Department of Biostatistics University of North Carolina at Chapel Bill. Institute of Statistics- Mimeo Series No

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1 ON SEQUENTIAL NONPARAMETRIC ESTIMATION OF MULTIVARIATE LQCATION by Praab Kumar Se Departmet of Biostatistis Uiversity of North Carolia at Chapel Bill Istitute of Statistis- Mimeo Series No Otober 1983

2 ON SEQUENTIAL NONPAR~mTRIC ESTI~~TION OF MULTIVARIATE LOCATION Praab Kumar Se Departmet of Biostatistis Uiversity of North Carolia Chapel Hill. NC USA.e For the multivariate oe-sample loatio model (relatig to a diagoally symmetri distributio). sequetial oparametri (poit as well as iterval) estimators based o appropriate rak statistis are osidered ad their asymptoti properties studied. I this otext. asymptoti risk-effiiey of the proposed estimators ad asymptoti ormality of the assoiated stoppig times are established. These results rest o momet-overgee of the permutatio dispersio matrix to its populatio outerpart; these are studied as well. A brief review of some other robust proedures is made alog with. INTRODUCTION Let {~i;i>l} be a sequee of idepedet ad idetially distributed (i.i.d.) radom vetors (r.v.) with a otiuous distributio futio (d.f.) defied o E P for some p~l. where eee E P. It is assumed that F(.;~). F(x-e). xee P (1) where the P margial d.f.s of F. deoted by F[l].. F[p]. respetively. are all symmetri about 0; i fat. F is assumed to be diagoally symmetri about the or~gl. Thus. e = (el... e)' is the vetor of loatio parameters. ad we - p are iterested i the estimatio of e. Based o a sample (~l"" '~) of size. let! (T l.. T p ), be a estimator of e. The loss iurred due to estimatig e by! is deoted by L(.;!'~) = p(!'~) +. (>O) (2) where p: E P x E P... [0."') is a suitable metri ad is the ost of samplig. per uit observatio. Some ommoly adapted forms of p(.) are the followig: p(~.~) max{laj-bjl: l~j~jl} (~~X-orm) p(~.~) -1 P I I (MAD-orm) p Lj=l aj-b j p(~.~) (!-~) 'g(~-~) (QUAD-orm) (3)

3 where 9 is some give positive semi-defiite (p.s.d.) matrix. Correspodig to (2), the risk i estimatig ~ by! is give by AF(,;~) '" EL(,;!'~) '" EP(!'~) + (4) Note that the risk i (4) depeds o F through the distributio of! ad the metri p(.). If T is traslatio-ivariat (as will be the ase i our subsequet aalysis), the 0(F) '" EP(!'~) ad AF(,) '" AF(,;~) do ot deped o e. Suppose ow that there exists a positive iteger O' suh that 0(F) exists for every ~O ad 0(F)~O as ~. For every C>O, we may defie the (5) The, T 0 is the miimum risk estimator (MRE) of ~. Note that 0(F) geerally - depeds o F (through some other futioals of F), so that whe F is ot ompletely speified (as is the ase i pratie), o sigle O may lead to the MRE simultaeously for all F F, a lass of d.f.'s. However, suitable sequetial proedures may be iorporated to ahieve this goal i a asymptoti setup where is made to overge to O. Towards this, we assume that for some q(>o), as ireases, qo (F)... o(f): O<o(F)<oo (6) ad, further, there exists a sequee {d } of osistet estimators of o(f). [For the first two ases i (3), q is typially equal to ~, while for the last oe, it is equal to 1.] Note that for C~O, (4) may be rewritte as so that asymptotially, as C~O, we have o (8) (7) e, (9) Motivated by (8) ad the stohasti overgee of {d } (to o(f)), for every (>O), we may defie a stoppig variable N '" if{~o: l+q~-lq(d+-h)} (0) where h(>o) is a suitable ostat, to be hose later o. Based o the stoppig rule N ' TN of e, defied for every - We say that we osider the (sequetial) poit estimator >O, Ap(C) '" EP(!N'~) + EN ' is a asymptotially ~IRE of e, ad we deote the orrespodig risk by if >O (11) (12)

4 I the literature, this is referred to as the (first order) asymptoti riskeffiiey (A.R.E.) property. followig results: as ~O, where Besides this A.R.E., we shall also study the Nio... 1, i the first mea, (13) o ~ ( ) (TN -e) - N(O,r) (14) - - r - = lim ->-oo E{(T - -e)(t _ -- _el'} (IS) [! is also the dispersio matrix of the asymptoti multi-ormal d.f. of ~(T -8).] - _ - Uder some additioal regularity oditios, we will also have the asymptoti ormality of the stoppig time i.e., o -~ 0 2 ( ) (N- ) - N(O,y ), as do (16) where y(o<y<oo) is a suitable ostat (depedig o q ad F). The seod problem is to provide a ofidee set size, suh that I (E P ), based o a sample of p(e I Ie) > l-a(o<a<l) - _ - (17) maximum diameter of I ~ 2d(d>0) (18) where a ad d are preassiged. For this problem too, fixed sample size proedures may ot work out (for all F F), ad hee, oe may take reourse a sequetial sheme [viz., Chapter 10 of Se (1981)], where (17)-(18) hold i a asymptoti setup: d~o. Asymptoti properties of the stoppig times are studied here. I this paper, we speifially osider the ase of R-estimators of loatio; some geeral remarks pertaiig to some other robust (viz., ~I- loatio are made i the oludig setio. Asymptoti properties of these multivariate R-estimators are studied i Setio 2. ad L-) estimators of Setio 3 otais some asymptoti results o the permutatio dispersio matrix whih are eeded i Setio 4 i the derivatio of the asymptoti results i (12, (13), (14) ad (16). Setio 5 deals with the ofidee set problem. multivariate ase (i.e., p~2). detail i Chapter 10 of Se (1981) ad Jure~kova Throughout the paper, we osider the geuie The uivariate ase has already bee studied i ad Se (1982), amog other plaes. The last setio deals with some other robust estimators of e (e.g., M- ad L estimators), ad oly the geeral setup is disussed. MULTIVARIATE R-ESTI~~TORS: ASYMPTOTIC PROPERTIES For eah j (= 1,...,p), let epj = {epj (u),o<u<l} be a o-dereasig, o-ostat, skew-symmetri ad square-itegrable sore-futio, ad let ep. = {ep.(u) = ep~((i+u)/2),0<u<i}, l_<j< J J J -..J (19)

5 For a sample ~i = (X il,... '\p)l, l~i~, of size, we the itrodue a set of ~ by lettig a. (i) = E. (U.) or. (i/(+l)), l~i~; j = 1,...,p (20) J J l J where Ul<",<U are the ordered r.v.'s of a sample of size from the uiform (0,1) d.f. Further, for every real b, let R~ij(b) be the rak of IXij-bl amog IXlj-bl...,IXj-bl. for i= l,..., ad j = l,...,p. Let ~(~) = (S l(b ),...,S (b ))1 be defied by l p p Sj (b j ) = Ei=lsg(Xij-b.)aj (R~ij (b j )), 1.9.3' Note that for eah j (= 1,...,p), Sj(b) is \ i b, ad Sj(6 j ) has a distributio symmetri about O. Let the a j = ~(sup{b: (22) S.(b»O} + if{b: J S. (b)<o}) J for 1,2,...,p, ad let e = - ( 1'"'' p ), (23) The ~ is a traslatio-ivariat, media-ubiased, robust ad osistet esti- - mator of 6 [viz., Chapter 6 of Puri ad Se (1971)]. Let B = Diag(B,...,B l p ) be defied by lettig Further, let v = be defied by lettig for j,j' = l,,p, where F[jjl] is the bivariate d.f. of for j 'I j I 1,,po Note that the v.. do ot deped o JJ are depedet o F through the joit d.f. F[jjl]' j 'I j'. E= ((Y jj,)) = B-lVB- l = ((Vjjl/BjBjl)) The, uder quite geeral oditios [f. Puri ad Se (1971, Ch. --, (21) (24) (25) (Xij-6j,Xijl-6jl), F, but V j j I' j 'I j', Fially, let (26) 6)], we have for Note that the asymptoti ormality i (27) does ot eessarily esure (15). however, for some a>o (ot eessarily >1), Ellx.lla<oo, the, it follows from - -~ Theorem 2.1 of Se (1980b) that there exists a positive iteger (depedig o 2 a a), suh that EII ~ -611 < 00, 'If >. - a For our purpose, we assume that for every j (= 1,...,p), for some 0 (O~O<\) ad K (O<K<oo), I (d r /du r ) j(u) I -o-r ~ K(l-u),O<u<l; r = 0,1,2, ad F[j] has a absolutely otiuous desity futio f[j] (with a first derivative- fij] bouded almost everywhere), suh that (27) If, (28)

6 for 1... p. We write d = (4+2T)-1. T>O. The. by a diret multivariate extesio of Theorem 2.2 of Se (1980b), we may olude that for every (29) k < 2(1+T), lime{k/21is-ellk}<... Note that throughout the paper stads for the maxorm. The (15) holds uder (28)-(29). Atually. the followig represetatio [a oordiatewise extesio of (2.49) of Se (1980b)] is the key to the above results: (~ -e) B-ls (e) + ~ (30) _ - where. uder (28)-(29). as ~. -~I I~ II + 0 a.s. as well as i the kth mea. _ for every k < 2(1+T). Further. {S _ (8).>1} _ - is a martigale sequee [see Se ad Ghosh (1971)]. so that {lis (8)-5 (8) II: m>} is a sub-martigale. Hee -m - -usig the Kolmogorov maximal-iequality alog with the fat that for every m~. E(~m(~)-~(~))(~m(~)-~(~))'= (m-)~' where ~~ as ~. we obtai by some stadard steps that for every >0 ad >O. there exist a * ad a *(>0). suh that for every ~*. (31) (32) Cosequetly. by usig (30). (31) ad (32). we obtai that p{ I maxi <~* ~li~ -~ II > 1 < m: m- ~ _m - (33) for every ~**. r.v. 'so suh that Clearly. if {N} be ay sequee of positive iteger valued -IN +1. i probability. as ~. the by (27) ad (33), as (34) Defiig B as i (24). we let w = sup{-~ii~(~)-~(~)+~~ii: 11 II<-~lOg}. (35) The. as a diret multivariate geeralizatio of (2.37) of Se (1980b). we obtai that for 0 ~ (4+2T) -1. T>O. there exist two positive ostats l. 2 ad a iteger ~. suh that - p{w> 1(10g)I~=~}~2-l-T, ~~~ (36) This last result yields suitable estimates of B alog with their rates of overgee. Note that uder 8. = 0, 5.(0) has a ompletely speified distribu- ) ) tio, symmetri about O. ad hee. for every a(o<a<l) ad, there exists a C(j), suh that p{i.s~ (O}1 > (j)le.=o} > a > p{ls.(0)1 > (j)}. for j =,a. -,a) - ),a '1,...,p, where -~C() + T /2V'" 1<j 3'. T /2 beig the upper 50a% poit of the.a a )) - A') a (") A(") stadard ormal distributio. Let the 8E) = sup{b: s. (b»c ) }. 8 u ). ("), ).a. if{b: S. (b)<-c )} ad Jet ).a

7 B. = 2C(j) /((S(j)-S(j))} j = l... p (37) J.a U. L. The. ~ = Diag(Bl..Bp) is a traslatio-ivariat. robust ad osistet estimator of B. By (36). (37) ad some stadard maipulatios. we obtai that for - -1 every >0 ad 6 ~ (4+2t) t>o. there exist a ostat C (O<<oo) ad a iteger O' suh that for every ~O' PC! IB -BII >d < - l - t - _ - Some other properties of B are studied i the ext setio. _ RANK BASED ESTIMATORS OF r: ASYMPTOTIC PROPERTIES I (20), we replae the ~j by ~j ad deote the resultig sores by.a;j(i), l<i<. l<j<. Let the R.. be the rak of X.. amog X lj.... 'X J " for i = J 1J 1) l... ad j = 1... p. Defie ~ = ((v jj,)) by lettig The _ V is a traslatio-ivariat, robust ad osistet estimator of v, deby (37), we fied by (25) [See Puri ad Se (1971, Ch. 5)]. Defiig the B. J let the (38) E = ~~l~b~l ((Y jj,)) = ((Vjj,/BjBj')) (40) To study the asymptoti properties of r. osider first a asymptoti represe- - tatio for Y. By (28)-(29) ad Theorem of Se (1981), we obtai that for 6 < (4+2t)-1. t>o. where the W. are i.i.d. radom matries with EW V k [0.2+t), ad further II ~* II = O( -~-) _ a.s., as Atually, by the same proof it follows that for every l ad a iteger O' suh that for every ~O' = V ad Ell W. Ilk < 00, _1 ~, for some >O. (39) (41) >0, there exist a fiite PC! I~*II >d > - l - t /2 _ - 1 Also, for the i.i.d. radom matries {Wi}' by the Markov iequality, -1 II} -k II -1 Ilk -1-T/2 P{II I:i=l~i-~ > ~ E I:i=l~i-~ ~ 2 ( ) ; 2 ( )<00 (42). (43) As suh, from (38), (40), (41). (42) ad (43), we obtai that for adequately large, for every >0, p{llr -rll >d - _ C <00 3 Let us ow write _ U =. ~(V_ -v) _ ad Y _ = ~(B-l_B-l). The, by (41) ad the lassial etral limit theorem (o the W.), _1 ~ - N(~'~l) as ~ '(44) (45)

8 where ~l is the disper~io matrix of the ~i' The asymptoti ormality results o y demads some extra regularity oditios. We assume that i (28) I(d /du )!jl.(u)1 <K(l-u).O<u<l (where K<oo) ad i (29). we assume that for J - every >O. there exists a e>o. suh that 00 (46) for every j = 1... p. The. we may use the seod order liearity results of v -1 Huskova (1982). as adapted to the oe-sample ase, ad o1ude that for B a ~ represetatio similar to that i (41) holds. where the remaider term is o( ) a.s., as ~. This represetatio esures both the asymptoti multiormality of \B-l_B- l ) ad the "uiform otiuity i probability" of the \B-l_B- l ). ~ - ~ - Hee. uder these additioal regularity oditios. ~(~~l_~-l) - N(~'~2) (47) for some p s d Writig B- 1 = B ~Y ad V = \I + -~u. we obtai o. " _2' - _ usig (40). (45) ad (47) that uder the regularity oditios metioed above ~(r -r) = 2Y \lb- 1 + B-lU B (-~) - N(O.*) ; * p.s.d. (48) _ -- _p - Also, the asymptoti "uiform otiuity i probability" result: p{ max ~II r r II > } < m: Im-l ~e' _m-- e. follows from the above results. so that (48) exteds of iteger valued radom variables. where -IN +1. as well to a sequee i probability, as {N} ASYMPTOTIC PROPERTIES OF N AND an _ The asymptoti ormality of the R-estimator has already bee osidered i (27). Also. after (29). the momet-overgee result i (IS) has bee studied. ~. (49) I the sequetial ase, the study of (11), (12). (13), (14) ad (16) depeds o the form of the metri p(.) i (2); we assume that the followig holds: (a) For 0 = Ep(~.6). (6) holds with o(f) depedig o F oly through - - r (whih is a futioal of F). i.e., o(f) = o([), ad further, o([) satisfies the (Lipshitz-type) oditio that 10 ([1)-0([2) I ~ Koll[1-r211. V r l E2 (SO) where K is idepedet of the r. ad is the max-orm. 0 -J (b) With q defied as i (6), there exists a fiite. positive that wheever the expetatios exist. Kl suh E{P(~.~)}r ~ KlEII~-~112rq. (r,::l). (51).Note that (50)-(51) hold for eah of the three Metris i (3). Also. ote that i (9). we take d = o(r). The Lipshitz-type oditio (SO) alog with (44) esure. - that for adequately large, for ever~ e>o.

9 (52) Now, by virtue of (44) ad (52), we may virtually repeat the proof of the first part of Theorem 3.1 of Se (1980b) ad olude that (13) holds. Further, (13) ad (34) esure (14). To establish the A.R.E. property i (12), we require to establish some uiform itegrability properties, ad, for these, we eed to put some restrai o h i (10) ad 6 i (28). We let 6 < (4+2T)-1, T > 1 + 2h, h>o (53) so that 6<1/6. Note that by (13), ad (11), it suffies to show that lim '''OEN / o = 1. Hee, by virtue of (4) CT (54) or equivaletly, (55) By virtue of Theorem 2.2 of Se (l980b), wheever, for some a>o, EI'I~lla<oo, uder (28), (29) ad (53), Therefore, by (51) ad (56), lim E{(~ll~-~II)k} < 00, 'if k < 2(1+T) (56) lim {kqe(p(s,e))k} < 00, 'if k < l+t (57) - - With this, we may agai follow the lie of attak of the proof of Theorem 3.2 of Se (1980b), employig the Holder iequality for the two tails ad the uiform itegrability of 115 (e)ll k (k<2 +T)) for the etral part. For iteded brevity, - _ the details are therefore omitted. Fially, to prove where *... oo as (16), we ote that by (10), N ~ ~ = [(-lq)l/(l+q+h)], with probability C+O. Also, by (10), wheever, N>~, - l q6(in ) ~ N~+q, (N-l)q+l < - l q(6(fn -1) + (N-l)-h) where by (8) ad (59), takig (o) q+ 1 = [-lq6 (F)] + 1 ((NC-l)/~)q+l-l Note that for h>~,.by (8) ad (13), as (N/~)q+l_l ~ {6(fN ) - o (E)}fo (I) < {o(in -1) - o(r}}/o(r} + C-lq(N _l)-h(~)-q-l -lq(n _l)-h(o)-q-l (58) (59) (60) (61) (62) so that from (B), (60), (61) ad (62), as do,

10 (o)~{(n Io)q+l_l} - ( o )\6(r _ ) N -15(~))/15(~) pO) Further. by virtue of (49) ad (SO). uder (13), as +O. (63) (~)~{15(EN )-15(E)} - (~)~{6(~o)-15(E)} (64) Hee. wheever h>~ [i (10)] ad ~ (15(r " )-15(r)) is asymptotially ormal. by ~ 1 -- (63) ad (64), (0) {(N/~) q+ -I} is also asymptotially ormal. Fially. by the Slutsky theore~. (g)~{(n/g)q+l-l} asymptotially ormal implies that (o)~{n I -I} is also asymptotially ormal (with a differet asymptoti variae), ad this proves (16). Note that for (64), oe eeds the extra regularity oditios due to Hu~kova (1982); for other results these are ot eeded. I ay ase. her regularity oditios are ot very restritive. ad hold for the ommo types of sore futios; we shall ommet more o these i the last setio. SEQUENTIAL CONFIDENCE SETS Note that defiig v as i (25) ad ~(~) as i earlier. uder ~, while -l(~(~)),~-l(~(~)) - ~ (65) 2 Hee. if x_ stads for the upper 100a% poit of the hi-square distributio "'p.a with p degrees of freedom. we have from (65) ad (66). whe is large. Followig the steps after (36). we defie the (66) (67) for 8*(j) = sup{b: S (b» ~~ X- } L. j jj"l'.a 8~:~) = if{b: Sj(b) <--\;j~.a} 1... p. For every d (>0). osider the a stoppig time N d * = if{~o: m~x (8 *(j) -8*(j)) < 2d} 1931 U, L, - ad the proposed {sequetial) ofidee set is (68) (69) (70) (71) I f we de fie, for every d >0, * d (72) the our basi oer is to verify (17) for I Nd i (71) (whe d+o). ad to show- that. Nd:/d:" 1. i probabiiity. as d+o. These results follow diretly by usig (35) - (36) [where we let b \8 L l< -8) "ad ~(e*u -8)], alog the same _,,

11 I ie as i the uivariate ase [See Se ad Ghosh (J971)], ad hee, the details are omitted. We may also ote that supf~ll'(8 -e)l(l'l)-~: l#o} supf~lllr~r-~(e -e)1 _ (lll)-~: l#o} (73) where hi stads for the largest harateristi root. Now, by (30)-(31), ee -e)'r- l (8 -e) = -l(s (e))'v-l(s (e)) + 0(1) a.s., as ~. Hee, by (65) ad (73), for large Pf~~ (~I~)-~ll' (~-~) I ~-\h;(~)~,al~} ::: 1- a Motivated by (74) ad (44), we may defie a stoppig time N~ (for every d>o), by lettig NO = iff> : d - 0 (74) (75) I this ase, if we defie, for every d>o, O = mi{>: > h (l d- 2 } d p,a (76) the by (44), o 0 Nd/ d + 1, a.s., as d+o (77) while (17) follows from (74), (76) ad (33). The same method of proof of the asymptoti ormality of the stoppig time as i (58)-(64) works out here; we eed (48)-(49) i this otext, ad i tur, these require the more striget regularity oditios due to Hu~kova (1982). SO~ffi GENERAL RE~~RKS I this paper, oly the ase of sequetial R-estimators has bee treated i detail. I the uivariate ase, Jure~kova ad Se (1982) have osidered sequetial M- ad L- estimators. Multivariate geeralizatios of these estimators follow o parallel lies. The oordiatewise uiform itegrability of these estimators, established i Jure~kova ad Se (1982) goes to the vetor ase as well. For the ~l-estimators, based o the vetor ~ of sore futios, the sample (aliged) mea (sore-) produt matrix a be used to estimate the true mea produt matrix, ad uder the assumptio that EII~llr<"", for some r>4, results parallel to (41)-(42) hold for this ovariae matrix too. The estimator of the diagoal matrix with the elemets J~j(x)dF[j](x), l~~ is the (diagoal) matr~x with the margial estimators, osidered i Jure~kova ad Se (1982), ad hee, the overgee properties [parallel to (38) ad (47)] hold uder the same regularity oditios. I view of the fat that the sore futio ~ are take to be essetially bouded, the extra

12 regularity oditios of Hu~kova (1982) are ot eeded here. Similarly, for the L-estimators, the (uivariate) deompositio of Gardier ad Se (1979) goes through i the multivariate ase, ad this yields the asymptoti ormality as well as the rates of overgee of the estimated ovariae matries of suh L estimators. I this otext, the jakkife proedure for obtaiig this estimated ovariae matrix also work out well; we may refer to Se (1982) where the lose oetio of these estimators are disussed i detail. We have osidered oly the first order A.R.E. property. ~fuh more elaborate aalysis will be eeded to pursue higher order A.R.E. results. These are posed as ope problems. We olude this setio with a loser examiatio of the Hu~kovaoditio (46). [Note that will follow so that hex) It suffies to osider the right had tail G(a,d) =j {l-f(x)}-l+df(x-d), a G(a,-d) < G(a,O) = -l{l_f(a)} <00, by symmetry.] We write H(oo) = 00. The, we have 1 - F(x) = exp{-h (x)} is the failure rate (FR) d>o, >O, a>o '" >O, d>o, ad hex) = (d/dx)h(x) ad H(x) is ireasig i d (>0), suh that lim sup{h(x+d)/h(x)} < '(l-)-l, V O<d<d ~ - e Towards this, we ote that if (78) while the lower tail G(z,d) = f exp{(l-)h(x+d)}d(l- exp{-h(x)}) a-d Hee, to verify (46), it suffies to show that for every >O, there exists a x with (79) (80) (81) lim suph(x)/h(x) < < 00 (82) x-- - the for all x (suffiietly large), H(x+d)/H(x) ~ exp(d), for every d>o. Cosequetly, hoosig d (>0) suh that exp(-d ) > 1 -, (8l) follows from (82). Hee, we may verify (82) as well. Note that H(x)+» as x+», so that for the etire lass of d.f. 's with bouded or o-ireasig FR, (82) holds trivially (with = 0). For distributios with ireasig failure rates (I FR), ote that to verify (82), it suffies to show (by itegratio) that lim sup{x-llog(_log{l_f(x)})} < < 00 (83) JC"'OO - ad this geeral oditio holds for almost all IFR d.f.'s (iludig some of the extreme value d.f.'s). Thus, (46) [via (82)-(83)] may ot be regarded at all very restritive. Fially, ote that the oditio that I(d 2 /du 2 ).(u)1 < K(1_u)-2, J - V O<u~l, holds for all the ommoly adapted sore futios (iludig partiularly the ormal sores ad the Wiloxo sores).

13 REFERENCES [1] Gardier, J.C. ad Se, P.K., Asymptoti ormality of a variae estimator of a liear ombiatio of a futio of order statistis, Zeit. Wahrsh. Geb. 50 (1979), Verw. [2] liu~kova, M., O bouded legth sequetial ofidee iterval for parameter i regressio model based o raks, ColI. Math. So. Jaos Bolyai, 32: Noparamet. Statist. If. (1982), [3] liu~kova, M. ad Jure~kova, J., Seod order asymptoti relatios of ~I-estimators ad R-estimators i two-sample loatio model, Jour. Statist. Pla. Ifer. 5 (1981), [4] Jure~kova, M. ad Se, P.K., M-estimators ad L-estimators of loatio: Uiform itegrability ad asymptotially risk-effiiet sequetial versios, Seque. Aal. 1 (1982) [5] Puri, M.L. ad Se, P.K., Noparametri Methods i Multivariate Aalysis (Joh Wiley, New York, 1971). [6] Se, P.K., A ivariae priiple for liear ombiatios of order statistis, Zeit. Wahrsh. Verw. Geb. 42 (1978), [7] Se, P.K., O almost sure liearity theorems for siged rak order statistis, A. Statist. 8 (1980), [8] Se, P.K., O oparametri sequetial poit estimatio of loatio based o geeral rak order statistis, Sakhya, Ser. A 42 (1980), [9] Se, P.K., Sequetial Noparametris: Ivariae Priiples ad Statistial Iferee (Joh Wiley, New York, 1981). [10] Se, P.K., Jakkife L-estimators: Affie struture ad asymptotis, Istitute of Statistis, Uiversity of North Carolia, Mimeo Report No [11] Se, P.K. ad Ghosh, M., O bouded legth sequetial ofidee itervals based o oe-sample rak order statistis. A. Math. Statist. 42 (1971), r

ON NONPARAMETRIC SEQUENTIAL POINT ESTIMATION OF LOCATION BASED ON GENERAL RANK ORDER STATISTICS*

.t~. ON NONPARAMETRIC SEQUENTIAL POINT ESTIMATION OF LOCATION BASED ON GENERAL RANK ORDER STATISTICS* by Praab Kumar Se Departmet of Biostatistis Uiversity of North Carolia at Chapel Hill Istitute of Statistis