'.
1
,
WEAK CONVERGENCE OF BOUNDED INFLUENCE
REGRESSION ESTIMATES WITH APPLICATIONS
by
Raymond J. Carroll and David Ruppert
University of North Carolina
Abstract
We obtain weak convergence results for bounded influence regression
M-estimates and apply the results to sequential clinical trials, with
special reference to repeated significance tests in the two-sample
·e
..
problem with covariates.
Key Words and Phrases:
Regression, Robustness, Sequential Analysis,
Repeated Significance Tests, M-Estimates,
Weak Convergence.
AMS 1970 Subject Classifications:
Primary 62E20; Secondary 62FlO, 62G35.
Raymond J. Carroll's research was supported by the Air Force Office
of Scientific Research under Grant AFOSR-80-0080. David Ruppert's
research was supported by the National Science Foundation under Grant
MCS78-0l240.
2
1.
Introduction.
Our primary concern is the comparison of two treatments in a
clinical setting,' although our results are quite general and this problem
...
is only one application.
To fix the discussion, in our major
we
~xample
assume that patients are recruited sequentially and assigned at random
into one of the two groups.
Important covariable information is recorded
at the time of recruitment.
The response variable is numerical (e.g.,
systolic blood pressure, cholesterol level) and is observed either
instantaneously or at a fixed time (from recruitment) in the future.
We
assume that there is either a maximum sample size or a target sample size
N (see Section 3) .
•
-e
..
In most experiments of this type, the investigator will want to
study the data periodically or as ·it is collected and (for ethical
reasons if no other) stop if obvious differences are observed.
In this
instance a sequential analysis is called for, possibly based on repeated
significance testing (RST) (see for example Siegmund (1977, 1979)).
The
usual methods for such tests would be based on least squares estimates
and F-tests for a linear model.
Such an analysis has two potentially
serious difficulties:
(i) if there are outliers or if the error distribution is
heavier-tailed than the normal distribution, the estimates
will be inefficient and the tests will have low power;
(ii) particular design points (which may be "outliers in the
".
design") can be extremely influential, even though the
observed response is not an outlier; this could cause
difficulties in interpretation at the very least.
3
To overcome the first problem, Huber (1977) and others have
suggested the "classical" M-estimates, which give bounded influence to
outliers in the response but unfortunately give unbounded influence
•
to outliers in the design.
Recently, there have been a number of
proposals to handle problems (i) and- (ii) jointly (Maronna and Yohai (1978));
these are the "bounded influence regression" estimates (BIR).
Our model takes the general form
(1.1)
y.1
= -1c. 8 0
(i = 1,2, ... ) ,
+ Yoz.
1
where the {z.}
are i.i.d. with distribution
function symmetric about
1
.
•
·e
•
zero.
The parameter YO is a scale factor to be explicitly defined
later.
For our purposes it is convenient and not unrealistic to assume
that the design
vectors {c.}
are also i. i. d. and independent of the
.
-1
errors; modifications of this assumption are discussed in Section 5.
In the two-sample problem, we will write ~ = c0-aS*Y;-'where 11 is the
intercept and a is the treatment effect.
An individual design point will
take the form (1±l £*), where the sign is determined by the treatment
group.
All the estimatesmentiorted so far can be generated in the following
manner.
Let (Sn (pXp) , Yn) be estimates of scale which converge to
(SO' yO); particular choices of Yn are "Proposal 2" (Huber (1977)) or
"MAD" (Andrews et al. (1972)). Define 8 as the solution to
-n
"
.
(1. 2)
n
~
~
o = i=l
I c·g1(lc.s C.12)l/J(g2(lc.s C·12 (y.-c.8)/y).
1 - 1 n-1
-1 n-1
1 -1n
I
I
I
4
Particular choices of gl' g2' and
~
give important special cases, e.g.,
x
.Least Squares
gl (x)
=
g2 (x)
=
1
~(x) =
Classical M
gl(x)
=
g2(x)
=
1
~(x)
bounded
BIR
Ixg1 (x) I
~(x)
bounded
bounded ,
.
Special cases of BIR estimates which are particularly noteworthy include:
Mallows (unpublished)
and
Maronna and Yohai (1978)
g2(x)
=
1 ,
S
n
=
Schweppe (unpublished)
gl(x)g2(x)
=
1 ,
S
n
=I
Hampel and Krasker
(unpublished)
gl(x)g2(x)
=
1 ,
gl (x)
•
·e
=
I
l/x
.
•
In Section 2 we state the major weak convergence results, apply them·
to the two-sample problem in Section 3, and prove them in Section 4.
Extensions are discussed in Section 5.
2.
Major convergence results.
Except where noted, we discuss only the special case that Sn
(and we take So = I with no loss).
not
~
is monotone.
= So
Our two results depend on whether or
If so, our technique is based on Yohai and Maronna
(1979), while if not it is based on Carroll (1978).
Since there are two
results, we have chosen to list groups of possible assumptions.
In the applications we are going to assume that if N is the maximum
-e
sample size, then one takes at least
~
observations before stopping,
where KN -+ 00, KN/N -+ O. We make use of this convention in this
section. The proofs and assumptions can be modified to the case
~
= constant,
but at a cost of notational complication.
5
Define the process {G (.)} by
N
(2.1) .
Theorem 1.
Assume Al-A7 and 8l-B5 bel-ow.
Then
(2.2)
Theorem 2.
•
·e
.
Assume Al-A7" 84 and Cl-C5.
1
[Ns]
sup IElGN(s) - N-~ L
OSssl
i=l
(2.3)
Then
,
c. gl(c ·}t/J(g2(c .)z.)
-1
-1
-1
1
p
I+
°.
Clearly, Theorem 1 gives a weak convergence result for the process
p
[Ns]GN(s)/N in D [0,1], while Theorem 2 considers the more interesting
process GN(s).
In the next section these results and their applications
are discussed.
AI.
t/J is bounded, odd, absolutely continuous, and Lipschitz of order
one.
-e
6
-e
.,
A4.
There exists A > 0 such that for every 1.!.1:s; AO'
O
sup
Iq I, Ihl:s; AO
r + 0,
.
-e.
A7 •
lim
E+O
(This actually follows from AI-A4, but it is convenient to state it
. separately.)
B1.
W is monotone nondecreasing and nonconstant, and if D(u,z) =
(w(u+z)-w(u))/z, then
lu/:s;a, Iz/:S;b imply D(u,z) ~ d > O.
B2.
p{lzl:s; e:} > 0 for all
B3.
The minimal eigenvalue A.
--
mIn
is positive.
B4.
E> O.
of the matrix
7
-e
> 0, there exists Q,N
such that N ~ NI implies
B5.
For every
C1.
In addition to AI, l/J has two bounded continuous derivatives except
E:
l
on a finite set B.
C2.
l/J is constant outside an interval.
C3.
The distribution function of g2 (c) z is continuous at all b
C4.
For every
£
> 0 as N -+
A
p{ I~n - ~o
C5.
For every
p{
Remark 2.1.
E:
IYn
I
>
E:
> 0 as N -+
-
Yol
>
£
E
B.
00
for some KN s n
S N} -+ 0 •
00 ,
for some K
N
s n
S N} -+ 0 •
Al holds for the common M-estimators.
B1
is as in Yohai and Maronna (1979), while CI-C3 are needed by Carroll
(1978).
Remark 2.2.
A2-A3 are not unusual for BIR estimates.
A4, A5, and A7 are
smoothness conditions corresponding to those of Bickel (1975).
Remark 2.3.
When l/J is monotone, B2-B5 are specially adapted versions of
assumptions in Yohai-Maronna.
In an unpublished manuscript, they have
also shown C4-C5 when g2 (£) :: 1, l/J is monotone, and scale is estimated
by Proposal 2.
8
-e
3.
Two-sample problem.
For the two-sample problem, the treatment effect is a' .§. 0 =
where
~'
= (0
1 0... 0).
ex.
We consider the situation in which we are
allowed to take at most N observations and let the process start by
taking at least
~
observations.
The usual RST methodology for testing
HO: ex. = 0 vs. HI: ex. ;t 0 follows this form: for each K ~ n ~ N, one
N
computes the usual F- or t-statistic based on n observations and compares
it to a cutoff point (possibly depending on n,N).
HI is chosen and
experimentation stopped if for any K ~ n < N the test statistic is
N
"large"; at the final observation (n = N), a new cutoff point is defined.
Note that in the previous section we have shown that if W is
·e
..
Brownian motion,
where "-..>" denotes weak convergence in D[O,l].
Let· {CN(o)} be a
sequence of nonnegative functions in D[O,l] converging (uniformly) to a
continuous function C(o) and let D be a positive number.
If YO' El ,
and EZ were all known, then a general class of RST' s which fits into
the framework outlined above would take the following form:
Take an initial sample of K observations. Reject H
O
in favor of HI if either of the N following obtains:
k
I-
(ii) N 2 a' 13
--
-N
I>
DA
00
(where Aoo =y 0
The constant Aoo can be est1mated o One reasonable estimate is
9
-e
-\J(s)
a
= Y[Ns]
where
(3.1)
A
'
• 1jJ' (g2(c
c.
f3[N S ]l/Y[N S ])
- 1.)(y.1 1-
and
(3.2)
= [Ns]
-1
[Ns]
\'
.,
.{
L.
c. c . gl (c .)
i=l - 1 - 1
-1
-e
..
The procedure we propose is the same as above but with AN(s)
replacing Aoo
•
Before analyzing the power of this test, we need the
following:
(3.3)
(We will analyze the power of these tests for contiguous alternatives so
the term a'[email protected] below will depend upon N, though this will not be explicit
in the notation.)
-e
Theorem 3.
k
SUppose that N;z
~' .@. 0 -+
that either (2.2) or (2.3) hoZds.
and
n (finite) ~ that
C4
and
CS
ho Zd~ and
Suppose further that 1jJ satisfies AI-A?
10
-e
(3.4)
lim Elclg2(c)
e:-+O
"
-
-
sup
1'iJ'(g2(c)(1+r)(z-ct)) - 'iJ'(g2(c)zl
Ir I:=;e: I It 1:=;£
--
=0 •
Then the power of the RST converges to
(3.5)
p{IW{s) + sn*' > C(s) for some o:=; s :=; 1 or IW(I) + n*' > O}
h
were
n*
,,-1
= n ( YO2 a . "-1,,
£..1 £..2 £..1
Remark 3.1.
I·
I
~)-~.
The quantity (3.5) can be computed from Anderson (1960).
In
comparing different tests, the relevant quantity is
-e
"
Remark 3.2.
Siegmund (1979) has considered RST's with (essentially)
and has derived approximations in the normal case which are better than
(3.5) .
Remark 3.3.
The approximations (2.2) and (2.3) can be used to construct
analogues to classical F-tests (see Schrader and Hettmansperger (1980)
for another approach).
large values of
One could test H : K' B = 0 by rej ecting for
O
11
-e
4.
Proofs.
The proofs of Theorems 1-3 are broken into a number of steps.
..
Lemmas 4.1- 4.4 we make the assumptions of Theorem 1.
and y
n
In
Al so define y
= (J -1
= (Jn-1 •
Lemma 4.1.
Define
~i(r)
Then the process
= -e1.gl{e- 1.)l/J(g2(c- 1.)z.(l+r)).
1
[Ns]
I
i=l
~.
1
(r)
on D{[-AO,A ] x [O,l]} converges weakly to a Gaussian process which is
o
almost surely continuous.
-e
Proof.
The finite dimensional distributions converge by Al and A2.
By
AI, A2, A4 and the remark following Theorem 3 of Bickel and Wichura
(1971, p. 1665), HN is tight (in the notation of their equation (3),
Define the continuous process
HN(r, s)
= ~(r,s) +
=
(NS-[NS])~[Ns]+l(r)N
-!z
HN(r,s) + Opel)
uniformly in (r,s), from whieh the lemma follows.
Lemma 4.2.
--
Define
Then for aU E> 0 , M> 0 there exists B
= B(E, M) such that
o
12
1
IrlsM, ItlsM, BN-~sssI} < £
P{lvNCr,s,t)! > £ foX'some
Proof
0
Fix £>0, M>0
0
0
For any B>O, 0>0, A2-A3 and the fact that tV is
Lipschitz show •
op Co)
•
From Lemma 4.1 we can choose B large and 0 small so that
1
p{{suplvNCr,s,.Q)I : BN-~S s so, IrlsM} > £} < £/2 .
Hence it suffices to prove the lemma when 0 S s S 1 for arbitrary 0;
·e
since M is arbitrary we need
~nly
prove the lemma for the process
generated by
With this new process
Var(VNCr,s,.!J) -+ O.
V~
we need only prove tightness since CA2 and A7 )
For any !*, by Al-A3
This means we need only prove tightness of 'V;CO,O,!*), which follows from
A4 as in the proof of Lemma 4. 1 .
Lemma 4. 3 •
FoX' every M > 0",
13
Proof.
Fix E,M > O.
From B3,
sup{P( I£~I=O): I~I = I} < 1 ,
so that for every D > 0 there exists neD) with
sup{p(o<I.£.!I<n(D)): I!I = l} < D/2 •
(4.1)
Choose Q > 1 to satisfy the 1nequality of B5 and P (I Z I>Q) < E.
Define
2
As in the proof of Theorem 2.2 of Yohai and Maronna (1979), n 1@
for some K s n
N
for some
~
S MN
I!I = 1.
II
I~n
implies
I>
~2
EN 1M
(e,
n-
k
2
EN 21M )
= n -1
supl~ln-1
O
S M
n
l
i=l
I{lz.
I>
1
By the strong law of large numbers (SLLN),
Q}
IN- 3/2
k2
so that R (~, EN 21M) > 0
n
Since ~ is monotone and skew symmetric, with
probability at least 1-2E by B5,
R
-n
> E
14
·e
lim sup R :s; EM sup II/J I (a. s.)
nl
O
n-+oo
•
Since I/J is monotone, applying Lemma 1 of Yohai (1974) shows that for every
1:
2
L > 1 (since N2/ QM + 00)
:s;
sup
Ie1=1
Egl(c)lcell/J(g2(C)Q2(1-Llcel))p(lz.l:s; Q)
- --
-
--
(a.s.)
1
Since I/J is monotone nondecreasing and nonconstant, lim inf I/J(x) < O.
x+_ oo
Therefore from (4.1), gl (£) > 0, (A2), dominated convergence, and Fatou's
.,
Lemma, one can choose Land Q sufficiently large so that the right side of
(4.2) is at most Aoo < 0 and Aoo is independent of E.
One completes the
o
proof by choosing E* sufficiently small.
Lemma 4.4.
N
For every E > 0 there exists M=M(E»O, L(E) such that for
sufficiently large
.
A
P{nIS
-n
Proof.
A
I-n
'e
1:2
>L for some MN
1:2
:s; n :s; N} < 4E •
Choose Q as in B5 and define
Let L = M/Q.
n S IN-
IN-
1:
2
As in Theorem 2.2 of Yohai and Maronna (1979),
1
> L implies RN(~' LN'2/n , n) > 0 for some
I~I = 1.
When
Q-l :s; On S Q (with probability at least l-E from B5) uniformly in
lei
= 1,
then by Bl,
15
By the SLLN and B3,
lim sup A (n,_6)
2
n~
·e
~
dMA.
mIn
p{
Izi
~
Q-l} (a.s.) •
Therefore, it suffices to show that
"
sup
IA (n,N,a)/
1
J!I=1
=
oP (1)
MN~~n~N
for some M.
We have for
I~/
=1
k
and MN 2 ~ n ~ N
n
k
sup
MN2~n~N
Assume M > 1.
IN-~ '-1
I
gl (c. ) c. 1jJ (g2 (c. ) z. a ) / •
1
1
1
1 n
1-
Then from B4 there exists M* independent of M (we may
take M* < M) so that
I} ~ M*} <
Hence
£
•
16
·e
or
Therefore for M sufficiently large
hence for some M** not depending on M
·e
"
Thus, since 00 = 1,
sup
1~1=1
IAI (n,N,~)1
S
k
MN 2 SnS N
sup
OSssl
OslrlsM**
I N-~
[N,S]
L
i=l
gl(c.)c. ~(g2(c.)z.(1+r))1 =
1
1
1
1
° (1)
p
o
by Lemma 4.1.
Proof of Theorem L
Fix e: > 0 and a >
o.
Then Lemma 4.4 shows
that when the sup in (2.2) is taken over KN/N S s Sa,
(2.2) is 0p(a).
Since a is arbitrary, it suffices to prove (2.2) when the sup is taken
over aSs S 1 for any a > O.
/\
!=Oo[Ns] (..@.[Ns]- [email protected])N
_k
2,
(by Lemma 4.4 and B4).
complete.
In Lemma 4.2 set
/ \ 1
r=[Ns] (o[NS]/oO - l)N-"2, both of which are Opel)
Then use AS; since aSs S 1, the proof is
o
17
·e
Lemma 4.5.
Assume A1-A7 and Cl-CS.
TheT'e is a function hex)
x + 0 foT' which the foUowing obtains:
fOT'every
smaU" almost surely as 'n + 00 and uniformly in
E
+ 0
> 0 sufficiently
10 -0'0 I~E,
-
I~ ~O I::;;E
+ oc i g 2(£i)1JJ' (g2(£i)zi) (~- ~o)
- g2(c
.)z.1 1JJ'(g2' (c
.)z.)(o-oo)}1
-1
-1
1
Proof.
~O=o
Set 0 =1,
0
without loss.
Define
d(E)
A(E)
= {I£I g2 (c)
~ d(E)}
L(E,O, .f~) = {lg2(£)0(z-£~ - KI >2e:d(e:)
Using C2 choose a > 1 such that
constant on [a,oo).
1JJ
for all
is constant on
Ke:B}.
(_00, -a]
and
Choose e: is so small that
2a/(1+e:) - e:d(E)/(l-e:)
~
a
and
2aE + e:d(E) (1+e:) ::;; 2e:d(E) •
--
Now suppose that
£1. A(E).
10 - 11 ::;; E, I.§. 1
::;;
E, (z,£)
Then if g2(£)z ~ 2a, it follows that
€
as
L(E,O,.§.)
and
18
·e
and
so that t/!(gZ(S)O(z -£.§})
= t/!(gZ(.0 z );
that t/!(gZ(.00(z - £.§}) = t/!(gZ(£)z).
If
similarly gZ(.0z
Ig (.0 z
z
I<
S
-Za implies
Za, then
S ZaE + Ed(E) (l+E) S ZEd(E) ,
so that gz (£) z is in the same interval between points of B as is
gz (c)o(z - £.§), so that we may use a Taylor expansion to bound the
summands in (4.3).
If
10 - 11 > E, I~I > E, (z,£)
I.
L(E,d,~), or£
€
A(E), then we can
use the fact that t/! is Lipschitz and constant ort (_00, -a]
to bound the summands.
CO{A
nl
[a, (0)
This suggests that we can bound (4.3) by
+ A + A
+ A }, where Co depends only on t/! and
nZ
n3
n4
AnI
=n
A
= n
nZ
and on
-1
n
I
i=l
-1
n
.I
1=1
{gz(£iHlo-11 +
l£i"~I)}z
HI£ilgZ(£i)
S
deE)}
gZ(£i H1o - 1 / + l£ill~I)r{(zi'£i) lL(E,O,~}
• {I£..i tg 2 (£i)
An3
= n -1
deE) }
s d(E)}
19
-e
We must prove the bound of the lemma for each term.
for AnI and A
by applying the SLLN and using A3.
n3
This is easy to do
For A , since l/J is
n4
S=
Lipschitz and A3 holds, we need only prove the lemma when
l/J is constant off a compact set, there is a constant M
l
An4(~=O)
:s;
la -
lln-
l
n
iL g2(£i)
Since
with
IZiII{I£ilg2(£i)~d(£),
so from A3 the lemma holds for A .
n4
O.
g2(£i) Izil:S;M l }
Finally, for some constant M ,
2
for some K£B} ,
so that A3, C3 and the SLLN and dominated convergence complete the
o
proof.
Fix £1' £2 > O.
Proof of Theorem 2.
Using the fact that 2:
is of full
1
rank, and applying AI, A2, B4, C4, C5 and Lemma 4.5, we see there exists
M , M such that N
2
l
~
M implies
l
(4.4)
(4.5)
P{N!-z{~Nlan- aol
+ l2: -
l
l
N-
l
r
i=l
c~
-1
g l Cc·)l/J(g2(c.)z·)I}
-1
-1
1
20
n
(4.6)
p{IL~ln-1 i~l
£i~)
,£'i g 1 (c i) {1J;(g2(£i)cr(Yi -
+ crc i g 2(£i)1J;' (g2(£i)zi)
- 1J;(g2(£i)zi)
(li - [email protected])
- g2(£i)zi 1J;'(g2(£i)zi)Ccr-cro)}1 > 4h(E 1 )(lcr- cr o l+ [email protected][email protected])
II
Choosing .@. = lin
I
cr==cr we see that if N
n
~
M the probability that the
1
following event obtains for all K ::;; n ::;; N is at least 1 - 2 (E + E ) :
2
l
N
I
II
n 1(3
-n - ~o N
(4.7)
-~
II
For (4.7) to occur we must have (for E small) n I(3n 1
1:
f 0 IN- 2
::;;
4M ,
2
proving that
(4.8)
II
Placing n(_S -_(3 o)N-
Proof of Lemma 3.1.
1:
2
into (4.6) with cr=cr.
n
I
(3=(3
-
n
completes the proof.
This follows by applying the technique of Lemma 4.3,
o
B5, Lemma 1 of Yohai (1974), and (3.3).
Remark.
AI, A3, and B1-B2 imply (3.4) by dominated convergence.
Proof of Theorem 3.
(4.9)
0
We will first show that for j = 1,2
sup { ILj N (s) - Lj
I:
p
KN/N::;; s ::;; I} + 0 •
21
First consider j
1-
= 1.
From A2, C4 and C5, with probability at least
t:./2,
(4.10)
IZlN(s) - [Ns]-
1 [Ns]
l c'. c ·gl(c- 1·)g2(c
.)t/J'(g2(c .)z·)1
i=l - 1 - 1
. -1
-1
1
sup
It/J' (gZ(£i) (l+r) (Zi - £i t))
Ir 1st:. , 1!.lst:.
- t/J' (gz (c
.) z.)
-1
1
I•
One applies (3.4) and the SLLN to prove (4.9); a similar proof using A7
and Lemma 1 of Yohai handles j = Z.
·e
Since L:
and L:
l
Z
are of full rank,
the power becomes
...
.
1
/\
for some ~/N s s s 1 or N~ la' ~NI > ~(l)D} ,
from which (3.5) follows by Theorem 1.
5.
o
Extensions.
(a) It is possible to extend the results to classical M-estimators.
2
In Theorem Z we need E 1£1 <
00.
For Theorem 1, A4 and A7 must be
modified so that Lemma 4.1 holds.
(b) We have dealt with the case in which the design vectors {c.}
-1
are i.i.d., both because it is often reasonable and because it simplifies
the exposition.
If these vectors are (i) constants or (ii) generated in
Section 3 by blocking on each group of two patients, it is still possible
22
to apply our techniques.
..
Detailed and reasonably simple conditions can
be worked out, both because of the nature of BIR estimates and because
our results depend in large part on easily generalized standard SLLN and
weak convergence techniques.
The major difficulties lie in (4.1) and
(4.2) and the terms A , A
in Lemma 4.5.
n3 n4
(c) It is possible but extremely messy to extend the results to
estimating So by Sn.
(d) In LeJ!lllla 4.5, the major use of sYmmetry is that it assures
E£gl (£)g2c.~)zlj.!' (g2(.0 z)
=
O.
For Section 3, the sYmmetry is unnecessary
if g.(c) = g.(lcl) (j = 1,2) and the 'patients are randomly (with
J -
probability
J-
~)
assigned to each group, for then
(e) The approximation (2.3), while only of first order, can in fixed
sample problems give some information about ,the dependence of the design
on the rate of convergence through Berry-Esseen results for
N
~
L.
i=l
-1
l: 1
,
c. gl(c
·)lj.!(g2(c
.)z.)
•
-1
-1
1
-1
Since the' {£i} are LLd., when YO is estimated in an appropriately
smooth fashion simultaneously with
~
(as in Huber's Proposal 2), the BIR
estimates are minimum contrast estimates of a vector parameter.
Iflj.! has
a derivative then Berry-Esseen results and Edgeworth expansions are
available (e.g., Pfanzag1 (1973) and Bhattacharya and Ghosh (1978)).
the Huber function lj.!(x)
=
For
max (-b , min(x,b)) when the design is bounded,
it should be possible to prove a version of (3.8) in Bickel (1974) and
obtain an Edgeworth expansion.
(In the location case with scale not
estimated, Jureckova (unpublished) has verified Bickel's (3.8).)
23
REFERENCES
.
Anderson, T.W. (1960). A modification of the sequential probab~lity
ratio test to reduce the sample size. Annal .of Math. Stat~st. 31, 1,
165-197 .
. Andrews, D.F., Bickel, P.J., Hampel, F.R., Huber, P.J., Rogers, W.H., and
.
Tukey, J.W. (1972). Robust Estimates of Loaation: Survey and
Advanaes. Cambridge: Princeton University Press.
Bhattacharya, R.N. and Ghosh, J.K. (1978). On the validity of the formal
Edgeworth expansion. Ann. statist. 6, 434-451.
Bickel, P.J. (1974). Edgeworth expansions in nonparametric statistics.
Ann. statist. 2, 1-20.
Bickel, P.J. (1975). One-step Huber estimates in the linear model.
J. Amer. statist. Assoa. 70, 428-434.
Bickel, P.J. and Wichura, M.J. (1971). Convergence criteria for
mu1tiparameter stochastic processes and some applications. Ann. Math.
statist. 12, 1656-1670.
Billingsley, P. (1968). Convergenae of Probability Measures.
John Wiley and Sons, Inc.
New York:
Carroll, R.J. (1978). On almost sure expansions for M-estimates.
Ann. statist. 6, 314-318.
Huber, P.J. (1977).
Robust statistiaal Proaedures.
SIAM, Philadelphia.
Maronna, R.A. and Yohai, V.J. (1978). Robust M-estimators for regression
with contaminated independent variables. Unpublished manuscript.
Pfanzagl, J. (1973). The accuracy of the normal approximation for
estimates of vector parameters~ Z. Wahr. und Verw. Gebiete25,
171-198.
Schrader, R.M. and Hettmansperger, T.P. (1980). Robust analysis of
variance based upon a likelihood ratio criterion. Biometrika 67,
93 and 102.
Siegmund, D. (1977). Repeated significance tests for a normal mean.
Biometrika 64, 177-189.
Siegmund, D. (1979). Sequential X2 and F tests and the related
confidence intervals. stanford University Teahniaal Report No. 134.
Yohai, V.J. (1974). Robust estimation in the linear model.
Ann. statist. 2, 562-567.
Yohai, V.J. and Maronna, R.A. (1979). Asymptotic behavior of M-estimators
for the linear model. Ann. Statist. 7, 258-268.
REPORT DOCUMENTATION PAGE
•
4.
...._
UNCLASSI lED
_._--:::=~~~---
READ INSTRUCTIONS
BEFORE COMPLETING FORM
TlTL.E (and Subtitle,
it.
TECHNICAL
....
Weak Convergence of Bounded Influence Regression
Estimates with Applications
J,7:;-.--;A7:U-;T7:H-:::O:-;:R:-;(-:.):------~-_·--------_·-----·--------f~-~o---,--~=----,---------._----_
9.
8.
CONTRACT OR GRANT NUMBER( .• )
Raymond J. Carroll and David Ruppert
AFOSR Grant AFOSR-80-0080
NSF Grant MCS78-0l240
PERFORMING ORGANIZATION NAME AND ADDRESS
10. PROGRAM ELEMENT, PROJECT,
AREA & WORK UNIT NUMBERS
T-A~
I--:-:---------------------------If-:-=------------·-.II. CONTROLLING OFFICE NAME AND ADDFlESS
12. REPORT DATE
Air Force Office of Scientific Research
Bolling Air Force Base
Washington, DC 20332
1--.
13.
April 1980
NUMBER OF PAGES
---
23
14. MONITORING AGENCY NAME .& ADDRESS(iI <li/h',ent /,om COlltro/l/nll Ol/ice)
15.
SECURITY CLASS, (0/ thi, ,el'0")
UNCLASSIFIED
,-s-;;.-·DECLASSIFICATION/DOWNGRADINGSCHEDULE
1-;-;;16,.-.-;D:::-IS;:-:;T~R:::-I B::::U-:-:T;:-I-=O~N -;:S';'T-:-AT~E;:-:M~E:-:N:-::T::-(~o.,..' -:-t1'-:-;s'-:R:7"-I'-o,·,·j-------
-.----.---'-------------------1
Approved for Public. Release -- Distribution Unlimited
17. DISTRIBUTION STATEMENT (01 the aba/,act e"to,ed III :::/1·/-:-0-C":"k-2""'::O-.':':/i:"'"d"::/':':/t:"'",,,-e-n-/·'·"r-on-,-::R-e-p-or-t)------------·---
18. SUPPLEMENtARY NOT.ES
17;19;-.-K;;;;'E';';Y"";:W:;';O:;";R:;';O::::S;:""(':::C::':o:":n7.
..-rj-'-."-'d:-l:-d:"'"e'~'I':':I/:-)''";b-)'-:b-:'o-c-::k-'-ll-ll/-::'')-P-'
1-----------------·-t '::'n-u,,--:"on--:"re-v-e-rs-e-,'";j-:dc- . .-:-:jI:"'"'-H'-c-,'s-.-·
Regression, Robustness, Sequential Analysis, Repeated Significance Tests,
M-Estimates, Weak Convergence
20.
ABSTRACT (Continuo on reverSf) side If
neCe8S8Tj'
·"o-e-'-=-)·----------·------·----
anc( Idenllfy by block--;;-"m-.
We obtain weak convergence results for bounded influence regression
M-estimates and apply the results to sequential clinical trials, with special
reference to repeated significance tests in the two-sample problem with
covariates.
DO
FORM
1 JAN 73
1473
EDITION OF I NOV 65 IS OBSOLETE
SECURITY CLASSIFICATION OF THIS PAGE (When Data
,
j
I
Ent~
.-J
UNCLASSIFIED
•
...
••
.
'
~
•
.'
.
·e