-e
ASUMPTOTIC PROPERTIES OF LIKELIHOOD RATIO TESTS
BASED ON CONDITIONAL SPECIFICATION
by
Pranab Kumar Sen
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1316
December 1980
ASYMPTOTIC PROPERTIES OF LIKELIHOOD RATIO TESTS
BASED ON CONDITIONAL SPECIFICATION*
Pranab Kumar Sen
Department of Biostatistics
University of North Carolina
Chapel Hill, NC 27514, USA
~~~arl'
The problem of testing a general parametric hypothesis following a
preliminary test on some other parametric restraints is considered.
tests are based on appropriate likelihood ratio statistics.
These
The effect of
the prelimianry test on the size and power of the ultimate test is studied.
In this context, some asymptotic distributional properties of some likelihood
ratio statistics are studied and incorporated in the study of the main results.
1.
Ill,!: rod'!.s,t:h0n •
In a parametric model, the underlying distributions are of assumed forms and
the parameter
fZ belongs to a parameter space
ZikeZihood function and let
hypothesis
H :
O
~ E W
w be a subspace
against
H :
l
~
•
n . Let Ln (6), 6
of n . For testing
~
~
E
n
be the
the null
the usual UkeUhood ratio test
W ,
(LRT) is based on the (log-) ZikeZihood ratio statistic (LRS)
L(0)
=2
n
and the null hypothesis
10g{6
I"W
sup
E
n Ln (6)/6s~Pw
rw
f'J
H is rejected when
O
Lnf'J
(6)}
(1.1)
LCO ) is significantly large.
n
This unrestricted LRT possesses some (asymptotic) optimal properties when
fZ is not restricted to some particular subspace of
*
n.
Work supported by the National Heart, Lung and Blood Institute, Contract
NIH-NHLBI-7l-2243-L from the National Institutes of Health.
-2-
n*(~
In certain problems of inference,
identified form some
~ E
setting
HO:
e~traneous
Q
*
E W (~
n
a subset of
n,
may be
considerations 7 and one may like to test
~
W , given that
n)7
*
En.
This may conveniently be made by
n*)
Q E n*\ w* ,
so that the corresponding
LRT is based on
(1.2)
and
L
n
H is rejected for significantly large values of
O
L
the restricted LRT based on
(asymptotic) optimality properties too.
inconsistent.
Q.
f n* ,
Ln
then
*
En,
~
L
n
Q.
E
L (0)
n
usually performs better than
n
under suitable regularity conditions, given
to the assumption,
When
n*
and,
possesses some
On the other hand, if, contrary
may become inefficient and even
Hence, one may not advocate the restricted LRT unless one
~
has high confidence in the assumption that
E
n* .
In a variety of practical problems, though some
n*
may be framed
from certain practical considerations, there may not be sufficient grounds
to enforce a restricted LRT.
At the same time, considerations
*
possible gain in power (when Q. En)
LRT over the unrestricted one.
of the
advocate the use of the restricted
Often, as a compromise, in such a case,
H* :
• •
a pre Z~~nary
test is made of
O
e
n*
(against
~ E ~£
*
HI:
e! n*)
~ ~ ~£
an d
an appropriate LRS is used (depending on the acceptance/rejection of
HZ).
For this preliminary test, consider the LRS
(1.3)
L*n
is rejected when
where
the distribution of
L*
n
(under
is ~
L*n,a* 7 the
H*) and
O
e
E
w is based on the test statistic
L
n
=
r
100a*%
point of
a * (0 < a * < 1) is the ZeveZ of
significance (size) of the preliminary test,
HO:
upper
L*n
Then the actual test for
L , where
n
*
Ln , i f
< Ln ,a*,
Ln(0) , i f L*n >- Ln a*
,
(1.4)
e'
-3and
L
are not independent (even under
n
H and asymptotically), it is clear from (1,4) that the distribution of
O
H )
(even under
of
(Ln ,
LCO )
n
'
O
L*
n
generally depends on
and
a*
L
n
as well as the joint distribution
Ln !actual1y, through the bivariate distributions of
L*) and (L(O) L*}].
n
n ' n
We are primarily concerned with a systematic
Ln • In particular,
study of the asymptotic properties of the test leased on
the effect of the preliminary test on the size and power of the ultimate
test is the main objective of our study.
In some special problems (arising
in the classical analysis of variance tests for some linear models with
normally distirbuted errors), the effect of
preliminary tests on the size
and power of some ultimate tests has been studied by Bechhofer (1951) and
Bozivich, Bancroft and Hartley (1956), among others.
Some nonparametric
procedures are due to Tamura (1956) and Saleh and Sen (1980), among others.
Sen (1979) has recently studied some asymptotic properties of maximum
likelihood estimators (MLE) under conditional specification.
results are incorporated here
in~he
main study.
These
In this context, the
(joint) distribution theory of correlated quadratic forms (in normally distributed
random vectors), studied by Jensen (1970) and Khatri, Krishnaiah and Sen (1977)
pa1ys a vital role.
Along with the preliminary notions, the proposed precedure is outlined
in Section 2.
Section 3 deals with the asymptotic distribution theory of
various statistics involved in the proposed testing procedures.
These
results are then incorporated in Section 4 in the study of the asymptotic
size and asymptotic power function of the test based on
L
n
in (1.4).
Some general remarks are made in the concluding section.
Bearing in mind that, typically, a mu1tisamp1e situation may be involved in
a preliminary testing problem, as in Sen (1979), we conceive of the following
-4mo d e.
1
k ( >- 1)
Let t h ere b e
~il""'~in.
be
ni
J.
i nd
den
t samp 1 es; f or t h e 1.th samp 1 e, 1 et
epen
independent and identically distributed random vectors
(i.i.d.r.v.) with a distribution function
~
where
E
EP
,
a = (aI' • •• , at) ,
E
~
8 ,H.,a
l
t
all
p
the
,
,
t~l
for some
for every
log Ln (a) = log Ln (X ,a) =
~
~
~ = (~ll""'~nk)
may not depend on
!2.
fi(~;~)
is associated
a
~
E
Q and
(with respect to
n.
~
and
1
L L
log f.(X
.. ,a) , a
1
~1J
i=l j=l
n = n +"'+n
1
k
~
~
E
(2.1)
Q
Suppose now that a subset
be specified by
w=
b(~)
{~ E
Q:
h(~)
= (hl(~), ••• ,hr(~»' =
An unrestircted MLE (~ ) of
~
a
~
,@ )
log L (X
H :
O
~ E W
~
for some r<t,
against
is an element of
HI:
~
(2.2)
n~~
, the restricted MLE is an element of
f
w.
Q, such that
= ~s~PQ log L (X ,a) ,
n~~'-
~
Q} ,
satisfies some regularity conditions, to be specified later on.
We are primarily interested in testing
while,
F
i
Then, the log-likelihood function is defined
~).
k
where
i=l, ••• ,k,
i=l, ••• ,k , but each element of
by
w (c Q)
• Note that
admits a density function
some sigma-finite measure
where
for
Further, we assume that for each
Fi(x,~)
i(=l, ••• ,k),
Fi(x;~),
l)-dimensional Euclidean space and
(~
Q ~ Et
with at least one df.
edf)
(2.3)
w, such that
(2.4)
Suppose now that a subset
where
Q* (c Q)
be specified by
g satisfies certain regularity conditions, to be specified later on.
Then, by (2.2l, (2,5) and the definition of
w* = {~
E
Q:
h(~) =
Q,
w* (= w n Q*) , we have
£(~) = O}
(2.6)
e'
-sA*
~
Let
and
~* be ~espective1y MLE o~ !l under
~ E
""l1
n*
and
!l
*
E W
Then, parallel to (1,1), (1,2) and (1,3). we have
L (0) ;::; 2 log L (X ,~ ) - 2 log L (X ,~ )
n
n ""'n ""'n
n ""'n ""'n
L
n
= 2 log Ln (X""'n ,~*)
- 2 log L (X ,~*) ,
n
n ""'n n
6
A*
2 log L (X ,6 ) - 2 log L (X ..6 )
n .....n""'n
n ""'n'n
(2.7)
(2.8)
(2.9)
For latter use, we also introduce the following statistic
Tn* = -Ln
+ L*
n
A ) - 2 log L (X ,~Y* )
= 2 log Ln (X.....n ,~.....n
n ""'n ""'n
(2.10)
Finally, we formulate the test function
follows.
'e
Let
a(O < a < 1) and
n
, corresponding to (1.4) , as
aO(O < a O < 1)
a * be defined as in (1.3)-(1.4) .
and
V
be positive numbers
Then, we take
1, i f L* < L* *, L
n
n
n,a
V
n
=
or L*
n
~
~
L(O)
L*
n,a*"."
L
n,a
~
n
L(O)O
n,a
(2.11)
0, otherwise,
where
L
n,a
and
L(O)o are respectively the upper 100a% and 100a O% points of
n,a
the (null) distributions of
and (2,11) is
L
n
and
L(O)
n
'
The size of the test of (1.4)
the~efore
(2.12)
and its power is given by
en (6)
= E6 {Vn }
.....
,
6
E
n
(2.13)
We are primarily concerned with the study of (2,12)-(2.13). For this
-6purpose, we introduce the following regularity conditions [adapted from
Aitchison and Silvey (1958) and Sen (1979)J:
[AI] n
n), for at least one
(both in
t
E , and for every
is a convex, compact subspace of
~l
+~2
i(=l" •• ,k),
(2.14)
dFiC~;60)
i
th
n and every i(=l, ••• ,k), Z.(6)
J. "'"
~ E
[A2] For every
~O
exists, where
f
EP
is the true parameter point,
log f.(x;6)
J.""''''''
Note that for the
density, the Kullback-Leibler information is
(2.15)
where
~
1.(6,6
)
1
0
t"'o.I
,....,
0, V6
E
,....,
[A3] For every
6
"'"
E
n
and the strict equality sign holds only when
n and i(=l,.,.,k), log
differentiable with respect to
s
s
ICas/a6ala6b2) log
where sl
~
0, s2
£,
K
~
~
fi(~;~)
is (a.e,) twice
e"
and
fi(~;~)1 ~ GsCx), V~ E EP , ~ En,
0, sl + s2 = s = 1,2 and 1
~
a, b
~
(2.16)
t, and where
G (x)dF.(x;6 ) < 00, Vi(=l,.",k) and s = 1,2 ,
s"""
1"'''''''0
(2.17)
Further,
lim .max
o~O
logf.(X..
J. ""'J. ;6)1
"'"
J.,a, b
]
~o
[It is also possible to avoid (2.18) by some alternative conditions, as in
Huber (1967) and Inagaki (1973), but, those will require in turn, some other
extra conditions on the first derivatives.]
[A4] For every i(=l, •• "k) and
~
En,
o
(2.18)
-70, 't/ a , b=1, •
We then define for each 1(=1,
t
,
•
t
•
,
t •
(2.19)
,k)
(2.20)
[AS]
~o
(1)
~e
(k)
are all continuous in
' ... '!!e
~
in some neighborhood of
and
I (n./n)~(i)
*
~e
[A6]
is positive definite (p.d.)
~o
i=l].
....0
(2.21)
lim (n-In.) = Pi(O < Pi < 1) exists, for every i(=l, ••. ,k) and
n+O
].
k
L
i=l
Pi
=1
•
n~,
Note that under [A6], as
(2.22)
'e
[A7]
h(Q)
possesses continuous first and second order derivatives with
~,V ~ E
respect to
n.
Let then
£~
[A8]
=
«(~/a~)h(~»)
(of order txr)
(2.23)
£e
is of rank r«t)
2(Q)
possesses continuous first and second order derivatives with
....0
[A9]
respect to ~ ,
V~
E
n.
Let then
D =
....e
....
JA,IOJ 12
e
....0
rAIl] t
>
« (~/aQ)g... ~»)
(of order txs)
(2.24)
is of rank s «t) ,
s+r
and the following matrix
(of order (s+t+r) x (s+t+r»
Note that
~,
f, E, f*, E*, f**
and
E**
are all symmetric matrices.
Finally, we denote by
Then, under the assumed regularity conditions, when ~ 0
A
""'1l
is asymptotically
ob tains,
(2.30)
-9Ln* ~
..-'k
r,
n
Ln:_ •
and
"""""'"
To study the asymptotic nature of (2.12) and (2.13), we need to study
(0)
first the asymptotic jo~nt distributions of (tand (L , T),
*
n
n
when the null hypotheses
For
L~O)
and
HO~
*
H
O
~
and H
O
n
mayor may not hold.
L:, we may directly use the results of Sections 3 and
4 of Sen (1979) with the allied restraints in (2.2) and (2.5), while for
we need to put the dual restraints in (2.6).
Let us denote by
(3.1)
~e-l)Be
= (:e*e* ~O
~O ~~O
(:e*e* - Be-I) = Q*e*'(-R*e*)-l.Q*e* , (3.2)
~O
~~O
~O
~~O
~O
- -l --1
-1= (:ee - ~e )~e <:ee -~e ) = QEi (-.Be) .Qe'
--
~O
= ~*
~O
(P
~e
~*
-
~O
~O
~O
-P**)B
~e
~e
~O
~O
~O
~O
~O
(P
-p**)
~eNe
~O
(3.3)
~O
(3.4)
~O
Then, proceeding as in Section 3 of Sen (1979) (with direct extensions for
the multiple restraints under consideration), we ob tain that under the
regularity conditions of Section 2,
L(O) = A'A(O)A
n
~n~
~n
+ 0 p (1)
L*
= /J; n~
'A*A
~n
+ 0 p (1)
L*
= A'A*A
+ 0 p (1)
n
n
"'0'"
Tn
=
"'n
A'M +
~n
~
0
p
(under H )'
O
(3.5)
(under H~),
(1)
(under H )
O
,
(3.7)
(under H )
O
,
(3.8)
where
A(O)i
~
A(O)=A (0)
~~O-
~
- - -~ !!e A=A
o
, ,A*B
. ", "'6 A*=A*
#'OJ
,..",
,
(3.9 )
~O
and
,A
. ", i"'8 A*=O
,..", •
~O
(3.10)
--*
Ln ,
~
From (2.30) and (3.5)- (3 .10)
we conclude that under the regularity conditions
of Section 2,
L(O)
n
-*
L
n
and, further,
here
2
Xr
V
Xr +s
~
2
L*n
I n and
2
*
n V Xs
(under H )'
O
V
and
I
-
nV
L
2
Xr
(under H*)
O
~
(3.11)
(under H )
O
(3.12)
are asymptotically independent under H ;
O
stands for a r.v. having the chi square d.f. with q degrees of
freedom (DF), and we denote its upper 100cx% point by
(2.11) and (3.11)-(3.12), we have, for
T
n,cx
2
X (cx).
q
Then, by
n~,
Xr2(cx), L*
n"'*
u.
-+
(3.13)
On the other hand, in general,
(3.14)
L(O)
so that
n
under H )'
and
L*n are not, generally, asymptotically independent (even
However, joint distrfuutions of correlated quadratic forms,
O
developed in Jensen (1970) and Khatri, Krishmaiah and Sen (1977) may be
incorporated here in the study of the joint asymptotic distrfuution of
L (0) and L*.
n
First consider
n
the case where
H
O
holds.
Consider the following
prd:> ab ility density function (pdf)
cx l ! cx 2 ! ~
rs;
a~ Ibl+a~ 'b2+a~
1jJ (.\! ;,e) L /.\! ;,e)
c
(3.15)
~
where
£
£
= (ul,u2)~0,
~u. bi~l
:2
JI {e
i=l
the Laguerre polynomials
(b l ,b
=
L~
;Lui
2
»Q,
~ =
(cx ,cx 2), m = (m ,m) ,
1
/ ~}, Q~~<~, b>O
are defined by
(3.16)
e-
-11-
a~
and the
are suitable coefficients,
Then, by (2.30), (3.5), (3.6), (3.9),
(3.14) and Theorem 2 of Jensen (1970), we conclude that under
regularity conditions of Section 2, for every
H and the
O
~~Q,
(3.18)
~(r,s))
where
is defined by (3.15).
In the above development, we have confined ourselves to the case where
an appropriate null hypothesis holds.
But to study the nature of (2.12)
and (2.13), we need to consider the case where
H or
O
H* may not hold.
O
For this purpose, we consider some local alternatives and study the
asymptotic behavior of the various LRS under such alternatives.
conceive of the following sequence {K }
n
"-
We
of alternatives
(3.19)
K :
n
where
Xl
and
X
2
are r- and s-vectors of real arguments and
true parameter point.
H :
O
Xl =
Q,
X2 =
Then, under
Q.
Then,
*
Xl
and
£a
*
X2
H*
O'
Xl = Q;
is the
X2 = Q and
Again, we basically follow the steps in Section 4
of Sen (1979) and define
-Xl =
H '
O
~O
*
Xl'
~O
*
Xl'
*
~l'
* -
X*
2 and
*
~l= ~e Xl' X2 =
~O
~O
£e
*
~2
*
*
Qa X2 , Q~O
e ~2
are both t-vectors, while
vectors, respectively.
by letting
~O
*
~l
and
A*
~2
*
~e X2
~O
(3.20)
are r- and s-
Under {K } in (3.19) and the regularity conditions of
n
Section 2, we have as in Section 4 of Sen (1979),
(3.21)
(3.22)
-12Thus, by (2.30), 0,1) (3.2), (3,9) and (3,21),..(3,22), we conclude that under
{K } and the regularity condit~ons of Section
n
L(O)
n
2
X
q,6.
where
2, marginally,
-+ 2
V Xr ,6. 0 and
(3.23)
stands for a r,v. having the noncentral chi-square d.f. with
q DF and noncentrality parameter
6., and
,
, *
*'
*
*
6. = Xl ~e Xl = Xl .Qe ~l = XI~1 = -XIE e Xl
~O
~O
~O
o
*'-
6.*
*'X2 ~e
*
(3.24)
,
*
**
X2 = -X 2Be X2
~O
~O
(3.25)
In a similar manner, it follows that under {K}
n
and the regularity conditions
of Section 2,
(3.26)
e'
X
where letting
B
v*
~e ~
~O
(3.27)
so that
-X'B e X
(3.28)
~O
Since
~
L
n
=
-L +L * , -L
n
n
n
L*~O, by (3.22), (3.23), (3.26), (3.28) and the
~O,
n
Cochran theorem, we conclude that under {K },
n
-
2
Ln V-+ Xr,u/I; 6. = 6.*-6.*
(3.29)
and furhter that
Tn
and L* are asymptotically independent under {K }.
n
n
(3.30)
The situation is somewhat different with the J'oint distribution of (L(O) L*)
n
'n
-13under {K }.
n
4It
Firstly, by (3.14), (3,21) (3,22) and (2.30), they are not
generally (asymptotically) independent.
Secondly, we have the non-central
case where Theorem 2 of Jensen (1970) may not apply,
However, we are able
to use the results in Khatri, Krishnaiah and Sen (1977)
and these provide
us with some exact as well as asymptotic expressions,
Note that by [All], (2.30), (3.21), and (3.22), we can use a
A
(not necessarily unique) non-singular transformation on
+
L(O) = z(l) 'z(l)
n
where
~n
0
~n
p
~n
suitable
and write
(1),
(3.31)
Z(l) and z(2) are r- and s-vectors and, under {Kn } ,
~n
~n
(3.32)
where
--
k and
Furhter,
L*
'f*
depend on
* Be*
* ~2'
!!e ' Q*e ' Qe** ' ~l'
~O
is non-singular.
~O
~O
~O
and
Be**
.
~O
Let then
(3.33)
I
.-R = ~r+s
The choice of
2
~
~
*
-1 Land
~
B
-1 rr ' •
.:c.<.
(3.34)
= .~
is arbitrary and the convergence rate of the infinite
expansion (to follow) depends on
-1/2
~O = I !-~ 1
2
and
'f*
Let then
exp{-1/2 trace[(!-B)
-1
B]}.
(3.35)
and for every k>O,
(3.36)
Finally, let
(3.37)
-14where
i
=
(jl~
coeff~cients ~i7 i~Q
j2) and the
Some formulae for the computation of the
Then~
Krishnaiah and Sen (1977),
by
~i
are suitable constants.
are also given in
(3.3l)~
Khatri~
(3.32) and by (2.11)-(2.12)
of Khatri, Krishnaiah and Sen (1977), we obtain that under {K } and the
n
regularity conditions of Section 2, for every
~
=
(xl,x2)~0,
lim p{L ~O) :S;x ,
n+O
l
(3.38)
x
~
=
f¢ * (~)d~ =
Q
where
¢j
is the d.f. corresponding to the pdf
11-gl
Note that by (3.34),
I-B
such that
~o=l,
by letting
is p.d.
=
Ik*l/o~o~
¢j'
j=1,2.
and we need to choose
In the central case (where
01'
02'
s=O) , we may take
r s IE*1 •
0102=
We shall study now the comparative performance of the three LRT's
based on
L(O),
n
Tn
L.
and
n
In addition to (2.11), we let
(0)
=
V
no.
{l.
=
{l.
V
nO.
n
~
L(0) ,
n,a
(4.1)
0, otherwise
~T
Tn
n,a
(4.2)
0, otherwise
and
v*no.'/(
L(0)
L*
n
"
{::
~
L:,a* ,
(4.3)
otherwise
where, by virtue of (3,13), in the asymptotic case, we may replace the exact
critical values
L(O)
n,a
and
I n,a
2
by X (cx)
r
and
*
* * by X. s2( a).
Ln~a
Note that under the regularity conditions of Section 2, for any (fixed)
and
a *:
o<
0.*
< 1,
Ee{V * *}
no.
'"
~l
as n~ ,
...15so that by (2.11), (4.1) and (4.3), for every (fixed)
Ie,..
~ Q*] ~
V
n
and
0
is asymptotically equivalent to vnNo.
(4.4)
."""
Ac
~*
~
~ u
Th e picture is different when
(or on a shrinking boundary of
v~~),
In fact, this is domain where we would like to study the behavior of
v
na
and
v •
n
Towards this end, we consider
Q*).
the set of alternatives {K }
n
* and HO are respectively characterized by Xl=Q,
in (3.19), so that HO' HO
and
X2=Q
Xl=Q, X2=Q.
Note that by (4.1) and the results of Section 3,
E
I
(0) H } = a, whatever
e {vna
.....0
Also, note that by (3.29), for
--
O
X2
may be •
(4.5)
Xl=Q,
(4.6)
where the equality sign holds when
X2=Q.
Thus, by (3.29) and (4.2),
(4.7)
where the equality sign holds when
~=O
(i.e.,
X2=Q).
This explains the
lack of robustness of the restricted LRT based on In'
of course,
Xl=Q,
H :
O
X=o
and the size of the test based on I
when nothing is specified of
unless we feel that
~,defined
X2'
n
Under
is
a.
HO (i.e., Xl=Q, X2=Q),
But, under
this may be generally
~
a.
Thus,
by (4.6) is very close to 0, the use of
the restricted LRT may result in a significance level greater than the specified
level
a.
values of
e
The discouraging fact is that for the set of
X2
6,
leading to large
E (Vnalx1=Q) tends to 1, so that the restricted LRT may even
e
.....0
be inconsistent against such alternatives.
Now, by (2.11), (3.11). (3.12) and (3.18), we obtain that
-16(4,8)
where
is defined by (3,15)-(3,17) and the coefficients therein
~(",)
~*, ~(O)
and ~e
(and these may be evaluated by a procedure
"'0
suggested in Khatri, Krishnaiah and Sen (1977». Alternately, the right hand
depend on
a(l-a*) + a*~aO, which may be
side of (4.8) is bounded from above by
equated to the desired
a.
Actually, if both
very close to (but less than)
a
and
a*
a
and
a O are chosen
is small, then this upper bound
provides a close approximation, or even, in (4.8), the series in (3.15) may be
approximated by a few terms.
v =0
,./.,1
'"
case, where
but
Parallel to (4.7), we now consider the
Then, by (2.11), (3.19) and the
may not be Q,
'X2
results of Section 3,
Ee {vnlxl=O}
+
"'0
defined by (4.6) is
~,
~O'~i' i~Q
on A(0)
'"
,
as well as
*
~
<
x;(a*)}p{x~,~ ~ X;(O)}
(4.9)
~Okj~0~i[1-~1(x;(aO);r/2+jl)] [1-~2(x;(a*);s/2+j2)]
+
where
P{X;,~*
,
and
~e
"'0
~l'
~
~2
X2 '
o
(with the equality sign for
,
and
'X =0)
2 '"
are defined as in (3.38) and these depend
In this context, note that
(4.10)
-
2
(a ~)p{~
2-
,X ~~(a)}
r in -
~(~
is
0),
(4,11)
and the second term on the right hand side of (4.9) is bounded by
2 A*
a O"p{
, Xs,u
~
2
X (a,*)} •
s
Thus, unlike (4,7), though (4,9) is affected by
to 1 as
~
or
against
'X2 f Q, than the restricted LRT,
~*
blows up,
X2 fQ
(4.12)
it may not converge
Or, in the other words, it is more robust
Hence, from considerations of
e-
..17...
validity-robustness,
~
V
n
If, in particular,
may be preferred to
~(O)~e ~*
v
na
is a null matrix, then (4.9) reduces to
....0
2
2
22~{Xs,6* < Xs(a*)}P{Xr,~ ~ ~(a)
+
(4.13)
and the validity-robustness picture becomes more clear.
In this case,
we have
(4.14)
a o= -a=a, one may choose
so that on letting
a*
arbitrarily.
Let us now proceed to the study of the asymptotic power functions of
.~
the three LRT's.
As in (4.4), for any fixed alternative, there is not
much interest in studying these (as the limits dagenerate at
a
or
1),
and hence, we confine ourselves to local alternatives, as in (3.19), for
which the limits are different from 1.
First, consider the case of the unrestricted LRT
From
(3.21), (3.23) and (3.24), we obtain that
(4.15)
where
60
is defined by (3.24).
lim E{V
Il-+<x>
where
6
Similarly, by (3.28) and (3.29),
IK}
(4.16)
nan
is defined by (3.29),
For a comparison of (4.15) and (4.16),
we may note that by
...D )] ...1
. . eo
(4.17)
-18-
(4.18)
Thus,
(4.19)
(4.20)
Qe )
e ) li-el(£e
0 ~O ~O
where writing
I
=
(£e Q
~O ~O
=
[~ll ~12]
~2l
~22
,
we have
-1
-1
-1
-k12k22 and k* = (fll-f12f22f2l)
(4.21)
Hence, from (4.19), (4.20) and (4.21), we have
(4.22)
From (4.22), we immediately claim that
X2
Hence, if
H* :
O
power (against:
LRT
)0).
no.
example, if
):2
=
Q => "X ~ /),0, with = holding for f 12
=Q
grea t er th an or equa 1 t 0 th a t
The picture may be different when
X2
+Q
but
Xl
= Ir2 ,
(4.23)
v no. has an asymptotic
holds, then the restricted LRT
._
~
h(~._.) -- n-1/2Yl)
o.
.....
0
f th e unres t r1C
. t ed
H* may not hold.
O
then by (4.22),
/), = 0, /),0
For
> 0, so
that the unrestricted LRT performs better than the restricted one.
In general,
(4.24)
where
chI
stands for the largest characteristic root,
neighborhood of
£X 2 ,
this may not hold,
Clearly, in a
This explains the lack of
efficiency-robustness of the restricted LRT, when
H* may not hold.
O
e-
-19For the preliminary test LRT
lim E{V
n-+oo
n
v
n
in (2.11)7 we obtain that
IK}
n
(4.25)
2
+ ph S,w"*
2
where (XS,w"*'
has (jointly) a bivariate chi-square distribution
(non-central
given by (3.37), with the coefficients depending on
!:J.* , !:J.O
and
-R
•
~e
o
If, in particular,
A(O)i
A*
ro.J6 I¥
t""tJ
~O
is .9, then (4.25) reduces to
2
phr,w"0
(4.26)
so that by arguments similar to in (4.22)-(4.24), we conclude that (4.26),
lies in between (4.15) and (4.16).
In particular, if
a
o = a = a,
then
(4.26) reduces further to
--
P{x
2
,,0
r,w
(4.27)
I
which is an weighted average of (4.15) and (4.16).
(for
b(O)1!e ~*
In general,
not necessarily Q), the second term on the right hand
~O
side of (4.25) can be evaluated by using (3.38) and it may be concluded that
v(O) and v
, and
na
na
is more (less) efficiency-robust than V (v(O)) when H*
the asymptotic power of
further,
v
n
v
a
lies in between that of
n
n
O
may not hold.
From the results of Section 4, it follows that unlike
the case of the
unrestricted LRT, for the preliminary test LRT, the computation of the
size
when
.-
needs elaborate expansion as in (3.38).
is
O·
~,
The situation becomes simpler
the later case arises in many linear models, where
-20the design matrix permits thjs condition.
size and power affected by the
than
V
v n may be preferred to
hC.~)
being true, but
V
and
(0)
and
n
V na
But,
v(O)
na
or
V
na
V
n
and
have
n
is more robust
Thus, from validity-robustness
V
On the other hand, for
na
•
the asymptotic power of
.[x 2 , in which case,
V
= Q.
v na
V
n
H*
O
is better than that
H* may not hold
O
v n performs better than
Thus, from the efficiency-robustness point of view,
to
V
' although a different picture may emerge when
is close to
Xl
+Q,
H* '
O
of
~(!)
against departures from
na
point of view,
of
valid~ty
Also, both
V
na
.
may be preferred
na
The actual computations of the asymptotic size and power function of the
three LRT depend on
may however be done.
:Be~O
as well as
X2 •
In some simple case, this
For example, for testing for the intercept parameter
(when the regression parameter mayor may not be equal to 0) in a simple
regression model (which includes the two-sample location model as a
special case), this comparative picture is very similar to the nonparametric
case dealt with in Saleh and Sen (1980).
over
1.
V
na
Aitchison, J. and Silvey, S.D.:
Bechhofer, R.E.:
V
n
may be seen from the numerical values presented there.
( or V(O))
na
Maximum likelihood estimation of
parameters subject to restraints.
2.
A definite advantage of
Ann. Math. Statist. 29, 813-828, 1958
The effect of preliminary test of significance on the
size and power of certain tests of univariate linear hypotheses.
UnpubZished Ph. D. Thesis, Columbia University, 1951
3.
Bozivich, H., Bancroft, T.A. and Hartley, H.:
Power of analysis of
variance tests procedures for certain incompletely specified models, I.
Ann. Math.
4.
Statist~
Huber, P.J.:
27, 1017-1043, 1956
The behavior of maximum likelihood estimators under
nonstandard conditions.
221-234, 1967
R---
P~oc.
5th BerkeZey Symp. Math. Statist.
~ob.
1,
e-
-215.
Inagaki, N.:
Asymptotic relations between the likelihood estimating
function and the maximum likelihood estimator.
~
Ann. !nst. Statist. Math.
25, 1-16, 1973
6.
Jensen, n.R.:
The joint distribution of traces of Wishart matrices
and some applications.
7.
Khatri,
e.G.,
Ann. Math. Statist. 41, 133-145, 1970
Krishnaiah, P.R, and Sen, P.K,:
distribution of correlated quadratic forms.
A note on the joint
Jour. Statist. Ptann. Inf. 1,
299-307, 1977
8.
Saleh, A.K. Md.E. and Sen, P.K,:
Nonparametric tests for location after
a preliminary test on regression.
9.
Sen, P.K.:
Asymptotic properties of maximum likelihood estimators
based on conditional specification.
10.
Tamura, R.:
~
.-
Ann. Statist. 7, 1019-1033, 1979
Nonparametric inference with a preliminary test.
Math. Statist. 14, 39-61, 1965
--
(to be published), 1980
Butt.