•.
'
.
~'"
..
ON THE ASYMPTOTIC DISTRIBUTION
'
OF THE
LIKELIHOOD RATIO IN SOME PROBLEMS
ON
MIXED VARIATE POPULATIONS
by
J. Ogawa, M. D. Moustafa and S. N. Roy
p~,.~,~
This research was sponsored
the
Office of Naval Research undert~ ~ct
No. Nonr-855( 06) for research :ili';r~babil­
ity and statistics at Chapel HilL/and partly
by the United states Air Force through the
Air Force Office of Scientific Research
of the Air Research and Development Command,
under Contract No. 18(600)-83. Reproduction
in whole or in part for any purpose of the
United States Government is permitted.
Institute of Statistics
Mimeograph Series No. 180
August, 1957
..
..",'1
i
i·
.Aw ''0 bO. ·a.n--A
ON THE ASYMPTOTIC DISTRIBUTION
•
OF THE
LIKELIHOOD RATIO IN SOME PROBLEMS
ON
MIXED VARIATE POPULATIONS*
by
J. Ogawa, M. D. Moustafa and S. N. Roy
Institute of Statistios, University of"NorthCarolina
Chapel Hill, N. C.
Summary and Introduction
Let the likelihood function of the population under consideration
be P(X tHo) and P(X J H) under the hull-hypothesis Ho and the alternative H respectively, and put
A. = max P(x
1Ho )
/ max p(x 'H)
Then it is well-known that, under certain conditions, the random
variable -2 logeA has the X2-distribution with suitable degrees of
freedom in the limit as the sample size tends to infinity, provided
the null hypothesis H is true.
o
S. S. Wilks ~6_7 stated this result early in 1938, and gave a
sketch of a proof based on J. L. Doobls work ~9_7.
*
•
Later, in 1943,
This research was sponsored partly by the Office of Naval
Research under Contract No. Nonr-855(06) for research in
probability and statistics at Chapel Hill and partly by the
United States Air Force through the Air Fbrce Office of
Scientific Research of the Air Research and Development
Command, under Contract No. 18(600)-83. Reproduction in
whole or in part for any purpose of the United States
Government is permitted.
2
A. Wald ~5_7 obtained the same result starting from somewhat stronger
assumptions.
However, as far as the present authors are concerned,
they have not so far seen any complete proof of this proposition
along Wilks' line, or, in other words, a complete proof based upon
Doob'e paper.
Thus it seems worth while to give a completely
rigorous proof of this proposition along the lines stated.
In this
note, the authors are mainly concerned with the asymptotic distribution of the statistic -2 log X defined, as before, for testing nulle
hypotheses of certain types on a mixed variates population which was
discussed by one of the authors in another paper ~3_7.
Toward that
end, in section 1, the authors present a rigorous proof of the fact
that -2 log X on Doob's assumptions has an asymptotic X2.distribution~
In section 2 they establish a theorem which guarantees the consistency
and the uniqueness of the solution of the maximum likelihood equation
and some discussion on the consistency of the maximum likelihood
estimate.
In section 3 the authors explain the problem of testing
hypotheses in the 3-variates (X,Y,Z) population, where X,Y are
continuous and Z is categorical.
Then in section
4,
the validity
of Doob'e conditions and the assumption of Theorem 2.1 will be
verified in the case of the population which was explained in
section 3.
1.
Doob's Theorem and Wilks' Theorem.
We shall start with the
definition of the nth approximation of the maximum likelihood
•
estimate of a Earameter or simply the maximum likelihood estimate
of a parameter calculated from a random sample of size n.
For the
sake of simplicity of explanation, we shall begln with the single
parameter case.
For each value of Q in a non-degenerate point set
3
~ , wh~ch
is called the parameter space, let f(x,e) be a probability
density on the infinite interval -co < x <
00.
Let x be a chance
variable whose distribution is determined by the probability density
f(x,9 0 ), which is called the true density.
Without any loss of
generality, we can assume that 9 ... 0, and put
o
f(x) ;: f(x,O) •
Suppose that for each set of numbers xl' x 2' ••• , xn ' n'" 1,2, •••• ,
it is possible to find a value of 9 in
1. e., a function of
Jl,
Xl' x 2' ••• , xn ' such that
n
1T
(1.1)
j.l
f(x.,9)
J
We call 9 the nth approximation of the maximum likelihood estimate
of Q or simply the maximum likelihood estimate of Q calculated
fro~
the first n observations.
In the following, we shall assume that the likelihood function
Jr f(x.,9)
has a relative maximum at 9 ... ~ for fixed xl ,x , ••• ,x •
2
n
J
j-l
Proposition 1.1 (Doob)
tQ\ ~
Bpal > 0
For each value of Q in some neighborhood
of Q ... 0 (which is the true value of e), let
f(x,Q) be a probability density in the infinite interval
-00
•
< x <
(D.
Let the true distribution of x be determined by
the true probability density f(x).
Suppose
(i) that log f(x,9)
can be expressed in the form
2
(1. 2)
log f(x,9) ... log f(x)
+
9.a(x)
+
~ ~(x)
+
y(x,9) ,
4
where a(x) rex), a2(x) rex), and ~(x) rex) are Lebesque measurable and
integrable over
x
< 00 and where
~
y(x,Q)
09
(1.3)
exists for
(ii)
<
-00
I
Q }
~
8
2
~ al , a2
• y (x,Q)
0
that if
(1.4)
p(x). L. U. B.
~I
0<I91::a22
then ¢(x) f(x) is integrable over
(iii)
(1.5)
and is continuous at 9 • O.
> 0,
Yo(x;Q)
9
-00
J'
I
< x < co;
that if 5(x,9) is defined by
f(x,Q) • f(x) [ 1 + Qa(x) +
then
(1.6)
l'~m
1
-2
Q ->0 Q
f
f. L~(x)
+ a(x)] + 6(X,Q)] ,
OJ
6(x,Q) f(x) dx • 0 •
-00
Then
2
a (x) f(x) dx +
(1. 7)
f" ~(x)
-co
•
Suppose that
(1.8)
0
2 •
f
-co
.
to
a 2(x) f(x) dx > 0 .
f(x) dx • 0
•
If the maximum likelihood estimate of Q is consistent, i.e.,
(1.9)
lim 15 F(
I 1\ I>
Q
*
s) • 0
n->oo
for every s > 0, then it follows that
1
1
(1.10)
lim 15
F
«(1
-/\
n 2 Q < A) = lim
n->oo
n->co
EF«(1
-1\
2
n
e<
t2
x)
.--1--JA ;'2 dt
f2n
iHfo
"
-00
for every constant A, uniformly in A.
In the foregoing statements, it is to be observed that (1.7)
and (1.10) are the conclusions, the rest being all premises or
assumptions.
For the sake of convenience of later use, we shall state the
same theorem in the multiparametric case and shall prove it.
Proposition 1. 2.
For each value of Ql • (Ql' ••• , Qs ) in the
I = /1'E.i=~ Qi
a1' 8 1 > 0 of ~ = .Q , let f(x,~)
be a probability density in the whole space Rm of the m-dimensional
neighborhood 'Q
~
Euclidean space. where
Xl • (Xl' x 2 ' ••• , x m)
m
denotes a point of R.
Let the true distribution of X be determined
•
*
For the definition of the probability measure PF see Doob ~2 7.
Since we have not assumed the measurability of the maximum likelihood estimate of e, it is necessary to express the consistency
in terms of the outer measure 1F •
iHfo
fF stands for the inner measure.
6
by the. probability density
'-.
...
t(x)
= t(x,O).
......
Suppose
(i) that log
f(~t~) can be expressed in the form
where
a (x)
s -
and ~(~) t(!), i(.!) ~,(~) t(.!) and ~(.!) t(!) are all Lebesque
measurable and integrable over Rm , and where
(1.12 )
exist for
Q
at
@ ...
I~ {~
a 2 :5 aI' 8 2 >
° and are continuous functions of
.Q
(ii) that if
i .. 1,2, ... ,6
(1.13)
,
,
1
(iii)
that if
f(!,~)
5(!,~)
is defined by
• f(!) [ 1 +
~ S!(!) + ~ 'D(!) + S!(!) S!' (!>]~ + 6(!'!!~'
I
then
(l.lS)
-
dx = 0
'!hen we have
fam
(1.16)
-
D(!) + S!(!) S!'(!)J f(!) dx = 0
•
Suppose that the matrix
(1.11)
V-
-J: ~(!) f(!) ~
am
is positive difinite and symmetric.
estimate
1\
~
If the maximum likelihood
is consistent, i.e.,
1\
lim 15F(
n->oo
J
~\>
&) • 0
for every e > 0, then
(1.18)
.
'r VI1-'21
S
(2n)'2-
dj: ,
8
for every constant vector
Proof:
Since
f(!,~)
Rm
(1.19)
Q
uniformly in
•
~
is a probability density, we get
{
for all
~
t(!,~)
d! • 1
in the neighborhood 1~
t~
a • Thus we have the relation
l
~.J!!(!)t(!)d! ~'[~(!).!!(!!)!!'(!)]t(.1!)d!'~
+
m
R
(
m
R
+ ~_o(!,~)f(!)~
=
Rm
If we choose
~
0
in the neighborhood in such a way that
e
= 1,2, ••• ,s ,
then the
( 1.20)
Dividing (1.20) by e and letting e
.~
0, we obtain on account of the
oondition (1.1,) thart
-
a.(x) f(x)
Rm
*
For two vectors
~
~
-
- = 0,
dx
and,£, the notation
i
= 1,2, ... ,s ,
~
<
E
simultaneous inequalities al<b l , a <b ,· •• , as<b
2 2
s
stands for the
9
or, in vector notation,
(1.21)
=
{!!(!) f(!) dx
Rm
Dividing (1. 20) by
6
0
2 and letting e
->
0, it follows on account of
relations (1.15) and (1.21) that
f
(1.22 )
m
D
R
Next, choose
~
i i(!) +
a~(!lJ
=
f(!) cJ:!
0, i .. l, •.• ,s •
as follows:
~'
= (0
\.!t
&
••• 0 eO •••
o eO
... 0)
Then, in view of (1.21) and (1.22), the relation (1.19) can be
expressed as
(1.23)
•
I L~
2
m
ik(!)+ai (!) "k(!)Jf(!)cJ:! + {
Rm
R
Dividing (1.23) by e 2 and letting e
(1.24)
->
.. 0
0, we get
~R L~ik(!)+"i(!)"k(!)_7
f(!!:) cJ:! • 0
m
B(!!:,~)f(!)cJ:!
for irk •
Now we can assume, without any loss of generality, that
(1.25)
f~ik(!)
-
Rm
f(!) cJ:! •
~iBik ~i
'
> 0
,
i .. 1, .•. , s .
In'fact, if (1.25) does not hold, we shall introduce a new set of
parameters
(1.26)
i
defined by
.
10
where B is an orthogonal matrix such that
(1.27)
BI V B
III
D
~
and where
D~
. ['1 0l, '1 0.
~s J
o
>
Since we have assumed that the matrix V is positive definite,
there certainly exists such a B.
For such a new parameter systemi, the probability density
function becomes
f*(~,i) - f(~,Q) [
1 +
itBI.!!(~) + ~ iIBIL)(~)+.!!(~)i!t(~)]Bi'5(""Bij,
which can be written as
(1. 28)
and
*
i~
f (!,i) ,.. f (!,.Q)
[.tI.
*
1...1 r
l+z:'~ (~)""'~IL.
we shall write the logarithm of it as
(1.29)
It will easily be seen that
where
P*(!)+~*(~)E*tl
, <!t7'i,+5 *(~,i) } ,
11
~ bikYk(!'~) I
k-l
or
(1.30)
i
Since
is a linear combination of
likelihood estimate, then
i
2, if ~ has a consistent maximum
has also a consistent maximum likelihood
estimate.
Thus all conditions in the theorem are satisfied with
respect to
i. .
The logarithm of the likelihood function obtained from the first
n observations is given by
(1.32)
Ln(9)
-
= 1:
n
j=l
n
log f(x.) +
._J
QI1:
u(x.) + 1
2
j=l- -J
n
+ 1:
and since
! ..
A
~
(1.33)
,
Ln(~)
y(x.,9)
j=l -J-
n
~'2::. l~(_x)-Q
J=
,
was supposed to have a relative maximum at
we obtain the equations
1:
nAn
n
1\
a(x.) + 92::
p(2f.) + 1:
.x(2f~;~)
j=l- -J
j=l J
j=l J
=
0
,
12
where 1'(!,~) • (Yl(!'~)' ••• 'Ys(!'~»' This can be rewritten as
follows:
o ,
i ... 1,2, ••• ,6
,
or
(1.34)
1
.. -
!n
n
a ( x .)
j-l i -J
E
,
i.. 1, 2, ... , s •
Since
as n -> 00, and
1\
1
Ii
and
n 'Yi!X .,e)
E
j-1
~
l.i 1-
1
n
< - E
r)i(x . )
- n j=l
-J
also
->
with probability 1, as n
-> 00,
we can see that
...
K
i
<
co
,
13
In
(1.35)
= 1:..
In
If. 6'i
i
l:. n a. (x .) + R.
J= l ~ -J
~
i = 1, ... , s
,
where Ri -> 0 in probability as n -> 00.
Applying Doob1s theorem ["2_7 to (1.35) and the Central Limit
Theorem rl
L
(1.36)
-
7 to .1.
rn
lim PF
l:. nlai(x .) , we obtain
J=
-J
;n I'Ql~l ~ lJ. l',
{
n->CO'!l
= lim
n->co
•
8
••• ,
;n 1\Q ~
n
ass1
rn l: j=lal (!J.) ~ lJ.:l'
PF -
(2n)"2 (~ ... ~ )"2
1
s
J
II'
1'"'1
-co
n
rn l: j-la
... , -
['
t
t
:r. .
1
J
~ lJ.'
1'"'8
l
. tt1
2
--
e
S
(!.) ~ lJ. ~
J
t :L
(r+···'1":S)
1
-00
s dt l ... dt s
and consequently we get, by putting
lJ.'
-! =
~
'~,
i
"'i
(1.37 )
lim PF[rn; ~J
<
n->co
i
= 1,
... "s
1
=
-co
In other words,
rn
-00
1\
~
has asymptotic normal distribution around
the true value ~ =Q with variance-covariance matrix V-l •
Q.
E. D.
Using the above theorem, we shall now give a rigorous proof
of the following result, stated and proved in sketch by Wilks ~6_7,
and also proved rigorously by wald, on assumptions somewhat different
from Doob' s .
J
!r0position 1.3 (Wilks)
To test a composite hypothesis
(1.38 )
, Q
0=
r
against the alternatives H
r Ho in the
Qo
(r < s)
r
situation given in the
previous Proposition, the statistic
(1.39)
-2 log A •
-2 log
max
1t.n1f( x . fH
JD
1Tj~l
max
-J
f(.!j
0
)
I H)
2
has the X -distribution with r degrees of freedom, in the limit,
as n ->
CD,
no matter which simple hypothesis might be true under
.'
the composite null-hypothesis H •
°
Proof'
For the time being, we assume that the true value of the
par ameter
~
is
-9
0'
•
(0
0
0
0)
91 ••• Qr 9 r +1 ••• Qs
•
Let the maximum likelihood estimate of Q under H be
-
0
1'*
9*' • (910 ••• Qr0 1'*
Q +1 ••. Q )
r
s
,
/\
and let the maximum likelihood estimate of
Q
under H be
Then
~.
we have
(1.40)
• E. n1 log,f(x.
J=
IQ )+(Q"*-9
-J -
0
-
-
1 (\i--9 0 ) I Zj nlA(x.)(Q
1'* 0
) E. nla(x')~2(9
-9 )
0'
J= - -J
-
n
1\* 0
+ E. lY(x., Q -9)
J=
-J - -
-
•
,
to'
-J
-
-
15
and
(1.41)
'*
1~g max j =1 f(x- j 1H ) ... Ln~_~)
and we know by Prop.
1.2
that
(1.42)
and
(1.43)
In ~ _Qo).~
...
iii
JL
In
E n a (x ) + R(n)
j -1 i -j
i
i ... 1,2, ••• ,s
,
*(n)
(n)
where Ri
' R
converge to zero in probability as n ->
i
From
(1.40), (1.41)
we obtain, on account of
00
,
•
(1.42), (1.43)
*
(1.44)
+ Rn
(1.45)
+ Rn
where Rn* and Rn tend to zero in probability as n ->
we obtain
(1.46)
-2 log A ... -2
1\
[ Ln{~)
I'J
- Ln{~)
1(1rn
,.. E r ~
i=l i
where Rn converges in probability to zero as n ->
00.
00
-
•
,
Thus
E n a (~.)~
j=:l i J
2
+Rn
t
16
Since, by virtue of the Central Limit Theorem, the random vector
-
1
In
2:
n
a (x.)
j-l s -J
has asymptotically the normal distribution
n
-> 00,
N(g;D~),
in the limit as
it follows from (1.46) that -2 log A. has the limiting
2
X .distribution with r degrees of freedom, as n ->
00.
It is easily
seen that this limiting distribution is independent of the specific
value of
~
o
•
Q. E. D.
2.
A Theorem on Maximum Likelihood Estimate.
We have seen in
the preceding section that if the maximum likelihood estimate which
was defined by (1.1) exists and is consistent, then under the conditions posed by Prop. 1.7 or 1.2, the maximum likelihood estimate has
the asymptotic normality and further that -2 log A. has limiting
X2-distribution.
If we define the maximum likelihood estimate by
(2.1)
and if it exists at all, this must satisfy the likelihood equation,
i.e. ,
[
~
Ln(Q)]
~Q~ ~.~-
o ,
i-l,2, ... ,s.
17
We can state the following
Theorem 2.1
Under the same situation as Prop. 1.2, i f
y(~,~)
has mixed second order partial derivatives with respect to .....
Q and
;heyjare c:n:inUOUs funCt/iC("
sup 1 ~up
i,k ~1~a2
2.3
).
Rm
;:2~~::
finite eX/P6ctations such that
=» Q!71' 9
f(x) dx
-
k
i
<
min
i
~i
"".;
,
then it follows that the maximum likelihood equation has one and
only one consistent solution for sufficiently large values of n.
Thus, if the maximum likelihood estimate
A
~
exists at all, then this
should be consistent in this oase.
n
Proof I
In this situation since the likelihood function
is a continuous function of ~ in the closed set
f\
I-
Ii I~ a2,
it
j -1
f(xj,Q)
-
-
it attains
-
its maximum at Q • Q, and Q must satisfy the maximum likelihood
- -
equation, i.e.,
n
~
;\
_a(~j) + Q ~
j=l
-
n
n
~(x.) + ~
j-l -J
r(_xj,_Q) • 0,
j=l
where
,
and y. (x,Q) • 1. -
-
J
~
(I Q.
y(_x,_Q) •
].
Now we shall show by an iterative method that the equation (2.4)
has a consistent solution.
we shall define s s.quence of BUccsssiva approx1maticnsl ~ ( y j
J
18
by the following equations,
E n a(x,)
( 2.6)
g(v) E n v(x.,e(v-1»
+
j=l- -J
-
• 0, v • 1,2, .•••
j=l"&' -J -
Here again we shall assume, without any loss of generality, that
(2,7)
-
~ ~(!)
f(!)
dJ! •
DI;
•
n
1
Since - lim - Z ~(x.)· Dl;' with probability one as n ->
n->co n j=l J
""
00,
we
can choose n sufficiently large so that
1
n
(2.8)
- -n Ej=l ~·k(x.)
= ~,1 6'1
1 -J
1~
where eik
we get
->
+
0 with probability one as n
+
eik '
->
co, thus from (2.,)
-e
where e' • (el, ••• ,e ) and e. -> 0 with probability one as n ->
-
S
1
Hence by the Central Limit Theorem we obtain
(2.10)
00.
19
This expresses the fact that, for each i,
N(O'~i/n);
asymptotically as an
~io)
and therefore
is distributed
~o)
also is a consis-
tent estimate of g.
-
Next, we shall show that~(Y) converges, as v -> 00, to a
A
1\
solution 9 of the equation (2.4), and 9 has the same limiting
-
distribution as that Of~(o).
-
From (2.6) it .follows that
~(v)_tg(v'-l»
-
Z n A(X.) + Z n ry(x. ~("-1)_ ....(x. ~(v-2) 7
.l. -J'j=lt' -J
j=l'- .l. -J'-
-
or
Now since
-
where g*(v-l) stands for a certain point on the segment connecting
-
g(v-1) and g(v-2)
(2.12)
-
lim
n->co
in~, and by our assumption
.1 z n
n
.
. (x .Agi~( v-l) )
YJ.k
J-l
(2.3)
-J'-
with probability one as n
->
co, therefore, for sufficiently large
values of n, we obtain from (2.11) that
Clt
0
•
20
K 1"'(\1-1) "(\1-2)
< ~1-4
~
- ~
( 2.13)
I.
, ], • 1,2, .•. ,s
,
and consequently
t\Q (\I) - I\(V-l)(
<
Q
-
(2.14)
-
K "(v-l) /\(v-2)
t
Q
- Q
,
v"
a:
2,3, ....
and
-
-<
!! -~o)
(~\o)
I
,
1\(1) t\(o)
(2.15)
/
-Q
- Q
Putting
( 2.16)
H(n) •
~
it will easily be seen that H(n) -> 0 in probability aa n ->
•
CD.
Combining (2.14), (2.15) and (2.16) we get
(2.17)
I f we define
~
by
...
(2.18)
,
the right hand side of (2.18). converges absolutely for sufficiently
large values of n, and
!'
~
more, as is easily seen,
(2.19 )
tK H(n) I
turns out to be a solution of (2.4).
-
I~ ~(o) 1is
K
(l -~)
,
Further-
dominated by the quantity
21
Finally we shall show the uniqueness of the solution of (2.4) for
sufficiently large values of n.
Suppose that (2.4) has two solutions
1\
Q
~
and Q : then we have
or
(2.20.)
Letting n -> co, we get
K
(2.21)
unless
1
I ~ - ~*
I
~ t
'
-> 0 as n -> co, and (2.21) is a contradiction
to our assumption (2.3).
3.
Q. E. D.
Testing Hzpotheses on Certain l\fultivariate Populations, Some of
Whose Variates are Continuous and the Rest are Categorical ~3_7.
We shall consider the case of a 3-variate-(X,Y,Z) population,
where X,Y are continuous variates and Z is categorical. Suppose Z
range
can/over r categories and suppose that the conditional distribution
of X,Y, given that Z belongs to the i-th category, is a bivariate
normal distribution with the mean
and with the variance-covariance matrix
•
22
Suppose we have a sample of n observations, where n is fixed from
sample to sample, such that n individuals belong to the i-th category,
i
i a 1,2, .•• ,r, and hence £i:l n a n. Every individual belonging
i
to the i-th category has two measurements (X , Yij ), j a 1,2, ••• ,n i •
ij
We shall consider the following two cases separatelYI
I.
•
Z is a random variable.
i are random variables
subject to the restriction E : n = n (fixed).
i l i
II. Z is a way of classification. In this case n (i a 1,2, ••. ,r)
i
are fixed numbers subject to the restriction Ei : l n = n.
i
Case 1.
Let Pi (i
a
In this case n
1,2, •• "r) be the probability that an observed
r
individual belongs to the i-th category and hence Ei=l Pi
a
1.
In this case, the likelihood function is given by
where
•
I
Eij = (Xij,Yij )
and U stands for all
Eij
•
We shall be concerned with testing of null-hypothesas of the
following types.
23
lao
Conditional independence between X and Y, given Z.
This amounts to testing the null-hypothesis
The statistic -2 log A in this case turns out to be
- Z
r
i-1
"2
ni log (1 - r (i»
,
A
where r(i) is the estimate of the ordinary correlation coefficient
calculated from t he sample belonging to the i-th category.
We can
show that the quantity (3.4) is equal in probability to
r
1\2
Z ni r (i)
i-l
It will be shown in the next s€ction that the population
probability distribution in this case will satisfy all conditions
Hence the statistic given by (3.5)
has a limiting distribution which is the X2.distribution with
of Prop. 1.2 and Theorem 2.1.
degrees of freedom r.
~
Independence between (X,y) and Z.
This amounts to testing the null-hypothesis
0.6)
against H f Ho' where
five arbitrary nuisance parameters.
for all i
~
1,2, ••• ,r,
~
and V consist of
24
The statistio -2 log A in this case turns out to be
A
E n. log
i=l
/'2
A
vll v22 (l-r )
r
A
/12
Vl1(i) v 22 (i)(1-r (i»
J.
,
"
and this is equal in probability to
(3.8)
where
"*1
V
(J.10)
1\
1\
1\
-
'" (vll ' v22 , v12 )
/'
[;;11 ;12J
Ul •
and
.
l'
v
12
v
22
/'
U2 '"
-1
-
ell
1'12
v
/~2
..... 22
v
]
1(~1)2
~(~12)2
~11"2
!(~12)2
2
1(122 )2
2
1\12 . . . 22
v v
J\lll\12
v v
/'v 22"12
v
2
(3.11)
,
,
,
I\vll~22+(~12 )2
"
and r is the estimated correlation coefficient caloulated from the
data pooled from all the categories. This statistic (3.8) has an
asymptotic X2-distribution with S(r-l) degrees of freedom.
~
Independence between (X,Z) and Y.
It can be shown that this amounts to testing the null-hypothesis
(3.12)
against H ~ Ho •
2$
The statistic -2 log A in this case turns out to be
1\
r
v
22
~ n log ~
X2
i-l i
v22 (i)(1-r (i»
0.13 )
'
and this is equal in probability to
(3.14)
r
i~l
~
1\
r- ni ( 2(i) - ~2)
122
L
2
+
Thus this statistic has an asymptotic X2-distribution with (3r-2)
degrees of freedom•
.!s!: Total independence.
What we are interested in is to test the nUll-hypothesis
(3.1$)
HoI ~(i) = ~
against H
rHo.
0.16)
, V(i)· V where v12 = 0 for all i
= 1,2, ••• ,r,
The statistic -2 log A is given by
r
2:
i-l
n. log
J.
,
This statistic has an asymptotic X2-distribution with (5r-4) degrees
of freedom.
26
Case II
In this case we are dealing with r independent bivariate normal
populations.
The hypothesis corresponding to Ia will be that X and
Yare independent in all the r different bivariate normal populations,
that corresponding to Ib will be that (X,Y) will have the same distribution in all differentr populatioms, that corresponding to Id will
be that X and Yare independent and have the same distribution in
all the r populations.
10 has no analogue in case II.
For each of these problems, the statistic and the asymptotic
distribution (on the null hypothesis) are the same as the corresponding
ones in case I.
However, the asymptotic power of the test for any
that Q r
problem in case II would differ fromjthe corresponding test in
oase 1.
4.
Verification of the Validity
o~
Doobls Conditions for Some Mixed
Variates Population
We shall show in this section that all
condit~ns
and Theorem 2.1 are satisfied in the special case
of Prop. 1.2
whiO~,was
treated
in section ]_
For the sake of simplicity of notation, we shall put
i • 1, 2, • _., r
Assuming that
£0
is the true value of the parameter, and dropping the
27
categorical symbol i for the time being, we can express the logarithm
of the density function as follows:
(4.2)
- ..
~<.~_~O) '~(X, (~_~o)
log f(X,Y,Q) • log f(X,Y,Q 0
) + 0
(Q-Q '
) a(X,Y)
-
-
+
where
ai(I,Y)
.f d
log f(I,Y,B2)
'\ 0
A
t'ik
y)
Qi
/
~ _ ~o
(X y) • ( ) 210g f(X, Y'i'
'
() Qi
dQ
k
Q
, i • 1, ••• ,5
= Q0 '
,
+ y(X, Y,~)
k-l""
i
,
,
~
, • • • ,;;J
,
- -
(4.4)
Since
(4.5)
,
= Q4(X-Q~)-Q5(Y-Q~)
Q~Q4-Q52
(4.6)
,
28
,
Hence we can easily see that
ff
00
(4.7)
.!l(x.,,)
-00
Furthermore,
ttx,1.~o)
dx dy '.Q
•
29
,
~l,(x,y,~) .. ~,l(x,y,~)
"0 210g
..
f(x,y,,i)
09;a9,
==
29
,
22
(Q3 94-9 ,)
~-92
L94(7;-91 )-9,(Y-9 2 )_7 - e 9
3
.
"'.
\
.. - •
(9 9
3
~2,(x,y,~) .. ~52(x,y,~) ..
~ 210g
93 . 2 2 £-
4-e,)
() 9
ae
f(x,y,,a)i
\
2 5
9,(x-G1)+~3(y-92)_7
_9 2
4 ,
,
'
30
,
31
~
(x y Q) • ) 210g f(x,y,j)
t"44 "-
'"oQ 2
4
Thus, on account of the relations
we obtain the followingl
32
-Q$o
v12 •
.
62 ' v13
• 0 ) v14
Q~Q4-Q5
•
0) vIr:.- • 0 "
~
,
gO
(4.8)
v22 •
3
Q~Q4-Q5
2
'
v 23 • 0) v24· 0) v2r:'
0 ,
•
9
and
It can be easily checked that
ff
00
(4.10)
-00
00
LP(X,y). 5!:.(X,y)'5!:.'(X,y)]
f(x,y,~O) dx dy •
0
-00
.Next we have
(4.11)
Yi(x,y,Q) .. J
-
...
X(x'YJj~
'0
5~.
k=l
Q
i
.. ai(x,y,Q) - a.(x,y,QO) -
1.
-
°
°
)
.£
and
£~ik(x,y,~ * )-~ik(x,y,~ t7(Qk- Qk)
where ~* stands for a certain point on the segment joining
i ~ik(x,Y)(Qk-QkO)
k-l
.£0.
....
33
By
means of Cauohy's inequality we get
IY1(x,y,~) I:: I~ -~o Pl L~iJc(X'Y'~*)-~ik(x,y,~O)J
'/~
.
)
2] ,
and henoe
(4.12)
Since, as was shown before,
~'k(x,y,Q)
derivatives with respect to
~t
-
~
has continuous partial
again by using the mean value
theorem and Cauchy's inequality, we obtain
(4.13)
I
'Yi
(x,y,~)
}i - ~o ,
.... 1.:5
~[
51.: (~ik' (x, y ,~
kllll j=l
J
**2]
1~
)
where ~** stands for a oertain point on the segment between ~o
and Q*•
-
We
can find funotions
~i(X'Y) it k=-l
~ j=l
~ max{~ikj(X,y'~)I'
i -1,2, ... ,5
~
such that ¢i(x,y) f(x,y,jo) are integrable, and
I Vi (x,y ,2) I
{ j ..
~o I~
Next we shall show that the condition (2.3) is also satisfied.
For this purpose we must introduce the new parameter
i
such that
,
34
(4.16)
B' vB·
, B' B = I ;
D~
then
y*(x,y,i)
(4.17)
= y(x,y,E!)
• log f(X'YI~ - (i-i")'BI~(x,YJ
f(x,y,B
)
and
(4.18)
and
)2
"'0Y(~1'~
g:).
/
)
Ql'
:t
Y~J'(x,y,~)
...
Pl)'(x,y,~)
h!. (x,y,~) I:; I~ -~o I¢n'
where ~ll'(x,y) was defined in (4.14).
Thus we get
00
J
•. -00
....
*
0
Yok(x,y,i) f (x,y,~ ) dx dy.
1
-
5
Z blob~,o
)1'=1
1
*~
-
(x,y )
~ll'(X,y,£O)
,
and hence
(4.20)
~ I~ -~OI}a~)ib~lk
f J,1~~I(x.Y)f(x.y.~O)dxdy
00
-00
We can choose
I~ - £ 0
J
=
Ii -i
O
(
sufficiently small
K • max max K'k(i.) '< ~
i,k
~
i
where
~
60
that
,
denotes the minimum characteristic root os V.
Thus we have shown that in the case which was mentioned in
section 3, all conditions required by Doob and also the condition
(2.3) are satisfied.
5.
Concluding Remarks.
The exact role of the consistency condition
in this whole scheme, the precise relation between Doob's ~2_7 and
Waldls ~5_7 papers and also between the present paper and Wald's
paper ~5_7 are worth a more careful consideration which is now
under way.
The authors hope to be able to discuss these in a later
communication.
•
36
REFERENCES
..
...
1.
I
Cramer, H., Mathematical Metho~~ of Statistios, Princeton University Press, 1946•
.
Cram'r, H., Rando~ Variables and Probability Distributions,
Cambridge Tracts in Mathematics and Mathematical Physics No.
26, 1937.
2.
Doob, J. L.,rtprobability and Statistics,rt Trans. Amero Math. Soc.,
Vol. 36, pp. 759-175.
3.
Moustafa, M. D., "Testing of Hypotheses on a fultivariate Population, Some of the Variates being Continuous and the Rest
Categorical," U. N. C. Institute of Statistics Mimeo. Series
No. 179, 1957 •
•
"
;
Ogawa, J., "A limit theorem of Cramer and its generalizations,"
U. N. C. Institute of Statistics Mimeo. Series No. 163, 1957.
5.
Wald, A., "Tests of statistical hypotheses concerning several
parameters when the number of observations is large," Trans.
Amer. Math. Soc., Vol. 54·(1943}, pp. 426-482.
6.
Wilks, S. S., "The large-sample distribution of the likelihood
ratio for testing composite hypothesis," Ann. Math. Stat.,
Vol. 9 (1938), pp. 60-62•
..