Sequential procedures based on M-estimators with discontinuous score functions

253

Recdvad 14 Aptil1980; WW

manuscript received 26 February 1981

RecommeMll;cdby T.L. Lai bwiedLwidth sequential wmFi nce intervals and sequential tests for regres,sion ctu bsaed on lWstimatot8 are to the case where the score-functions generatins timetors have jumpdisconti the context of the asymptotic normality of the stopping variable, for the confidtact interval problem, it is observed’ that the jump-discontinuititi induce a slcmm rateof convergtnce, The proofs of the main theorems rest on the tweak converg~tiy of somt rtlated prosrrsse~and this is also studied.

AWwct:

AMS SwbjiectClds@Wion: 62LlU,62Ll2,6oF17. Key woe; Bounded-width Confidence Intervals; Jump-discontinuities; M-estimator; Regression (Location) Parameter; Score-functions; Stopping Variables; Sequential Test; Weak Convergence.

1. introduction

Let {XII irl} be a sequence of independent random variables (rx) with co(ntimousdistributionfunctions(d.f.) {& ia 1) where

the ~1 8.re kmwm (mgm~ion) constmts (not all equal to O), A is an unknow;i pwume!w and the ~~~~~w~ d,f, F betan of d* f*‘seIf aa1the ci are for the cu’not all while, to the classical location intend to construct (i) a so called ~e~~~~~o~ma& eprobubiiity 1 --a and width d2d md having the cov Ho : A ==A0(specified) vs. + This work was done +* Work of this author f S--

: A = AI (specified)> do,

this ~ut~~~rwas visiitin?file thi

0 .2‘i

arolina, Chapel I-Ml. and ~~~~~d~ns~~tut~,

having type I and type !I errors bounded by q and CY~, respectively, where d( > O), al (0 c aI c 1) md ~(0 e a2 < 1) are ail preassigned numbers, For unspecified F, for either of these prob%ems,no j&echample size procedure exists, but under an asymptotic(as d 10 or as Ai-- 40 4 0) set-up, a possible solution is to draw observations ser;tuentiallywith respect to some appropriate’ For the confidence interval problem, such procedures based on the imst squares and rank order estimators are due to Chow and Robbins (1%5), Qles Geertsema (196$),Sen and Ghosh (1971)and Ghosh and Sen (1972),am For the sequential testing problem, rank based pkrocedures are due to Sen and Ghosh (1974) and Ghosh and Sen (1976, 1977), among others. For the location model, a sequential confidence interval based on M-(estiiatstmsis due to Carroll (1977), while JureCkova and Sen (1981) have studied the general regression model, both the confidence interval and the testing problem, under less restrictive regularity conditions. In both these papers, the xotv=function (v/) [gemWing the Mestimators] is assumed to be absolutely continuous and some extra regularity conditions are needed for threuq~mptoticnormalityof the stopphg numbest. M-estimators corresponding to score-functions having jump-dkontinuities constitute an important class of estimators (the sample median belongsto the same). The object of the current investigation is to study the asymptotic theory of sequential confidence intervals and tests for LI based on M-estimators when the generating score-functions have jump-discontinuities. For the simple location model, JureCkovB(1980) has observed some qualitative difference in the Rateof convergence of P&estimators corresponding to absolutely continuous (y and p having jump-discontinuities. As will be seen later rtsn,the same feature is generally true for the regression model and analogous qualitative difference appears in the rates of convergence of the stopping times. Along with the preliminary notions, the sequential confidence interval procedure is presented in section 2. Sectio.st3 deals with some i~~ariunceprinciplesrelating to M-estimators corresponding to v having jr!mp-discontinuities, and, in section 4, * theseare then incorporated in the proofs of the main theorems of section 2, The iast section is devoted to the study of the asymptotic theory of sequentirrl tests for d based on M-estimators, where also, the results of section 3 are utilized in the derivation of the main results. As regards the asymptotic consistency of the proposed procedures and the nsymptotic normality the allied stoppi are concerned, aanlost sure representation of M-esti ors (in terms of s) is not really needed. The invariance principles studied in thingpaper serve se well and rest on comparatively less restrictive r ularity conditions. These results may have interest of their own. Section 3 is therefore of b&c interest.

y continuous on any bounded interval in E, it possesses first tives ((v\‘)and wj2jPrespectively) almost everywhere (a.e.), ~2 is a %epfunction, defined as follows. For some positive i teger p, assume there exist open-intervals Ej == j = 0, . . . ,p with flQ:= -00 earl <“xu~
p uj,

f ),

where the /$ are real numbers (not all equal). Conventionally, we let &(Uj)=+@j-*

forj=

+pj)*

1, ...,Pt

W)

Also, we assume that both ~1, ~2 (and hence, w) lare nondecreasing and skewsymmetric, so that Wj(-X)= -\$(x),

(2.9)

VX~E, j= 1,2.

The d.f. F in (1.1) is assumed to be symmetric, so l.hat F(x)+F(-x)-l,

VxeE.

(2.61)

Note that by (2.5) and (2.6), i w/j(X)W(X)=O, j= I,& so

(2.7)

that by (2:;) and (23, EdS,,(A)=O,

Vnr 1,

(2.8)

where Ed stands for the expectation when A holds. Further, for every n( r l), S,(C)is \

(2.9)

in t&3.

oreover, we assume that F possesses an absolutely continuous probability density function (p.d.f.) f having a finite Fisher information (2.10) Iso, jve assume that

0i

Ylr”

‘I(%) exists

for r = 1,

(2.11)

Qo

-00

(2.12)

nd

Note that (2. IO) ingruresthat 1I$ (A-)1dx< 00, ent condition that for each jf = 1, ..,&, in s bound& and

,/ Cf-0,

whereCf;= i c;.

(2.16)

Remark. If ry 0) then the assumptions (2.10-(2J3) are not needed; in this case (2.14) and the boundedness off’ in some neiihborhoods of the ~puffke. Moreover, if vI is con,stant outside a bounded interval Ia, b]CE (most of the functions suggested for M-estimators are of tlhis type), then, for (2.11)~(2.13), it

suffices to assume that v\~) is bounded inside (a, b). Regarding (2.8) and (2.9), &, (in (2.1)) may bc formally written as d, = (d ,if+ d ,‘*)/2, where

d n*=sup(t:S&)>Q}

and 6,+‘~-inf4t:s,(t)
(2.17)

Moreover, we mte tha! by (US), (2.15) and (2 hi), w(C,l’S,(A))-,~..~~(O,a~)

as n-,ao,

U; given by (2.15) depends on the unknown

(2.18; distri

j”zutisnfunction 17. Wa shall

estimate ai by the sequence s:= J aid&, 01 with

rtzl,

(2.19)

H P,,(x) = n - ’ 0

tl(X-Xi u(t)

= 41

mdflCi)*

.AE

therwise. Finwlly,

Pi=

1,29...,

1 -a/2,

O<#< 3,*Nstic

n 1~~correspondin (2.24) and d>O. It follows from (2.2 totic properties of Nd and o W%?&%/Y~9Y” Yllf

Y21

(>03*

(2.35)

then (2.26) may a&o bt!replaced by: as d i 0, nd vv&+1 inprcJb&lbility,

(2.29)

N’otethat for the lot tion and the equispaced regression models, (2.28) holds for (t) = t and t3, respectively.To prove the asymptotic distribution of the stopping assumption on the sequence {c~},“,I : variable Nd, we n lim u,, = il (e[1,oo)) and It=-+0

f.im m&xcf/C’~* = 0, w=+omLskrn

(2.30)

where

(2.31)

258

Theorem 2.2. Uflder (2.30), (2.32) and the assumptkmsof Theorem 2. I, (2.33) holds

for ally

(E E,

where 0 = t alIz a0/y and yz2is defined by (2.14).

Corollary 2.2. u, in additionto the hypothestbof Theotem 2.2,

(c;;‘c,- I)(w-‘m-1)-g holdsfor some fi, 0
(2.34)

&sn*~ j (mh)-

11
* mty

(qJ

:

Km,,, E;I= 0,

The proofs o!f the theorems and corollaries are presented in section 4. It may be noted that (2.34) holds for the simple location and the equispaeed regression models with #= 1 and 3, respectively; (2.30) also holds in these cases. The convergence properties in (2.26)-(2.29) hold without change even if v/2 However, Theorem 2.2 does not extend to the case v/2-0 (i.e. ~2 111 v/r): while in the 1) has a nondegenerate asymptotic distribution as d 10 (see case v= lyl nfin@?JQJure&ovB and Sen (1981)), in the current case, where ~2 nr(NJnd - 1) asymptotically normal as dl0. Hence, jump-discontinuities in the score function lead to slower rates of convergence of the stopping numbers.

3. Some invariance pthciples relating to M-esthatots The results of this section will provide the main tool for the proofs of Theorems 2.: and 2.2 (as well as the main results in section 5). Let {c&;rO, rtr0) and (d&&O, nz0) (with c*=dd=O, VnrO) be two triangular arrays of real numbers and let {AZ,,} be any sequence of positive inte such that k, -+= as n--w and that (3.1) and

for every hv= 42 and s = 0, 1,X. oreover, denote (3.3)

2

nk,

l

Let X,, X2, . ,. be i,i.d. random variables distributed according to d.E. ,F and let F, 1~ larity conditions of section 2. Consider the process

K'={(s,t):Osssl, ItI sk+)

(=.[QJ]x(-k*,k*))

(33

is a compact set in

and ~22is defined in 2.14); suppose that ~22>O (i.e., ~#O). Then Wn(s,t) belcxngs to D[K+] for every n. The following theorem states the weak convergence of W,l to some two-parameter Gaussian process and it also extends Theorem 2. It of JureCkovB and Sen (1980) to the case where the score functions have jump=*discontinuities4 Theaem 3.1.

K *} be the process defined ill (3.6)-(3.9). Then, urtder (3. l), (3.2) and for v and F satisfying (2.2)-(2.6) and (2.18) through (2.15), Wn converges weakly to the Gaudan process { U’(s,t) : (s, J) E K *} such that W(qt)=O, V(s,t)eK+and Let

( W,,(s, t) : is, t) E

EWa,t)~

Wisl,t’)==(sAs’)*g(t,r’),

(3.10)

V(s,t),(W)dC*,

where if t* t’>O, otherwise.

Pm& Write Q,>?k(0 = Q!# it) tc (X9), i== 1,2. Denote

-

Q!??it) whereQ!:j (0

W!?cs( 0 = {[Qltf!,&) where

Then

-

(3.11)

isgeneratedby

Vi

according 63.12)

asymptotic~e~a~~~rof I@“’ is s~4elydet~r~~~ by ~~~ g~~er~t~ by the discontinuous component v~, Hence, to prcwe the th~~em* it sul’fice~to show ~~~ (a) the finite dimensional distributions of Wfj converge to those of W and (b) ( Wfj> is tight. It fcAlowsfrom the definition of ~2 that

Thus

the

where I(A) stands for the in~cator function of the set A, Nate that

(3.19) and

wherleg(tl t’) is defined by (3.11); thus the li~~~tin covar~anc~of ~~~~coincide with those of W, Moreover, any linear combination of

(3.1’7)as a sum of iad distributions of

~~~~~~~rn.

“,2)then follows fram the classical c

261

e measure of A Analogous in ’ = { (~1, tz),

is a finite positive constant, y hlslds for increments in (82, ta)) , tl C=tp~ ts The tightness of { IV’,“‘} then and Wichura (1971), Theorem 3.1 is proved. l

f

independent increment sins but not in t, and hence, Wichura (1971)do not apply here directly.

cone,y = ytI+ ~21risgiven By (2.11) and (2.14), i,e., Y= 1 W”‘(x)dF(x)+ i (@j-Bj-l)f(aj)* -cS j=l Then, under the assumptiorwl of Theorem 3.1, the process ( W,*(s,t) cw@wrges weakly to the proms W = { W(s,t) : (s, t) E K *}.

(3.25) : (s, t) E

K *)

Proof. (3.24) follows from (3.1).(3.3) and from (3.5). Ccrrduy 3.2. Let 6, be the M-estimator defined by (2.17) with 4e sequence (2.30). Thm, under ry2 { era1r_ l satiing

(3.26) Remark 1. If pv2 0, the order on the right hand side of (3.26) is O&I--li2), as it

follows from Jure&ov& and Sen (1981). Theorem 3.2, Let X1, X2, . . be iid random variablesdt’slributedaccordingto B.f, F. Supme that Fand v satisfv the regularity conditionr(2.1)-(2.6) end (2.10)-(2.16). l

(3.27)

ed as in secfiobt2.

is a nonnegative submartingale, Therefore, by the ~olmogo~~vinequality (for subn~~rtin~~es),for every 8 > 0,

(3‘30)

~e8~din~ the monotonicity of 1y’,the su~re~um on the r~ht~~~d side of (3.3~) could be approximated by a supr~,~um over a finite set of c>ointssnd it follows from the assumptions that lim,,, EfIM,,(t)-EM,&) =O unifurmlyin tcEf -k$k*], Finally, the expectation term EMRB(t) may be replacedby .- tyC&K$z~similarlyas in corollary 3.1. To prove the Theorem 2.2 of section 2, we shall still need the following result concerning,the error of the appro~m~t~on of G$by si:

nT=max{k: C”,S 7X$.

m&x ncsksnr

c2 1(si - 6;) --(sf2- c&l --% 0, asn-+oo

(3.31)

(3.32)

(3.33)

ensure 3. Note that by the ~intc~hinelaw of la st2 --+cJ~ as.

asn-)oo.

e Theorem 3.2 implies

(3.34)

Ledma then fell

from (3.36) and (X37). s from Thecorem3. E that, if s>O and t2- dl#O,

(3.39) -42 where C has the stmd normal distribution. Under some ssumptions, (3.39) are replaced by sequences random variablese ore precisely, if s’n, tnlr tn2are random variables such that snA

s(&(O,1)) and

1tnl - tn216

.t( ~0)

(3.40)

for some constants s and r, then fWn(s,Sk)-

Wn(&rGtl)IqFz~

- tni I i/-

6

(3.41)

We shall recall to this result in the next section. 4. Proofs af Thewmis 2J and 2.2 Proof of Theorem 2J. For any fixed d>O and any 8, by (2.24),

NOW,it follows from (2.21)-(2.231, Theorem 3.2 and Lemma 3.1, that L,rc, -fL 28=2t,/2tQ/y

as n+m,

(4.2)

while, by (2. X), lim,i,, 2&Z& = 00and thus lim,,, P(& > n) = 0. Mareuver, it follows from the d~fi~~iti~n(2.2 ) of Nd thai Ivdis 1 in d) >O); this ether with the ct that L, >O with robability 1 implies that ivd--*a>a.s. as dlQ. To prove (2.26), dcf”,nen$& i==1,2, by

where by (4.2), CRyi_ 1&,g _ 1 -A 28 as &O, while by (2.25:)and (4.3), 2dCa_ 1 -+ 28( I+ 8) > 20. Hence, (4.4) converges to 0 as d 10. Similarly, P(C,$ CNdzf;I - s)--+O as &OS This proves (2.26). By virtue of Theorem 3.2, Lemma 3.1’)(2.21) and (2.26), it follows that

/

CNdtNd--% 28 as d 110.

(4.6)

Thus, (2.27) follsws from (4.5) and (4.6. To prove Corollary 2.1, we m;ay no&t that (2.26) and (2.28) imply (2.29). This completes the proof of Theorem 2.1 aad, Corollary 2. I. /

Proof of Theorem 2.2. It follswrs from il:hedefinition of .N& Pd {np(dCN, --S)/tky)

s.& {n/4(LNdCNd-2B)/852y}

(4.7)

and PA (n~‘4(tKNd- f -- ts),/ery) sPd (n~‘4(L+lC+r

-28)Ak2y).(4.8>

Hence, to prove (2.33), it suffices to shok that as d10, - 28),‘(29) I,V} -)@((26/y&1’2yy/K)

JS {&%&l~

(4.9)

andI

max()n*‘4(t,e,-L,~~C,,)): PUt

Cni = tYi, d,li =

ci/Cn, 1 sisn.

GiB2/C2 n n =vWWY n

IC~,A.$-llCS}=O,(l).

Then, regarding (3.20) and (3.21), PI’=x2n -+A2

as n-+q

(4.10) we

have (4.11)

where (4. t 2) It follows from Corcs!lary ZI.1and from Remark 3 thao

. -rCZ,,P;,, >1

--)AjO, 28).

.

.13)

vd, P.K. s&n /

262

ode1 (1.1) and the testing problem of (1.2). c”lo:d=O

vs* HI :d=8r6,

Bspecified,

Iv .11

as in Jtu+kavBand Sen (1981),we consider the following test. Define s: as in

.19), L, as in (2.33) and put

$8 = C*L,I211*/~ (&

v = q/y by (4.2)).

(5.2)

Let ar and a2 be the prescribed probabilities of the errors of the tyw 1 and type II, ra~ively, 0 < t%l + 4x2 < f, Let A and B be two numbers stick that )/att q(S)and continue sampling as l

Thus, the stopping number N (=N(&) is defined by N(~)=min(n~no(S):6S,(~~)~(.c~~“)$(~,a)}

(5.4)

and we stop sampling when N(S) observations are made with the acceptance or rejection of Ho according as i&f@) ( f 6)&j,

i+&) 5 b

or z a.

(5.5)

Jure&ovB and Sen (1981) studied the properties of the test in tihe case of an absolutely continuous function, i.e., when w= vi. They proved that the testing procedure terminates with probability 1 and derived the limiting OC function as 610. Unlike in the case of sequential confidence interval, it could be shown that the presence of jump-discontinuities of the score function does not change the limiting OC function of wh sequential test (excepting the contribution of yzl in y). The proufs f the stopping time of the sequential procedure a:ld (ii) of (i) the finitenes ova and Sen (1981) readily extend to the case lof t,u= w1+ ~2 Theorem 5.1 of Jur when the theorems, of se&n 3 of the current paper are incorporztted. Hence, we omit the details.

Bickcl, PA, and

J. Wichura (1971). Convergence criteria for m~~Itipar~meter stochastic th. Statist.

Carall, R.J. (1977). On the asymptdc

42, 1656-1670.

normality of stopping ti

&bins (11965).On the asymptotic theory of fixe$~wi~th sequential confidence intervals for the mean. AM. Math. Stutist. 36, 457-462.

Geertsema, .J.C. (1%8). S~~cquenlial confidence intervals bas& op r&n t&s. Arm, AW#~&. Sia&d. 39, 1016-1026. Oh&, M. and P..K, Sen i(X!V2),On bounded-length oonfidence interval for the regression ~~f~c~~nf based on a cHassof rank statiwics. Sankhy&f, Ser. A. 34, 33-52. Ghosh, 2111. and P.K Sen ( W6). Asymptotic theory of scrquentialtests based on linear function9 of order statistics. In: S. )IkedaCI.ail., &sU,Ewrys in Pr&abW_p aad Ststistim Ghosh, M. and PJL Scn (1977). SequentiN rank tests for rqrw&t. dT4nk Gleser, t,J. (8965). Qn the asymptotic theory of fixed size 3cqutntioitcon regression paranleters. Ann. Ikkth. S&t. 36, 43+‘?. Huber, Ia.J, C197?1). Robust re:$wssion: Asympt&ics, canjectutes and Montt C riu. Ann. Stat&t. 1, W-82 II. .hr&ovB, J. (Wi9). Asymptoticlinearityof a rankstatisticin rylressionparameter.Ann. Msrth, Skzfbsr, 40, 1889-1900. Ju~&kovA, J. (1973). Central U&t theorem for Wilcoxon rank statistics process. Arm. Stut&t, 1, ~~~-i~. Jur&‘ko\& 3. (19771. Asymptotil: relations of M-estimates and R-estimates in linear ic ;ion mdadel. Ann. Stat&t. 5, 464-472. JuredlRwB, J. (1979). Boundecl~length sequential confidence intervals for regression and Isc~rtian parameters. Prac. 2nd P’agut C’w$. on Asymptotic St&tie, pp. 239-250. Jure&ovb, J. (UN). Asymptotic*representation of M-estimators of location, Opmtio~fomr)rung ltnd Si:..Mik, Ser. S/at&t, 11, 61--Y’3. Jur~tCkavB,J. and P.K. Scn (1981). Invariance principles for some stochastic processes r&tins to Mestimators and f heir rolle in sequential statistical inference. Sunkhyd, SW. A, to appear. Sen, P.K. and M. Ghosh (1971).0n bounded len&thsequential confidence intervals based on onasaaple rank order sta:Mcs. Ann. Ma%, Statist. 42, 189-203. Sen, P.K. and M. Ghosh (19’74).Sequlentialrank tests for location, Ann. Stat&, 2, 340-552.