ON ALIGNED M-TESTS IN LINEAR MODELS
by
Pranab Kumar Sen
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1346
...
ON ALIGNED M-TESTS IN LINEAR MODELS *
by
Pranab Kumar Sen
University of North Carolina, Chapel Hill
For a simple regression model, some M-test based on an
alignment procedure are
are studied.
co~sidered
and their asymptotic properties
The theory is extended to subhypothesis testing in
some linear models.
~.
Consider the simple regression model
x.1
= e1
+ Sc. 1
+ e.,
where the
4It
X.
are observable random variables (r.v.),
1
S (slope) are unknown parameters, the c i
and the errors
e.
1
R.
(intercept) and
are independent and identically distributed (i.i.d.)r.v.
F, defined on the real
It is desired to test for
H :
O
treating
e
are known regression constants
with continuous, unknown distribution function (d.f.)
line
(1.1)
i=I, ••. ,n,
e
S
=
0
against
HI:
S
~
(or> or
<) 0,
(1.2)
as a nuisance parameter.
AMS Subject Classifications:
Key Words & Phrases:
62F35, 62J99
Alignment, asymptotic normality, asymptotic stochastic
li,nearity, linear models, M-tests, Pitman-efficiency, robustness
*Work
partially supported by the National Heart, Lung and Blood Institute,
Contract NIH-NHLBI-7I-2243-L from the National Institutes of Health.
2
A variety of rank tests for (1.2) is available in the literature
[viz., Hajek and ~idak (1967)].
the nuisance parameter
8
Since the ranks are translation-invariant,
is of no concern for these tests.
The situation
with the tests based on M-estimators is some what different.
testing for (1.2) is to estimate simultaneously
the test on the estimate of
e
8,
e
and
One way of
then to base
and its estimated standard error.
Alternatively,
as in Schreder and Hettmansperger (1980), one may employ some robust
likelihood ratio type criteris.
theory, one needs to estimate
In either case, to apply the asymptotic
E~
2
and
E~',
~
where
is the score function
(presumed to be absolutely continuous) generating the estimates.
Actually,
in Schrader and Hettmansperger (1980), following Huber (1973), it has been
assumed that
~
has bounded derivatives (at least up to the second order).
In addition, for testing
H against one-sided alternatives, these
O
likelihood-ratio type tests may not be fully efficient.
developments on
~he
In recent
·v
~
asymptotic theory of M-estimators, Jureckova (1977,
1980), Yohai and Maronna (1979), Jure~kova and Sen (1981) and others have
eliminated the boundedness conditions on the derivatives of
~.
possible (finitely many) jump-discontinuities of
~
and incorporated
This should also be
the case for the dual testing problem.
The object of the present investigation is to focus on a simple alignment
E~',
procedure which eliminates the need of estimating
incorporates a
larger class of score functions and provides a computationally simpler and
asymptotically equi-efficient M-test for (1.2).
Along with the preliminary
notions, the proposed M-tests are considered in Section 2.
properties of these tests are studied in Section 3.
Asymptotic
In the concluding
section, the results are extended to some general linear models.
l~ Ali~ed ~-!~~!§.
For every
(M (!), Mn2 (!»by letting
n1
!
=
(t , t 2 )
1
€
2
R ,
define
M
(t)
"'It ....
=
3
(2.1)
(2.2)
where
W is some score function.
We assume that
W(x) = W (x) + W (x) ,
1
2
where
WI
x
(2.3)
R ,
€
R, and
is absolutely continuous on any bounded interval in
for the step-function
open-intervals
J
W'
2
we assume that for some
r = (a r , ar+1)'
0
~
~
r
p, with
0), there exist
p(~
_00
= a
O
< ••• <a
p < a p+1 = +00,
such that
Vx
where the
we let
~
WI
d
W2
=_ ~(d r + d r - 1
)
'
For
p
~
1, conventionally,
Further, we assume that
both
are non-decreasing and skew-symmetric, so that
W.(-x) = -W.(x),
J
1 ~ r ~ p.
(2.4)
r=O, 1, ... ,p ,
r
are real numbers, not all equal.
r
w2 (a r )
and
J
€
J
Vx
and
symmetric about 0, so that
Let . (~ , ~)
n
n
j=1,2.
The d.f. F in (1.1) is assumed to be
J W(x)dF(x)
=
o.
R
be the M-estimator of (8, 6), i. e., (~ , ~)
n
n
of the system of equations
is a solution
M (t) =0, or
""I1 "'"
"'"
(2.5)
Also, let
of
"'"
8
n
be the M-estimator of
M (t, 0) = O.
n1
Since
M (t, 0)
n1
e
is
when
\
6= o
in t,
"'"
en
1. e. , "'"
8
n
is a solution
may conveniently be
written as
~[sup{t:
M (t, 0) > O} + inf{t:
n1
M (t, 0) < O}) •
n1
(2.6)
The solution in (2.5), however, may have to be obtained by the trial and
error method.
If
Q(x)
be defined by
Q'(x)
= W(x),
then the Schrader and
4
Hettmansperger (1980) procedure rests on
8n ) -
2YO- 2{ln l(Q(X. 0
i=
1
Q(X - ~
i
- ~ c ))}
n i
n
(2.7)
where
J ~2(x)dF(x)
Y
= -J
R
su~table
needs to substitute
Y<
0 < 0 ,
0
and it is assumed that
~(x)f'(x)dx
(2.8)
R
00.
In practice, their testing procedure
estimators of
00
and
Y in (2.7) and rests
on the asymptotic chi aquare distribution of (2.7), with one degree of
freedom (D.F.), when
large
n,
H
O
holds.
Also, for the solution in (2.5), for
we have [viz., Yohai and Maronna (1979)]
(2.9)
where
(2.10)
and hence, using suitable estimators of
°0
and
Y,
an asymptotic test
for (1. 2) can be based on (2.9).
The procedure to be considered here is quite simple and does not require
the estimation of
Y or
(~ n , ~ ).
An = Mn 2(8n ,
n
Let
(2.11)
0) ,
T
n
=
An (c*s
)-1
n n
(2.12)
We propose to use
T
as a test statistic.
the properties of
T
and show that for local (contiguous) alternatives,
n
n
In the next section, we study
the asymptotic relative efficiency (A.R.E.) of
the earlier test is equal to 1.
its computational simplicity.
T
n
Thus the A.R.E. of
with respect to either of
T
n
is not affected by
Further, like the test based on (2.9),
may be employed for both one and two-sided alternatives.
T
n
5
absolutely continuous probability density function (p.d.f.) f
having a
finite Fisher information
I(f) =
f
(3.1)
{f'(x)/f(x)}2dF (x)
R
Note that (3.1) insures that
f
If'(x)ldx <
00.
Concerning the constants
R
c
i
in (1.1), we assume that
(3.2)
lim -1 *2
*2
n C = C :
n-+=
n
lim -c
n-+= n
(Note that
c
n
= n -lIn1=
. lc .. )
1
~
0 < C* <
00
I-I
c
= c exists, where
(3.3)
,
<
00
•
(3.4)
Then, as a basis of our subsequent analysis,
we quote the following result due to Jure~kova (1977):
Under
H :
O
6
=
0,
for every
0 < K<
K:
00
,
1
sup{n-~IIMn«8, 0) + ,t) - Mn«8, 0) + yn,tYll:
_1
11,t II ~ n ~}
-+-
P
0 , as n
-+-
00
,
(3.5)
where
(3.6)
Further, from Yohai (1974), Jure~kova (1977) and Jure~kova and Sen (1981),
ir follows that under the assumed regularity conditions,
1
I n~<8 n - e)1
~ A
In (tJ n -
= 0 (1)
p
e, ~n
- 6)1
when
H holds ,
O
= op (1)
•
(3.7)
(3.8)
6
From (2.6) and (3.5) through (3.8), we obtain that under
H :
O
S = 0,
as n
+
00
,
n-\tn 2<'8n ,
0)
(3.9)
M (8, S) has the same (J'oint) distribution as M (0)
where under (8, S),
under
*
H :
O
8
"'Il
= S = 0,
"'Il ""
and further under H* '
O
(3.10)
Thus, by (3.9) and (3.10), under
S=
H :
O
°,
(3.11)
so that by (3.3) and (3.11),
(3.12)
S2
The consistency of
n '
as an estimator of
and Sen (1981), and hence, by (3.12), under
T
n
2
v
~
00' follows as in Jureckova
H '
O
(3.13)
+N (O, 1).
1
V
By (3.13), the asymptotic critical values of
T
n
can be taken as the
appropriate percentile points of the standard normal d.f.
asymptotic power properties of
local alternatives
{HO}
n
°
H:
n
T ,
n
To study the
we confine ourselves to a class of
specified by
S = n-~ A,
where
A(~
0) is fixed.
(3.14)
Note that under (1.1), (3.1), (3.2)-(3.4) and (3.14), the contiguity of the
7
(sequence of) probability measures under
..
v
..
as in Hajek and Sidak (1967, ch. 6).
as
to those under
H follows
O
(3.9) and this contiguity imply that
n -+ 00,
(3.15)
Also, by (2.1), (2.2), (3.10) and the contiguity established before, under
{HO}
n '
(3.16)
where
(3.17)
and
{HO}
n
y
is defined by (2.8).
Thus, by (3.15), (3.16) and (3.17), under
'
A jC*
n
n
2
N (/,.yc*, °0)
•
1
V
-+
Finally, the stochastic convergence of
S2
n
(3.18)
2
(to 00) under
, under . {HO}
aforesaid contiguity insue that
n
H and the
O
as well.
Thus,
by (3.18) and the above, under
(3.19)
The efficacy of . {T}
n
(in the Pitman-sense) is therefore equal to
2 *2 j 2
°0
(3.20)
Y C
and this agrees with the efficacy of the test based on
~ n or the
likelihood ratio type test, considered by Schreder and Hettmansperger (1980).
Hence, the aligned M-test based on
T
n
is asymptotically equi-efficient
(for Pitman-type alternatives) with respect to either of the other tests.
8
linear model
1
~' =
where
q-vectors.
(SI' ••. ' Sq)
Let
r + s = q.
with
q(~
for some
~'= (~i' ~2)'
1)
~I
where
$;
i
$;
and the
and
(4.1)
n ,
~2
£i
are
are specified
rand
s
vectors,
By a canonical reduction, we may consider the following
linear hypothesis
(4.2)
~I
where
is treated as a nuisance parameter (vector).
The aligned M-test for (4.2) is very similar to that in Section 2.
We partition' £1
we have
~I
Xi
as
= ~i£iI +
(£11' £12)' 1
ei , 1
$;
i
$;
n.
$;
i
$;
Let
n, and note that under
HO'
be the M-estimator of
(under H ) i.e., it is a solution of
O
(4.3)
Let then .
(4.4)
1,2 ,
C
""'Ilrs
*
fn = fn22 - fn2IfnllfnI2 '
(4.5)
(4.6)
(4.7)
and
(4.8)
If we assume that
9
lim -1 C*
n-+OO n "'Il
= £*
exists and is positive definite
(4.9)
and (3.2) and (3.4) hold coordinate arise, then under the assumed regularity
conditions on
on the
wand
F
and the additional concordance-discordance condition
£i' due to Jure~kova (1977), we are able to make use of her Theorem
4.1 and proceeding as in Section 3 obtain the following:
(i)
Under
HO'
2
T
n
has asymptotically the central chi square distribution
with s D.F.
Under
(ii)
o
{H }, where
~2
n
= n -~~,
has asymptotically
a non-central chi-square d.f. with s D.F. and noncentrality parameter
(4.10)
By virtue of these results, it follows that
2
T
n
has the same A.R.E.
as the likelihood ratio type test considered by Schrader and Hettmansperger
(1980).
4It
On
the other hand,
unrestricted M-estimator
2
T
n
~ of
does not need the computation of the
g
in (4.1) nor the estimator of y •
REFERENCES
[1]
Hajek, J. and ~idak,
Academic Press.
[2]
Huber, P.J. (1973). Robust regression: Asymptotics, conjectures and
Monte Carlo. Ann. Statist. 1, 799-821.
[3]
Huber, P.J. (1981).
[4]
Jure~kova, J. (1977).
[5]
Jureckova, J. (1980). Asymptotic representation of M-estimators of
location. Math. Oper. Statist., Ser. Statist. 11, 61-73.
[6]
Jure~kova, J. and Sen, P.K. (1981).
[7]
Schrader, R.M. and Hettmansperger, T.P. (1980). Robust analysis of
variance based upon a likelihood ratio criterion. Biometrika, 67,
93-101.
z.
(1967).
The Theory of Rank Tests.
Robust Statistias.
New York:
New York:
John Wiley.
Asymptotic relations of M-estimates and R-estimates
in linear regression model. Ann. Statist. 5, 464-472.
v
..
Sequential procedures based on
M-estimators with discontinuous score functions. JoUP.
Statist. PZan. Inf. Z, in press.
10
Ann.
[8]
Yohai, V.J. (1974). Robust estimation in the linear model.
Statist. 1, 562-567.
[9]
Yohai, V.J. and Maronna, R.A. (1979). Asymptotic behavior of M-estimators
for the linear model. Ann. Statist. 1, 258-268.