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INVARIANCE PRINCIPLES FOR SOMESTOCHASTIC PROCESSES RELATING TO M-ESTIMATORS
AND THEIR ROLE IN SEQUENTIAL STATISTICAL INFERENCE
by
Jana Jureckov~
Charles University, Prague, Czechoslovakia
and
Pranab Kumar Sen1Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1213
February 1979
,
INVARIANCE PRINCIPLES FOR SOME STOCHASTIC PROCESSES RELATING
TO M-.ESTIMATORS AND THEIR ROLE IN SEQUENTIAL STATISTICAL INFERENCE
byv ~
JANA JURECKOVACharles University andPrague, Czechoslovakia
PRANAB KUMAR SENI
University of North CarolinaChapel Hill, NC, U.S.A.
ABSTRACT
Weak convergence of certain two-dimensional time-parameter
stochastic processes related to M-estimators is studied here. These
results are then incorporated in the study of the asymptotic properties
of bounded length (sequential) confidence intervals as well as
sequential tests (for regression) based on M-estimators.
AMS Subject Classification Nos: 60FOS, 62G99, 62L99
Key Words &Phrases: Asymptotic linearity, bounded length (sequential)confidence interval, converage probability, 0[0, 1] space, M-estimator,operating characteristic, sequential tests, stopping number, weakconvergence, Wiener process.
lw~rk' ~f this author was partially supported by the NationalHeart, Lung and Blood Institute, Contract No. NIH-NHLBI-71-224S fromthe (U.S,) National Institutes of Health.
2
1, INTRODUCTIONc; •
Let Xl~"'~ Xn be independent random variables (rv) with
continuous distribution functions (df) Fl , "', Fn, respectively, all
defined on the real line R. It is, assumed that F.(x) =F(x _I',°c.),1 1
X E:R, i ~l, where F is unknown, cl " .. , cn are given constants
and 1',0 is an unknown parameter. An M-estimatop ~ of ~on [c.f.
Huber (1973)] is a solution (for ~) of the equation
r~=lci 1l!(Xi - ~ci) = 0 .
where 1l! is some specified saoPe funation. Various asymptotic
(1.1)
,..,properties of I', have been studied by various workers; we may refern
vto Huber (1973) and Jureckova (1977) where other references are also
cited. In this context, one encounters a stochastic process
M ={M (~) =t lc.[1l!(Xi -~d.) -1l!(Xi )], ~E:R} (where dl , .•. ,dnn n 1= 1 1
are suitable constants) and the asymptotic linearity of M (~) in ~n
,.., vplays a vital role in the asymptotic theory of I',n [see Jureckova
(1977)]. In the context of sequential confidence intervals for 1',0
oas well as sequential tests for ~ based on M-estimators, we need
to strengthen the asymptotic linearity results to random sample sizes,
and, in this context, some invariance principles for certain two-
dimensional time-parameter stochastic processes related to {M} aren
found to be very useful, With these in mind, in Section 2, we formu
late these invariance principles for {Mn} and present their proofs
in Section 3. Section 4 deals with the problem of bounded-length
(sequential) cQnfidence i.ntervals for 1',0 based on M-estimators and
some asymptotic properties of these procedures are studied with the
aid of the invariance principles in Section 2, The last section is
3
•
devoted to the study of the asymptotic properties of some sequential
tests for ~o based on M-estimators .
2. SOME INVARIANCE PRINCIPLES RELATED TO M-ESTI'MATORS
For technical reasons, in this section, we replace the X. byl.
a triangular array {(Xnl , ... ,Xnn); n ~l} of rv's and assume
that the -following conditions hold.
(A) For every n (~ 1), Xni , i ~ 0, are independent and identically
distributed (i.i.d.) rv with a continuous df F, defined on R.
(8) ~: R+R is nonconstant and absolutely continuous on any
bounded interval in R. Let ~(l) be the derivative of ~ and
(2.3)
(2.2)
(2.1)
F) < 00,
Ioo[~(l) (x)]2dF (x) <00,
_00
2 foo (1) 2 2o <01 = [~ (x)] dF(x) -Yl (1jJ,
assume that
Yl(~' F) =Joo~(I)(X)dF(X) ;0,_00
(i)
~ e (ii)
'l(iii)
_00
(2.4 )
(C) Let {c ., t ~ 0; n ~ O} and {d., i ~ 0 ;nl. nl. n ~ O} be two
triangular arrays of real constants, such that if {kn} be any
sequence of positive integers for which
um{.max c~i/t. 'sk c~.} = 0,n+oo l.sk ) n J
n
um{. max d2
. /t.'<k d2j } = 0,n+oo iSk nl, .)- n n
n
um{max c2id2. /l.'Sk c2.d~.} = 0 .n+oo isk n nl. ) n n) J
n
02 =1. d2 . SO*2 <00 , Y n ~l.n jSkn nJ
k too as n +00,n
then
(2.5)
(2.6)
(2,7)
(2.8)
4
Conventionally, we let cnG = dnO = 0, V n ~ 0 and let
n ~o (2.9) !
Mk(M =l~ OC l·{1jJ(X . -6d .) -1jJ(X .)}, k ~O, 6 ER.On 1= n nl nl nl (2.10)
Let K*={t, ~): O~t~kj' O~161 ~ki}(= [O,ki]x [-ki,ki]) be any
compact set in R2
(where 0 <ki, ki <00) • We define
Wn = {Wn (t, M, (t, 6) EK} by letting
Wn (t, M = {Mnn(t) (~) - EMnn(t) (6)}/ (<11An), (t, 6) E K*
2 2net) =max{k: Ank ~tAn}' t EKi = [0, ki]; kn =n(ki) •
(2.11)
(2.12)
Then W belongs to D[K*] for every n ~ 1. Finally, let W= {wet, 6),n
(t, 6) EK*} be defined by W(t, 6) = M~ (t) , (t, 6) E K* , where
~ = {~(t), t EKp is a standard Wiener process. Then, we have the
following
Theopem 2.l. UndeP (A), (8) and (C), {Wn } aonvepges weakly to W.
We are also interested in replacing in (2.11), EMnn(t)(s) by
a more explicit function of (t, s). For this, we assume that the
following holds.
(8') In addition to (2.1) and (2.2), 1jJ(1) is absolutely continuous
(a.e.), denote its derivative by 1jJ(2) and assume that
lim foo\1jJ(2)c.x +t) _1jJ(2) (x) IdF(X) = 0 , (2.13)t-.Q
_00
(D) F admits of an absolutely continuous probsbi1ity density function
(pdf) f with a finite Fisher information 1(,£) = f (,£, / f) 2dF
where ft (x) = (d/dx)f(x).
Let then
Y2(1/I, F) • [1/1(2) (x)dF(x) tC1/l(l) (x) I-f' (x) !f(x)}dFC>:] . (2.14)
..
5
so that by (~,2), (D) and the Schwarz inequality,
Y;C1lJ, F) s I (f) [I1lJUlJ2dF <00. Also, let
~n(t) 2h (t) =1..' 0 c .d . I A ,n .1= nl nl n t e: K~ •
1(2,15)
Note that by C2,8}, (2,12) and the Schwarz inequality,
h2(t) SA-2(~~(t)c2.d2.)(~~(t)d2.)StD*2 <00 V t e:k* C2,16)n - .n 1..1=0 nl nl 1.1=0 nl ' 1 •
Let us now define WO = {WOCt, fl) = {M (t) (M + fly(t/J, F)L~(ot)C .d . }/Ca, A ),n n nn 1= nl nl n
Ct, M e:K*} and W = {W Ct,fl) =WO(t. M - 2162Y2Ct/J, F)h Ct)lal
,- n n· n' n
(t, fl) E:K*}. Then, we have the following.
Theopem 2.2 • .undep CAl, (B'), (~) and (D),
to W and if, in addition,
{Wn
} weakly aonverages
Ii h ( ) h {~WO}m t = 0, V t £ k*l' ten,n~ n n
aonvepges weakly to W.
In some applications, we may encounter a process W* ={W*(t, fl),n n
-e (t, fl) £ K* }, where, defining the M (Mnnas in (2.10) and {nCt) }
,"
as in (2.12), for every (t, fl) E: K*,
W~ (t, fl) = (al An) ... 1{Mn (t)n (tl Cfl) - EMn (t)n (t) (6) } . (2.17)
(~,18)
(2,19)
The weak convergence of {W~} to some appropriate Gaussian function
depends on the interrelationship between the elements of different
rows of the triangular arrays {c.} and {d.}. Suppose thatnl nl-2~n(tAs)
lim A I..i 0 c (t) i d ( ). c ( ). d ( ). =g (s, t) s, t £ K*n~ n = n _ n .5 1 ns 1 n s 1
exists for every (5, t) and is a continuous function of s, t,
o Ss, t SKi. Further, we assume that there exists a positive number
M, such that
~qi OCck·dk · _.c .d .) + L~ lck2
.dk2.J / CA
2k _A
2) SM,
L~ = 1 1 ql ql l=q+ lIn nq
uniformly in 0 S q <k S kn I n ~ 1. It is also possible to replace
C2.l7) - (2,18) by alternative conditions, Then we have the following.
Theorem 2.:5. Under CA), (B), (~) and (2,18) - (2 t 19) ,
6
{W*} aonvergesn
weak'ly to a Gaussian funati-on W*. where W* = W* (:t, .1), t, t.) EK*
has mean ° and
EW* (~ , .1) W* (t , A.) = At. ,g (5 , t L V (s, .1). (t • .1') £ K* (2,10)
FinaUy, if
Under (A), (B'), (e), CP) and (2,18) - (2.19), in (2,17), we may also
tn(t)rep'laae EMn(t)nCt) (A) by .1Yl(~' F)li=O Cn(t)idn(t)i
1 2 ) tn (t) 2 *- ¥ Y2(~' F li=O cnCt)idn(t)i' for (t,.1) EK •
g (5 , t) = S 1\ t, 'tf (5, t) EKi ' then W*:: W•
In Sections 4 and 5, we shall encounter a single sequence {c., i ~o}l.
and will have cki = c i ' dki = ck/Ck where C~ = L~=lc~i' for k ~ 1.
In such a case, (2,18) and (2.19) are not difficult to verify. In
fact, here, g(s, t) = S 1\ t, Y s, t EKi.
3. PROOFS OF THEOREMS 2.1, 2.2 AND 2.3
First, consider Theorem 2.1. Let us define W~ = {W~ (t, .1), (t, .1) £ K*}
by letting
° tn(t) (1)W (t, .1) =Ml' ° c .d i{~ (X.) -Yl ($, F)})/OlAn ' (t, .1) EK*, (3.1)n l.= nl. n nl.
where net) is defined by (2.12). We like to approximate W by WOon n
Note that for every (t,.1) EK* ,
E{WO(t,.1) - W (t, .1)}2= [tni(to)c2.var{~(x . -.1d .) -~(X .) _ d .~(1)(X .)}]lo2A2n . n . L. = n1 n1, n1 n1 n1 n1 1 n
InCt) 2 (1) 2 2 2S i ° c iE{~(X i-.1d i) -~(X.) -.1d.~' (X .)} lOlA= n n n . n1 n1 n1 n
(3.2)
I* 2 2 2 (1) 2 2 2= [. ('t)c .d i.1 E{['/J(X i-lid .) -~(X .)]/.1d i -~ (X.)) IOIA.-] ,n. nl nn nl n1n 'nl "b
where t~(t) extends, over all B; ~ sn(t) and dni ~ O}, Note that by
(2,'2) and (2,4), for every h >0, f h- 2{'/J(x +h) _~(x)}2dF(X) =_00
7
Jih.,.If1//1) (x + t)dt] 2dF (x) S flO
h·1 (fhIl/J(l) (x + t)] 2dt)dF(;lC) ..
_00 ° .00 °frl/Jo,)(X)]2dF ()C)«(0) as h ..O! A similar case holds for h <0,
.. 00
Hence, by Fatou "s lemma~
lim(h'"2 [l/J(x +h) ",l/J(x)]2dF (x) = Joo[l/J(l) (x)]2dF (x). (3.3)h~ .
_~ ~oo
Similarily,
lim fooh-1[l/J(X +h) .. l/J(xnl/J(l) (x)dF(x) = foo[l/J(l) (x)]2dF (x). (3.4)h~ .00 _00
From (2.6), (2.8), (3.2), (3.3) and (3.4), we obtain that
lim[E{W (t, 6) _wOrt, 6)}2/t62] = 0. V (t, 6) e:K*. (3.5)h~ n n
By (3.5) and the Chebychev in equality, for every (fixed) m(~ 1) and
(t ., 6 .) e: K* ,IS j s m, as n" 00
J J
max IW (t., 6j ) -WO(t., 6.) 1£"'0.lSjsm n J n J J
(3.6)
tm °Also, by (3.1), for any £= (£1' ... , b ) ;0; /.. 1b.W (t., 6.) =
-m ~ J= J n J Jk (1)
I.noe . . {l/J (X.) -Y1(l/J, F)}/al , where the e. depend on b,1= 1J n1 n1 ~
t l , .•. , tm' 61, ... ,6 " {c.} and {d.} and by (2.5) - (2.8),m n1 n12 kn 2
max e ill ._oe ...0 as n"oo , (3.7)Osisk n J- nJ
n
while the {l/J(I) (Xni
) -'VI (l/J, F) }/a1
are i.Ld.rv with °mean andv.
unit variance. Hence, by a central limit theorem in Hajek and Sidak
tm °(1967, p. 153), we conclude that lj::l1bjWn(tj, 6j ) is asymptotically
normal, Further, by (2.7) and (3.1),
EW~Ctj' AjlW~(:t.v 6t ) "6j 6t (tj Att ) = EW(t j , 6j )W(tt' 6t ), (3.8)
for every j, t=l, ,,,,m. Hence, for every m(~ 1), (tj
, 6 j ) e:K*,
8
From (3,6) and (~,9), we conclude that the finite dimensional distri
butions (f,d.d,) of {W} converge to those of W. Hence, to proven
Theorem 2.1, it remains to show that {W } is tight. For this, forn
a process x, we define the increment over a block B = (t: 10 $! $ 11 }
by
x(B) =x(!l) -x(:l:ll' t 02) -x(tol , t 12) +x(!o)' (3.10)
where !j = (t j1 , t j2), j =0, I, Also, let Bo(1o) be the block
{~: 10 st s~o + o,V (0 > 0) • Then, proceeding as in (3.2) - (3.5), we
obtain that
(3.11)
(3,12)
(3.13)
where A(B) stands for the Lebesgue measure of the block Band
~.: In = 0, uniformly in 0(> 0) and (t, t1) E K*.
Thus, from (3.11) and the results of Bickel and Wichura (1971), we
conclude that {W - WO} is tight, Hence, it suffices to show thatn n
{WO} is tight. Toward this, we note that wO (BAt, t1)) = 0 [W (t + 0) - W (t)],n nun n
where
(\'n(t) (1) )W (t) = L' ° c .d .{ljJ (X.) -Y1(ljJ, F)} lalA , t EK*l'n 1= n1 n1 . n1 n
. 2 ° 2 3Hence, E[Wn(t) -Wn(S)] s(t-s), so that E[Wn(Bo(t, t1))] $Q =
[A(Bo(t
1t1)]72, for every 0 >0 and (t, t1) EK*, and hence, the tightness
of {WO} follows from the results of Bickel and Wichura (1971). Q.E.D.n
•
9
To prove Theorem 2.2, we note that for every (t~ 6) E:K*,
-11 ( . ~n(tl 1 2. ~n(t) 2 IA EM (t) M + I1Yl(l/J, F)L' 0 c .d i -~ Y2 (l/J, F)L' 0 c i d .n nn .1= n1 n 1= n nl.
=A-1n:~(t)c .E{l/J(X . -l1d .) --l/J(X ) tl1d .l/J(l)(X .) _~2d2.l/J(2)(X .)}In 1=0 n1 . n1 nl ni n1 . n1 2 n1 n1
(0<8<1)
~¥2D*t sup r'l/J(2 l (x +h) _l/J(2) (x) IdF(x)h'lhl~6d*. n _00
(3.14)
..
where d* = max Id . I 0+ 0 as n 0+00. Hence, by (2.6), (2.13) andn l<'<k n1-1- n
(3.14), sup{lw (t~ 11) -W (t, 11)1: (t,l1) £ K*} ~ ° as n-+ oo , son n
that Theorem 2.2 follows from Theorem 2.1.
For Theorem 2.3, in (3.1)~ we take for every (t, 11) E:K*
W~o(t, 6) =6(Li~~)cn(t)idn(t)i{l/J(l)(Xni)-Y1(l/J, F)})/al An •
(3.15)
Then, as in (3.2) - (3.5), lim E{[W*(t, 6) -w*o(t, 11)]2} =0, V (t, 11) E:K*,n-+oo n n
I *0 pand hence, as in (3.6), max w*(t
j, 6.) -w (t., 11.) _0 as n-+ oo •
l~j~m n J n J J
*0The convergence of f.d.d.'s of {W } to those of W* followsn
[under (2.18)] as in (3.7) -.(3.9). Further, in (3.11), we may replace
Wn
and W~ by W; and W:o, respectively; (2.19) insures the same
inequality. Finally, by (3.10) and (3.15), for every (t, li) E: K* ~
W* (B .1' ( t, li1) = <5 [Q* (t + t5) - W* (t)] ; (3 •16)nun n
~ (t) = (L~~~) cn(t) i dn (t) i {l/J(l) (Xni ) - Y1 (l/J, F)}J/a1An· (3,17)
A* ~*( 2 -2{~n(t) 2 2 ~n(s) 2 2Note that E[Wn(t) -wn·{s)] =An Li=O cn(t)idn(t)i +£1=0 cn(s)idn(s)i
- 2Li~~i\s)cn(t)idn(t)iCn(S)idn(S)i} which, by (2.19), is bounded from
10
above by M(t - sL V t ~s. Thus, E[W~(BO(t, 6))]2 ::;M03 =M[A(Bo(t, 6))]%
for every (t, 6) EK* , 0 >0, and this insures the tightness of {W:O}.
Q.E.D.
4, SEQUENTIAL CONFIDENCE INTERVALS FOR 60 BASED ON M~ESTIMATORS,
"
Let us consider the simple regression model: o 0Xi =6 c. + X. ,
1 1i ~ 1,
where the X~ are i.d,rv .with a continuous and symmetric df F,l.
defined on Rl , the c. are known regression constants and we want to1
provide a bounded-length confidence interval for the unknown parameter
60, Our procedure rests on the M-estimators [viz, (1.1)]. Since F
is not specified, no fixed-sample size procedure exists and, therefore,
we take recourse to sequential procedures.
Let us define C~ = r~=lc~ and assume that
lim n-lC2 =C2 exists (0 <C <00) ,n-+co n
lim{ ~x c~/C~} = 0n-+co ls1sn
Define an M-estimator of 60 as
(4,1)
(4.2)
.,t. = l.(~(l) + 6(2)) .n 2:n n '
~(l) = sup{a: S (a) = ~ ci1Ji(x - ac.) >o}, 6(2) =inf{a: S (a) < O},n n il 1 n n= (4.3)
1Ji is non-constant, nondecreasing and skew-symmetric on
is "'" in a ERl , and hence, 6 exists), and, further,n
We assume that
R1
(=> Sn (a)
we assume that
foo
2 20<00 = 1Ji (x)dF(x) <00 •
_00
(4.4)
v '"Then [c,f. JureckQva 0977)], as n -+00,
(4.5)
where defining Y1(W, F) by (2.1l,
v($, F) =(lo/Yl ($, F).
11
(4.6)
Thus, if v(W, F) is specified, then for a given de> 0), one can
define a desired sample s~ze nd by letting
(4.7)
(where Tal 2 is the upper 50a% point of the standard normal df) and
take
I* =[8 -d, '6 +d],nd nd
nd
so that by (4.5), (4.7) and (4.8), we have
lim P{80 EI* } =1 -a (0 <a <1) •d....O nd
(4.8)
(4.9)
Thus, for small d, I* provides a bounded length confidence intervalndfor 80 with asymptotic coverage probability 1 -a. In our case,
neither F nor V(W, F) is specified, and hence, the procedure in
(4.7) - (4.9) is not usable. For this reason. we take recourse to the
Chow-Robbins (1965) type sequential procedure. [See GIeser (1965) for
the sequential least squares procedure and Ghosh and Sen (1972) for the
sequential rank procedure.] Let us define
2 -l~n 2 ~ { -l~n A}2$ =n /.. lW (Xi -~ c.) - n /.. ll/J(X. - 8 c.) ,n 1= n 1 1= 1 n 1
and
~L =sup{a; S (a) >T /2C s },n n _. ann
8U-}l· -1¢{a: S (a) < -T 12C s }.~ . nan n
Also, we assume in addition that
limE{ sUJ? [l/J(X~ - t) -l/J(X~)]2} = 0h ....O t : It Ish 1 1
V '"Then [c.f. Jureckova (1977)], it follows that
(4.10)
(4.11)
(4.12)
12
(4,13)
[Actua11Y1 by (~,2) and the Kintchine law of large numbers l as n~~1
-1~n r 0 . r ~n 'i=~111 (Xi) -+. .111 ()C)dF(x) a~s'1 for r = 1, 2 (4,14)
Let w* =ma~{1(6 -tp)cil: 1 si :!>n}(s C 16 - 110 1 max Ic 11 wheren n n n 1<' < n_1_n
c . =ci/C 1 1 si sn), Then 1 by (2,5), (4.2) and (4,5)1 w* ~ 0n1 n n
as n ~~. On the other hand, [w* < E] (£ >0) insures thatn
where
r I r 0 r o· Iw ICE) = sU1 111 (X. -t) .. $ (X.) , l:;;i :!>n, (r = 1,2)n t It < 1 1
: -£ . (4.16)
are i.i.d.r.v. By (4,12) and (4,16) along with Markov inequality,
for £ >0 arbitrarily small, the right hand side of (4,15) can be
Hence, by (4.14) and (4.15),made arbitrarily small, in probability.
2 n 2sn ~ °0 , as n ~OO, (4,17)
(4,19)
(4.18)
Note that by (4,18),
show that P{6° £ IN } ~ 1 - ad
asymptotic properties of
:r has width s 2d, while,it remains toNdas d ... O. Our interest centers around the
=min{n ~nO: Ta/2
en :!>dCn }
where nO (~2) is an initial sample size and we propose
C\ 0 ~
IN =: Itt N :!> A s ~ N ]d ' d ' d
as the desired confidence interval for 11°.
v ...while the rest of the proof of (4.13) follows as in Jureckova (1977)].
With (4.7) - (4.9) and (4.13) in mind, we consider a sequential
procedure where a stopping variable Nd is defined by
Nd =min{n ~nO: ~u n .. .8L,n s 2d}
13
The01'em 4. t, Unde1' (4,1), (4,2), (4,4) and (4.12), as d +0,
Nd/nd +1, in p1'obabiUty~
2 'D 2 2 2d Nd ....... e =[Ta/ 2,,(IjJ, F1/C] = lim (d nd) C< (0) ,
d+O
(4.20)
(4.21)
(4.22)
Remark; In the Chow~Robbins (1965) procedure, (4.20)~(4.2l) have been
established up to an a.s, convergence as well as convergence in the
first mean; parallel results for rank estimates are due to Ghosh and
Sen (1972), But, the later authors needed considerably stringent
regularity conditions on the score functions for such stronger results.
Here also, under stronger regularity conditions on the c. and 1jJ,1
such a,s. results can be established. Hcwever, we do not intend to
pursue these results.
Proof of Theorem 4,t. For every £ >0 and d >0,
Then, by (4.18)
p{Nind >1 +d = P{Nd >n~£}
= p{i_ -~L >2d, V nO ~n ~nod } ~P{6-U,n ,n £ °U,nd£
....- b. > 2d}
°L,nd£
(4.23)
t
By (4,1), (4.7) and the definition of n~£, 2dC 0 +(1 +£)~2Ta/2,,(IjJ, F) >ndE
2T a/2,,[IjJ, F), so that by (4.13) and (4.23), p{Nd/nd >1 + d + 0 as
d +0, Similarly, for every E >0, p{Nind < 1 + d + 0 ad d +0, and
hence, [4,20) holds. Then~ (4.21) follows from (4.20), (4.1) and (4.7).
TO prove (4,22), we define z = {Z (t) =S (6o)/OoC, t £K*l '
n n net) nwhere
(4.24)
14
o 0t E K*l' Since X, =X, ... A c
i• i ~ 1 are
1 1
i.i.d.r.v., by the same technique as in the proof of Theorem 2.3, it
follows by some routine steps that under (4.1), (4.2)~ (4.4) and
(4.12), {Z} converges weakly to a standard Wiener processn
S = {set), t EKP. This insures that
sup Iz (t) 1=0 (1) and lim sup Iz (t) -Z (5) I =0, in prob.tEKi n p 6+0 0<sstss+6sl n n
Also, we now appeal to Theorem 2,3, where we take ck ' =c, and1 1
for 1 sis k, k s k, Thenn
Finally, by (4,1), (2.18) reduces to g(s, t) =5 At, V 5, t EKi.
Hence, from Theorem 2.3, (4.3) and (4.23), we conclude that for every
so that
n -+00 ,
(4.25)
(4,26)
e·
to>O (O<to<ki),
sup C I~ (t) - AOI =0 (1) , (4.27)t <t<k* n n p0- - 1
sup I -1 2 ~ 0 I ptostski Cn Cn(t) (An(t) -A) +Zn(t):V(IlJ, F) --0, (4.28)
sup IC- l {C2 ( ) (At ( ) _Ao) _c2
( ).~(sf'·AO)}1 P-+O as <HO. (4,29)toss<tss+6Ski n n t t n s
Similarly, by Theorem 2.3, (4.11), (4,12), (4,24), (4.26), (4.27) and
(~. 28),
t~Ut~k*lc~lCnCt){Cn(t)(AL,n(t) -6n(t~+TCt/2V(IlJ, F)}I £...0, (4.30)
0- - 1
to~~ki Ic~ICn(t) ICn(t/Au,n(t) • An(t)) • Ta / 2V('i>, p)}1 !'.. 0, (4,31)
1...1 2·~ 0 2 A 0 I p
sup * Cn {Cn(t) (AL,n(t) - A ) "Cn(s) (AL,n(s) - A )} --+ 0 (as 0 +0)toss<tss+oSk1 (4.32)
15
where by
By virtue of (4.20), (~.32) and (4.33), as d ~o,
-11 2 A ~ 2 A ACn {CN (A. - N· - 6L N ) - C (Au - 6L )} I
d d --U, d ' d nd lJ,nd ,nd
(4.30) .. (4.31), as d ~o ,
C (~- .. AL ) t. 2T /2v(I/J, F).nd lJ,nd ,nd ex
Also, from (4.20), (4.28) and (4.29), as d ~o,
L.o. (4.34)
(4.35)
"" 0 .p A 0 VCN (~ - I!. )1 v(I/J, F) '" C (6 - 6 )/v(I/J, F) -Zn' (1) ,
d d nd nd d
where Z (1) is asymptotically (as d ~ 0) N(0, 1). Finally, bynd
(4.32) - (4.33), (4.20), as d ~o,
(4.36)
Thus,
-11 2 A "" 2 "" A I pC {CN (6u N .. A._) .. C (l:lu - 6 )} -+- 0 .nd d ' d ~d nd,nd nd
A pV
N-+- v(I/J, F) as d ~o and hence, by (4.36) and (4.37),
d
(4.37)
lim p{1!.0 £ IN } = 1 - ex .d~O d
This conpletes the proof of the theorem.
Note that by (4.13) and (4.30) - (4.31), for every to >0, as
n -+00 ,
I ..1 {A } Psup C C (t) v (t) - V(I/J, F) - 0 •
t <t~·* n n n0- ~""I
We now appeal to Theorem 2.3 to present an invariance principle
(4.38)
(4.39)
relating to estimators of 'Yf(W, Flo Note that by (4.6), (4,13) and
(4.17), we have
"" "" ""B =5 Iv =2T /25 IC (A -6L )n n n ex n n 'U,n ,n
(4.40)
We intend to study the asymptotic behavior of {Bn(t)" 'Y1 (I/J, F), t E Kil·
(4.43)
(4.41)
p~o , (4.44)
16
For this, we note that by (4,14), (4,15), (4,16) and (4 1 27), we may
improve (4,17) to the following: for every to >0, as n -+ 00 ,
· . I 2 21 p 0. ~sup s 't) - 0"0 -+ ,tostski nu
where net) is defined by (2,12). [Actually, in (4.15) - (4,16), we
may replace w; by Sup{wn(t): tostski} which by (4. 27) ~ 0
as n -+00]. From (~,39), (4.40) and (4.41), we obtain that as n -+00,
sup *lc~lCn(t){Bn(t) -Yl(l/J, F)}I ~o, lJ to >0 . (4.42)tostSk1
This, in turn, insures [by (~.20)] that
BN - Y1 (l/J, F) ~ 0, as d +0 •d
To obtain results deeper than (4,41) and (4.43), we note that
by virtue of Theorems 2,1 and 2,3 (where as in (4.24) - (4,37), we
take cki =ci and dki =ck/Ck , 1 si sk; k ~1) and (4.30) - (4.31),
for every to >0,
sup {I {W*(t,t stsk* no 1
~ 0 ~ 0where an(t) =Cn(t) (6L,n(t) -6), bn(t) =Cn (t)(6U net) -6) and
c = Ta
/ 2V(l/J, F). Also, by Theorem 2.3, as n -+00,
{W;(t, an(t) -W;(t, an(t) +2c), to st Ski} E.... 2c{s(t), to st Skp.(4.45)
From (4.44) and (4.45), we have [by (4.11)]
1 2..... ~
O"lAn
{ [2Ta./2Cn(t)sn(j:.) "'Yl (l/J, F)Cn(tJ C.AU,n(t) - 6L,n(t))]' to st Skp
~ 2Ta
/ 2V(W, F){s(t), to St Skp , (4.46)
where we assume the regularity conditions of Theorem 2,2 and further
e·
that defining hn
by (2.14),
limhn(t) =0. V tostski.n-+co
(4.47)
~I
17
By using (4,39), (4 ~401 and (4,46), we have
{[Cn (t)Cn (t/V(1/J, F)O'lAnlIBn(t) ""Y1 (1/J, F)). to st SkpV-+ {~(t), to st ski}
(4.48)
(4.49)
so that from (4.46) and (4,48), we obtain that as n-+ oo,
{cur L~=IC1]-~cn(t)(Bn(t) -Y1 (1/J, F)), to st SkP~ {O'l~(t), tO:St :SkiJ,
(4.50)
for every 0 < to <ki < 00, From (4,21) and (4.48), we conclude that as
d ~O
2 tnd 4-~C [L.' 1c .] {BN -Y1 (1/J, F)}/O'1 -+- N(O, 1).nd ~= ~ d
Let us now define
(4.51)
(4.52)02 -ltn 2 0 1 tn 0 2s =n L.' 11/J (X. -f!. c.) - (- L.' 11/J(X. -f!. c.)) , n ~l.n 1= 1 1 n 1= 1 1
By (4.1) and assuming that ~1/J4(X)dF(X) <00, we may repeat the proof_00
of Theorem 2.1 and show that for every e: > 0 and K < 00,
(4.53)
(4.54)
Also, from (4.46), as n-+oo
iCy1 (1/J, F) /O'lAn){Cn (t) [sn (t) lao - Vn (t) Iv(1/J, F)]}, to:S t :S kp
V-{~(t): tostski}
By (4,1), (4,53) and (4,54)~
{(rI (1/J, F)/O'IAn){Cn(~) [s~Ct/ao "'VnCt )!V(1/J, F)]}, tO:St :Skil L{~(t), to:St Skp (4,55)
Now {nCn "" 1) ..ls~2, n ~2} is a sequence of U-statistics of degree 2
and the weak convergence of partial sequences to Wiener processes
18
follows from Miller and Sen (1972), If we let
2 [00 4 .. .(2 .20'2 = l/J ex) dF (x) ... C .l/J (x)dF (x)) ,
"",co "",00
(4.56)
then using the decomposition (2,181 and (3.1) - (3,5) of Miller and
Sen (1972) along with the proof of our Theorem 2.1, it follows that
{ 2 02 2(when (4.1) holds) (Cn(t) [Sn(t/OO -1]/Cn02, Zn(t), t EKil weakly
converges to {(~1 (t), ~2(t)), t EKil where Zn is defined after
(4.23) and sl and ~2 are independent copies of a standard Wiener
process. Thus, if we assume that
. (~n 4) 2 2hm L'=lc . /C =Con-+<lO 1 1 n« (0), (4.57)
then from (4.55)~ and the above discussion, it follows that an n~oo
{(y1 (W, F) /°1CO){Cn(t) [Cn (t)/V(W' F) -1], to ~ t S kil
V-+ {~ -~. "'1 (t) + (y1(l/I, F) ° 2/2°1COlt s2 (t), to S t ~ kp,
for every 0 <to <ki <00. By (4.20) and (4.58),
A *2C
n[VN /v(l/I, F) -1] ... N(O, a ),
d dwhere
(4.58)
(4.59)
(4.60)
2 2*2 COOl 1 2° = 2 [1 +402] •
y1 (l/I, F)
Note that by (4.7) and (4.18), as d +0, V n ~nO
P{Nd >n} =P{Cm/V(l/I, F) ~Cm/Cnd' V nO ~m ~n} , (4.61)
and by (4.21), (4.61) converges to 1 or 0 according as n/nd converges
to a limit less than or greater than 1. If we let n=nd +o(n~), the
right hand side will have a limit different from 0 and I, In fact,
using (4.32), (4,33), (4.56), (4.59) and (4.60), it follows from the
above that as d +0,
19
(4.62)
where ~ is the standard normal df and Co and a* are defined by
(4,57) and (4,60), This leads us to the following
Theorem 4.2. Under l4,l), (4.2), (4,4), (4.12), (4,56) and (4,57),
lim P {d,"l(Y'Nd/nd .. 1) sy} = ~(yCO/a*), V y ER,dfO 1::::.
0
where nd, Nd are defined by (4.7) and (4.12) and CO' a* by (4.57)
and (4.60).
5. SEQUENTIAL TESTS FOR 1::::.0 BASED ON M-ESTlMATORS
As in Section 4, we consider the regression model: o 0X. =1::::. c. +X.111
where 6. is specified. [In (5.1), we may let HO: 1::::.0 = 1::::.
0for any
specified 1::::.0 ' But, then, working with X. -I::::.Oc.,.11
i ~ 1, we reduce it
to (5.1).] Some sequential test for (5.1) based on linear rank
statistics and derived estimators have been considered by Ghosh and
Sen (1977). Because of the close relationship of M- and R-estimators,
led by their motivation, we may consider the following sequentiaZ
M-test.
~n 1As in (4.3), let Sn(;l) =[,i=ICi$(Xi -aci ), a ER, n ~l and
define s~ and Bn as in (4,10) and (4.40), Let then (0 <) ett' et2 « 1)
(0 <etl ~CX2 <1) be t~e desired type I and type II error probabilities.
Consider two numbers B~ et2/ (l ~ al )) and A(s (l - et2) letl ) (so that
o <B <: 1 <A <(0) and define a = logA, b = 10gB (O' 00< b< a< (0), Then,
we st~rt with an initial sample of size nO(=nO(l:::.» and continue sampling
as long as
Ds2 <6.8 S (k) <n n n 2-(5.2)
20
Define the stoppinfj va.ria,bZe NC=N(A)) oy
A8 S (~2l)/s2 ~ (P, a)}n n n (5,3)
(we allow NCAl = +00 if [5,2) holds for all n ~ nO CA) ). Then, we
stop sampling after having NCA) observations and accept HO: 60 = 0
o 1 2or HI: 6 = M> 0), according as 68N(6)SN(6) (¥) is S bSN(6) or
> 2- aSN(6) ,
Note that for every fixed 60 and M> 0) ,
P {N(6) > n}=P {bs2 <68 S (k2
) <as2 , V nO(6) smsn}60 AO m m m m
S P {bs2 <68 S (;.821
) <as2}
60 n n n n
{ 2 1 0 2}= Po bs < l:::.B S (;.82 - 6 ) <as .n n n n (5.4)
When
where
Hence,
60 = ;.821
, S (O)/OOC is asymptotically N(O, 1) [see Section 4]n n
Cn -+co as n -+ co , while 8n ~ YI (ljJ, F) (finite) and s~ £.... o~.
(5.4) converges to 0 as n -+co. If d =..60 +¥ >0, then [as in
e,Section 4, $ t and Sn(a) is \ in a] for every K(O <K <co), there
exist an n*, such that
(5.5)
by Theorem
p8n --+ Y1 ($, F)
of (5.4)
d ~ K/C , V n ~ n*n
which insures that S (d) sS (K/C ), V n ~n*, and hence,n n n-1
2.1-2.2, Cn [Sn(K/Cn) "Sn(O)]-+ -KYI ($, F) as n-+ oo , while
2 p 2and sn -..... 00, Since KCnY1(W, F) -+ co, the right hand side
again converges to 0, A similar case holds for d <0. Thus,
(5.6)lim P {N(A) >n} = 0 ,n-+oo AO
so that the process terminated with probability 1,
We like to study the OC function of the proposed procedure.
As in Ghosh and Sen (1977), we take recourse to the asymptotic case
where we let A-+ 0, We assume that
21
AO = 4>ll where ~ £ I ={u: lul:s; K} for some K > I, (5.7)
[Note that for fixed 40 (j 0), the OC will approach to 1 or 0 as 6 +0,
according as AOis < or > 0,] Further I we assume that. nO (ll) +00
such that
lim nO (s) = 00 but lim ll2no (ll) = 0 ,ll+O ll+O
(5.8)
,e
Pinally, all the regularity conditions of Theorem 4.2 are assumed to
be true here. Let Lp (4), ll) be the OC function of the sequential
test in (5.2)-(5.3) when P is the underlying df and llo =4>ll. Then,
we have the following
(5.9)
Thus, the asymptottc straength of the test is (L(O) =l-al , L(l) =a2
)
fora aU P.
Proof. POl" an arbitrary £(> 0), we define stopping variables £N•. (ll)1.J
as the smallest positive integer (~nO) for which
i 2 1 2 .b(l + (-1) £)00 <llYl (l/J, P)Sn(~8) <aoO(l + (_1)£) (5.10)
is not true, for i, j =1, 2 and denote the associated OC functions
by Lij ,p C4>, ll), i, j = 1, 2, Then, by virtue of (4,41) and (4.42), for
every £ >0, there exist an II (say, llO >0), such that for all 0 < II :s; llO'
Thus, i~ in (S,IO) we let £+0 and denote the limiting stopping
~ variable and OC functions by NOCM and 0 AL respectively, thenLp (4),
it suffices to show that (5,9) holds for 0 M, Towards this, weLp (4),10
e let nL\ =min{n ~nO(~): L\2C2 ~ l}, II > 0 and define {Z } as in afternL\ n
(4.23) •
22
Then, by steps similar to in (4.24)-(4,29)1 we conclude that
as I:>. -"0, for every O>e:>k*<oo1 '
when
(5.12)1 1 2 Psup IMS(¥)-S.... (<1>1:>.) +(<I>-2)I:>.Y1(l/J, F)C.... }~ 0,
e:~t~i iiI:>. (t) nl:>. (t) nl:>. (t)
Z ={Z (t) =S C<I>I:>.)/00C 1 e: st ~k*l}~ {~(t), e: st Sk*l} (5.13)nl:>. n/i ~I:>. (t) nl:>.
where
nl:>.(t) =max{k: C~ stC~}
Thus, when 1:>.0 = <1>1:>. ,
(5.14)
e: st Skil~ {Zn (t) - (<I> - ~)t/v. (l/J, FL e: st Skil·I:>. (5.15)
By (5.13), (5.15) and the results of Dvoretzky, Kiefer and Wolfowitz
(1953), the desired result follows.
Under more restrictive regularity conditions on the score function,
for sequential rank tests, Ghosh and Sen (1977) have studied the limit
of 1:>.2 EN (I:>.) (as 1:>.-..0). Similar limit in our case also demands more
restrictive conditions on l/J and the c ..1
However, under (4.1),
the limiting distribution of !:>.N(I:>.) is the same as that of the first
1exit time of {~(t) + (<I> - 2)tJ'v (l/J, F), t ~ e;} when we have two
absorbing barriers b",~,F) and aV (l/J ,F) •
We conclude with a remark that very recently Carro11(1977) has studied
the asymptotic normality of stopping times based on M-estimators where he
needs the assumption that the score function is twice bounded1y continuously
differentiable except at a finite number of points and that F is Lipschitz
of order one in neighbourhoods of these points. Our Theorem 4.2 remains valid
under weaker conditions and also the conclusions are valid for a more
general model.
23
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CHOW I Y.S. and ROBBINS 1 H. (1965), On the asymptotic theory of fixedwidth sequential confidence intervals for the mean. Ann. Math.Statist, 36~ 457-462.
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