMnB(1/2-es+e).. By (4,10) and Lemma 3.3 (Case (I)) L h.s. of (4.20)SMnB('!2-6'Eln Z Z(H(Xl): AoUAi)IB. i=1. EMn.B(112-6)n-B!2(1-r)(1+a2)+([B]+1)(1-a2). ' S; MnB(112-6+e).. To prove (4rt2), it is enough to show the following two inequalities:. (4e23) '.El￡ci{Ko(H7(Xl･))--EKo(H7(Xi))}IBSMnB(i/2-""e' ' i=1 (4･24) El te.,cm-i,.Ii.ll.. f{u(or-X))-Fd(ov)}Ko'(H(or)dFi(or)IB. f7. d4i. SMnB(1/2-e+e).. Since for any integer p(O<pSP). ･n n i=1 i=1. ' Z ElK,(H(X,))-EK,(H(Xl))IPSMZ E1K,(H(Xl,))lP =MnflKe(H(T))IPdLI(T). ' sMn(X"-"+.1[ll.-.]{u(1-u)}-p(ii2-6)du. e.

(9) Berry-Esseen Rates 9 SMn(i-a)+ap(i/2-6), so for any integer k()1) and for any set {pi, '･', pk} of positive numbers such that pi, ･'･, pk-i are integers and pk=:P-k-1 Z pd.. oX-1 Z ･･･ nZ nk " EIK,(H(X},))-EK,(HT(Xle,))iPi il=1 ik=13'=1 re $M " {n(1-a)+aPj(1/2-6)} $Mne(1/2-6+e), e' --'1. which implies (4.23).. To show (4.24), put Li(u)=Ko(u)X(u: Ao(`')(i=1, 2) where Ao(i'={u:1? '. n-aS.u<1} and Ae(2'={u:O<uSn-"}. Then, it suMces to prove (4.24) on replacing Ko by Li(i'--1, 2), respectively. We show (4.24) for Li only since the. t. s. restisanalogouslyproved.Let ' ' Ul,=:f{u(or-Xi)-Fd(or)}Lo'(H(or))dFi(or). (ISi, 7' <.. n, i=7･). Ulei==O (IS.i<.,.n). Then, for i4i'{env} and {en･,･} are independent and EUlej=O (ISi, 1･.Sn). We note that for any integr g(IS-gS[P]). nn nqn q. (4.2s) n-aZEIXc,UljlaEMn-aZ Z ZIE{H en,d}1. o'--1 i=1 e'=llflki=1 l=1. $Mte.,E[,l,(flu(yt-X))-H(ori)11Li'(H'(y,))ldH(or,)]] $Mnf",(flu(ovi--x)-H<[ori)IILi'(H(ort))ldH(bli)}dH(x) $MnZ*f･･･f(f#i.4r-1[.fi,{1u(yk.---x)---H(or,.)IIL,,(H(or,.))1]dLl(.)]. '. yklS"'Syka X dH( Nki)'" dH( orka). $ Mn Z *f' "f[ ,.i, { ,ij, H( orkt) ,fi, (1 - H( ork,))}][ ,Hl, IL,'(HT( ov,,))l]. Ykl$"'SYka. XdH(yki)"'dH(blkq) '. SM(q+1)Z*n f "'"' f (1"mug)j".q,(1'--ud)-(3/2-"'dur･･･du, lmn-ptSuiS"'Sva<1. m. ". s;Mnl'manaa(112'6)$Mnq(112--6)+e .. where orko=:"oo, blk,.,==oo and Z* denotes summation over the q! permutations (ki, ''', ka) of (1, ''', q), and Mh is a constant depending only on P.. If O<p<1, then for all any positive integer q such that g+pS-P. n nn. El Z c, U},Ia+P$ME{I Z c, UIAq Z lUl,,IP}.. Hence, by the method of p'$p). i=1. i=1 i=1 the proof of (4.25) we. have that for any P'(O<. n :.s--{; MnB'('!2-6+e), El Z ci U}jlB' i=1. which implies that(4.24) holds for Li(u). Hence, we have (4.24). So, the proof. of Lemma 4.4 is completed..

(10) 10 KYosHiHARA and H. NEGisHi Lemma 4.5. Under the assumptions of Theprem 4.2. n (4.26) EjSn")-ptn(t'--'ZciYi`t'IBS.MnB(i/2-6'e' (l:1,･･･,d-1). i=1 Proof. Let l(ISI$d-1) be fixed. To prove (4.26) it is enough to show that (4.26)for Kt holds with P=2r (r being an arbitrary positive integer) since by (4.4) (d("/dec(")Ki(u)(7'--O, 1, 2) are bounded and if (4.26) holds with P==. 2r then for any P'(O<P'S2r). nn i=1. ElSn(i)plten(i)-- : ciYt(i)13'S{EIS.(`)--'iet.(i)'m- Zl ciYi(i'12'}B'/2'.. i=1. Now, we proceed to prove (4.26) with P=2r. By (2.10) it is enough to. " l. show the following inequalities:. nn (4.27) EIn-'Z ZciZw(t)12'SMn2r(i/2-6+e); i--1i=1. 1. 'i. (4･28) E te.,ctrpi(n6+'1 -H(xl))K",,, 2's.Mn2r(ii2-6+e);. n (4.29) EIZctqi2K"t2,12'SMn2r(ii2-6+e); i=1. (4･30) Elt?.,ci(ne+'1-H(x2))K,i2r$Mn2r(ii2-a+e), (4'31) E .(i+1) te.,ci,.Z,.,.{u(Xi'X))---Fd(Xi)K,'(H(x))] 2'gM.2r(ii2-6+e).. e'$i. Let Bi= UtAJ. Let r=e/4(1+a`"2). Sincei6i/(n+1)-H(X)IS.n-i (i=1, ･･･, i=o n), so by (4.4) and Lemma 3.3 (Case (ll)) L h.s. of (4.28)SMn-2rn2'at(5i2-a)E[ ￡n [rpilx(H(x,): Bt)]2r. i=1. =<Mn-2rn2rat(512-6)n'-r(1-r)(1+at'2)+2r(1-at+2). SMn3rat(1-"a2)-r(1-at)+2rat(112-6)+re14. 5Mn2r(1/2-6+e), and 1. h.s. of (4.30)SMn-2'n2'"i(3/2-6)E[ nZ z(H(xl): Bt)]2r S.Mn-2'n2ra;(312-6)n2r(1-ai'2). i--1. =Mn2rat(lf2-6)+2rat(lda')EMn2r(1/2--6+e).. Similary, t-. L h.s. of (4.29) $Mn2r"t(5/2-"'E[ Z nrpi2z(H(Xle): Be)]2'. i---1. SMn2rat(512-6)n-2r(1-r)(1+at'2)+2r(1-at+2). SMn2rat(1!2-6)+4rat(1-a2)+2rr(1+at+2). SMn2'(1/2-6+e).. By (2.6). nn nn 2r (4.32) 1.h.s.of(4.27)=n-2'Z'''Zci,'"ci,.Z"'￡E["Z"'ikik] il"1 i2r"1 3'1=1 O'2r"1 k"1. nnnn 2r. sMn-2' Z ･･･￡ lZ ･･･ Z E[ " Z(`'i,d,]1･ it=1 i2r=1 2'i"1 e'2r"1 k"1. t.

(11) Berry-Esseen Rates 11 Since EZ(i'ij--k-O and EZ(`'wZ(`'i･j･ =O if 7･ti' or 7' 7S'i, so the expectation in. (4.32) is possibly nonzero only if each factor has both indices repeated in other. factors. Among such cases, consider only those terms corresponding to a given pattern of the possible identities ia==ib, ia=jb, 7'.=7'b for IS-aS-2r, ISbS2r. In general, there are at most 26r such patterns. For such a pattern, let g denote the number of distinct among ii, ''', i2r and p the number of distinct values among 1'i, ''', 7'2r. As a typical sum we consider. nqp (4.33) T({.},{b))=Z Z1Z Z E[" fian {z(e).,,,}btj]I ta. i--'1pti=1e':bwllvd=1 i=1e'--1. p where {bw} are non negative integers such that ai=: Z bij(i----1, ''', g). e' --"1. For brevity, let. l. t Wi,==R{Z(t'.,,,}bi, (1;-Si<=g,1$t;Sp). e'=1. l. ts. Since E[Z(t'ijlXi]=O, so if ui4vj for all i(IS-i<,..q) and J'(IS7'-f{;p) and if. q for some P(1;$P$P)i.,biB=1, e.g. b.B==1 and biB==O(igEa), then. g.. }btdE[Zp.gB1XyB] (4･34) .E[ " Wlep]=E[H*{Z(i'ptivj. i=1. =E[H*{Z(`)pivJ}b`'E[Z(`'papBIXpta]] =O where n* denotes the product of all (i, 7') (IS-i<...q,1$J'<..p) which is different. A is the random vector which consists of all Xpt, (1$i<=q) and from (a, P) and XB X),(ISI'-SI-p) being different from XyB. Similary, since E[Z(`'ijlX)]= O, so if pm 7k. vj for all i(IS-i.S--q) and 7'(1$7'-Sp)and ifp Z b.j =1 for some cr(IS-aSg), then O' --1. (4.35) E["wr,]=O. a. i--･1 Further, since IKt'(H(x))1>O only if H(x)GBi, so for any i and any 7'. f{u(x-X))-L(x)}Kt'(H(x))dFi(x) 5flKi'(H(x))ldFi(x)S-Mn"t(3f2-"'Ez(H(Xt):Bt) and l{u(Xl - X)) - Fd(.ISirG)}Kt'(H"(Xl ))IE Mnat(3/2-"'z(H(X): B,). 9. i. Hence, we have. (4.36) IZ(i'ijlSMna`(3X2-6'{Z(H(Xi):Bt)+Ez(H(Xi):Bt)}. 0n the other hand, by the method of the proof of Lemma 3.1. b. te.,E f{u(x-Xi) -- "Ef(x)}K,'(H(x))dF,(x) ic. $,.l,E(fl{u(x-X3)"'-Fj(x)1Ki'(.H(x))licdF,(x)} $Mnk(3!2-'6'aÈ[{ n: Fd(X)(1-Fd(Xle))}X(H7(Xi): Bt)] j' --1 -<- Mnk(3/2-6)a`ni-a`"2EZ(H(Xl): Bt) and similary. '.

(12) 12 K. YosHiHARA and H. NEGisHr ･ n F..,E[l{U(dXle-Xf)-"'F,(Xle)}Ki'(H(Xii))lk1Xl;] sMnk(312'-6)atni-at"2z(H(Xi): B,).. Hence, we obtain. n (4.37) ]E[Z(t),,klX,] e' --1. sM2kni-ai"2nk(3/2'a)ai{z(H(Xlt): Bi)+EX(H(Xi): Bt)}.. Now, we consider. nnq nnq. Z Z E[fi "VVIip]= IX : E[fi Wi,.-.iE[{Zì',,,.}b`p. 7. vp-1"ivp=1i=1 pp-1=lvp=1i=1. IXpi, "', &a, Xi, "', Xlyp-i]]. 1i･. nq. ==.Z ZI(p)E[Hwr,p-iE[{Z(i'p,y.}b`'IX'p,}]] vp--1=1 i--1. (4.38) +Sil:IhE[fiVVI,p.i{Z"'p,.,}b'p] vp-i=1k=l i=1. '. Znp-1n ZE[fl wr,,-,{Z(t).,,,}b`p] '+ vpn=1 k=1 i=1. '. =Ii+I2+I3(say),. where Z(t) denotes summation over all vk(ISvteS-n) which is different from any member of the set {pm,'", ptg, pi, '", vt-i}.. Since I3 is the same type as the left hand side of (4.38), so we consider Ii. and I2. As in (4.36) we have Ii=E[ fi {wr,p--2Zcp--i)E[{Z(`)#,p."}b`'P-i1X]et,]Zl(p)E[{Z(`).,p.}b`'1Xl,,,]}]. i--1 + Z] E[ "qq {VV},p-2{Z(`)#,.,}b""-iX(.)E[{Z(`).,,.}b`plX,,]}] + ￡. and. k=1 i=1 p-1 q E[ H {"[ua,p-2{Z(`)pipk}b`''-'Z(p)E[{Z(`'uiv,}b`"laXlptt]}]. k=1 i--1. I2 = t [E[ fi (1]Vi,p-2{Z(t)pt,pt,}b`"Z(p-i)E[{Z(t)p,p..}b`,"-'IXI,,,])]. k=1. i=1. qg. + X E[ H ( wr,,-2{Z").,.,･}"`,p-'{Z(i'.,.,}bt･p]. kl=1 i-il. q fiE[ I]Vl;,p-2{Z(l)ptyk'}bt'P-i{Z(l)#tptk}bip]. + Zp-1. ?. 1. kl---1 i=1. Thus, using the above method, repeatedly, we can rewrite T({a},{bi) as a sum of term of the form. nnq q. H = ,il.;, "',Iil.l ,E[ ,"., (,.ll. {Z(t)E[{Z(`'ptpt}b"IXIett]},,gi{ ,Z.-,Z(`'pt,p,}b"')]. where D is a subset of the set {1, 2, "', p} and D- is the complementary set of. D.. Now, using (4.34)--(4.37) and Lemma 3.3 and noting ￡ Sbij==2r, we have. x 'x.. i--1 e'--1. ,s.

(13) Berry-Esseen Rates 13. nn q. H'= Z ･･e Z IE[ H " {M2b"nb'`(3i2-6'at(Z(H(X.,): Bi)+EX(H(Xp,): Bt))} ×. Pl=1 Ag=1 i"ltED. " {Mnbtt'(3i2-6'at(z(H(X].,): Bi)+EX(H(X.,): Bt))b`t'}]1. tleT sM22rn2r(3i2-6)at X ･･･ S E[ fi (z(H(X".,): Be)+EZ(H(Xpt,): Bt)]. pJ=1 pg=1 i=1 -<.M22r+qn2r(3i2-6)a` S ･･･￡ [ rt EZ(H(X.,): Bi)]. Pl"1 #a=1 i'1 sMn2'(3!2'6)ev` fi qn { Z EZ(H(Xpt,): Bi)}. i=1 pi=1. $Mn2r(3!2-6)a`nq(1-'a`"2)SMn2r[(3/2-6)at+(1-a`+2)}.. '. Thus, we have. ･ n-2'T({.},{bl)S.Mn2r(i12-6+e), s. which implies (4.27).. The proof of (4.31) is analogously proved by the above method. Thus, we. havethedesiredconclusions. ' Lemma 4.6. U>zder the assumptions of Theorem 4.2.. n (4,39) EISn(`'-pt.(`'- Z cj Yi(i'IB;S MnB(!l2-6"e).. i=I Proof. As in the proof of Lemma 4.5 it is enough to prove (4,27)-(4.31) when l=a. By (4.4) and the definition of d. supld(i'/du(i'Kd(u)l$Mne (i--O,1,2). uEAd Hence, (4.27) follows from the above method. (See, also Lemma D in Serfling (1980: p.305)). (4.31) follows analogously. Since for any positive integer k. ElrpilteSMn-ic/2 (ISi<...n), and l&/(n+1)'H(Xle)ISn-i (ISi<=n), so (4.28)-(4.30) easily follow.. Thus, we have the lemma. Proof of Theorem 4.2. The proof follows easily from Lemmas 4.4-4.6. Theorem 4.3. Under the conditions ojC Theorem4.2. n (4･40) EISn-ESn-ZciYtlB;SIMnB(i/2-6"e'. ' i=1. Proof. Since. nn. ElSn kmESn- Zl ci Yi1B;-S M[ElSn-ptn' II[] ci YilB+1ESn-ptn1B],. t. i=r i=1 so (4.40) follows from Theorems 4.1 and 4.2. By the proofs of Theorems 4.1-4.3, it is clear that theorems remain true if. l. we replace the uniform boundedness condition of the regression coeMcients {c.i}. o. b. by Condition ([). Hence, we have the following theorem,. Theorem 4.4. Let S(O<6<1/2) and JELa be .fixed. Sumpose Conditions (I) and (ll) hold. Then, for all e(O<e<6) and P(2$P$6/(1-25)) (4.6), (4.17) and (4.40) remain true. Corollary. U>zder the assunzations of Theorem 4.4. (4･41) l{var S.}il2 --{var( ￡ c, Y,)}i!21SMnB(i12-6'e). i=1.

(14) K. YosHIHARA and H. NEGIsHi. 14. Proof. (4.41) follows from Theorem 4.4 since for any random variables U. and. V. 1{var U}i/2--{var V}i/21${var(U-V)}ii2.. g5. ProofofTheorem2.1. We now proceed to prove Theorem 2.1. Since for JEL6(O<6<1/2), we can rewrite.n Z ciYi as a sum of independent random variables i--･1. nM'i.,cif{u(TdX))-F,(T)}J'(H(x)dFi(x). ". i#i. +cd{J(H(Xi))-Etl(H(X)))} (1'--'1,'",n),. which have zero means and finite absolute moments of order 2+26 (cf. the proof. ii'. of (4.25)). Hence, by the Berry-Esseen theorem and Condition (M) we have. nn. suplP( Z ci Yi < T{var( Z ci Yi)}i/2)-￠(T)[ =O(n-"). i= 1･ i=1 As the rest of the proof is easily by Theorem 4.4, so the proof is omitted. (cf. Proof of Theorem B in Serfling (1980, pp.309--310)).. References. [1]. BERGsT6M,H,and PuRi,M.L. (1977). Covergence of remainder terms in linear rank statistics. Ann. Statist. 5 671-680.. [2] [3] [4] [5] [6]. CHERNoFF, H. and SAvAGE, I. R. (1958). Asymptotic normality and eMciency of certain nonparametric test statistics. Ann. Math. Statistics 29 972-994. HAJEK, J. (1968). Asymptotic normality of simple linear rank statistics under alternatives.'Ann. Math. Statist. 39 325-346. HusKovA,M. (1970). Asymptotic distribution of simple linear rank statistics for testing symmetry. Zeit. Wahcscheinlichkeitstheorie view. Gebiete 12 308-322. HusKovA, M. (1977). The rate of convergence of simple linear rank statistics under hypothesis and alternatives. Ann. Statist. 5 658-670, JuREsKovA, J. and PuRi, M.L. (1975), Order of normal approximation for rank statistics distribution. Ann. Probability, 3 526m533.. [7]. MAsoN, D.M. (1981). On the use of a statistic based on sequential ranks to prove limit theorems for simple linear rank statistics, Ann. Statist. 9 424'-436.. [8]. [9] [10]. SEN, P.K. and GHosH, M. (1972). On strone covergence of regression rank statistics. Sankhya A 34 335-348. SERFLiNG, R.J. (1980). Approximation theorem of mathematical statistics. Wiley, New York. YosHiHARA, K. (1978). Limiting behavior of one-sample rank-order statistics for absolutely regular processes. Zeit. Wahrscheinlichkeitstheorie verw. Gebiete 43 101127,. ". 1. l t.

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