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ON THE ASYMPTOTIC DISTRIBUTION OF F-STATISTIC UNDER
THE NULL-HYPOTHESIS
m
A RANDMIZED P BIB DESIGlil"
WITH M :ASSOCIATE CLASSES TmDER THE NEYMAN MODEL
By
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Sadao Dceda, Junjiro Ogawa and MDtoyasu Qgasawara
University of Nihon University Nihon University
North Carolina
Institute of Statistics Mimeo Series No.
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1~54
December 1965
SUMMARY
1:.
2.
3.
4.
Statement of the problem
Asymptotic per:nu.tation distribution of (e, e) and e
Alternative asymptotic permutation distribution of
(e, e) and e
Asymptotic null-distributions of the F-statistic
after the randomization
Acknowledgements
References
T,pis research was supported by the Mathematics Division
of the Air FO'.Uce-Qffice of Scientific Research Contract
No. AF-AFOSR-760-65
DEPARTMENT OF STATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
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SUMMARY
This article gives a probability- '1heoretical justification to the previous work [ 5 ] on the asymptotic null-distribution of the F-statistic for
testing a null-hypothesis in a randomized partially balanced incomplete block
design with m associate classes under the Reyman model, which is the lOOst
general case among the series of our preceding works along the same line
[1, 2, "
4, 5].
The theory of asymptotic equivalence of proba.bility distri-
butions developed by one of the authors [ 6, 7 ] seems to be effectively
usable for the present study.
The investigation in tntsarticle throws a
light on the asymptotic nature of the power function of the F-statistic under
consideration, Which will be worked out ip the forthcoming paper.
1.
Statement of the Problem
Throughout this article the notation and terminology are the same as those
of [ 5 ] unless otherwise stated.
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We are concerned with a partially balanced incomplete block design with
m associate classes, which has v treatment with the association, b blocks
of size keach, r
replications of each treatment, and the number of in-
cidence of any pair of treatments
A
u
if they are u-th associates.
Let the incidence matrices of treatments and blocks be
! and t
respectively, then the Neyman model assuming no interaction between the
treatments and the units is given by
(1.1)
x = 71 + !'t' + '1!l' + :rc + e,
Where x' = (~.' ••• ' x n ) is the observation vector, 't" = ('t'l'···' 't'v)
and ~'=(~1' • .. , ~b) are treatment-effects and block-effects being subjected
to the restrictions
v
1:
i=l
't'. = 0
J.
and
b
1:
t3
a=l a
=0
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respectively, and
= ('1.' ... ,
1t'
n ) stands for the unit-errors being sub-
1t
jected to the restriction
11£'
1t
=0.
.-II
II
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82 = x I ( ~ V#)x
t
u=l u
(1.4)
82 = x' (1- lk· :B - ~
e
u=l u
vlfJx
respectively, where
vu# = cu (I - 1k B) Tu # (I -k1B),
u
= 1,2, ••• ,
m
with
T#
u
=
cu =
Here,
~AU
#$'' B = ,1,:t.%,1,' and
k
rk _ pu
A #,u = 0,1, ... , m, with A
u
'
#=
u = 1, ••• , m.
G /v
are m + 1
00.
of tIle association algebra, and Pu ' u = 1, ••• , m and
orthogonal ide:rrq>otents
P,J = rk are the
cl1a.racteristic roots of NN', N being the incidence matrix of the design, with
respective multiplicity ~, ... , am' a o' for which
tl au = v-l.
We are interested in testing a partial null-hypothesis that some of the
hypotheses A #-r
u
= 0,
u = 1;.·.·., m, are true.
We can take, without any loss
of generality, the null-hypothesis
(1.6)
vThere
HO(h): Au#-r
= 0,
u
= 1, ••• ,
h is a positive integer not greater than m.
h
This hypothesis is called
the 'partial' hypothesis, and is equivalent to the hypothesis
2
~~l Au#-r
= 0.
-II
It
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When h = m, this reduces to the 'total' nUll-hypothesis H: 't' = o.
o
To test the null-hypothesis Ho (h), we consider the partial sum of
squares
S~(h)
(1.7)
instead of S~
given by
h
= x' (L Vu #) x
u=l
(1.4). Then the null-distribution of the variate
(1.8)
before the randomization is the non-central chi-square distribution of the
degrees of freedom
(1.9)
I-
a = 01+•• ~
~
with the non-centrality parameter
h
= 1l !HLC llAu*Yt'1C/ rP
t
u=l
h
= 1l'( L
Vu~1l/a2.
u=l
Whence its probability element is given by
- 1
(1.10)
exp
ii(h») d (
2
In the special case when
h
= m,
\
the variate
(1.8) reduces to
(1.11)
and the null-distribution of this variate before the randomization is the
non-central chi-square distribution of degrees of freedom v-l with the
non-centrality parameter
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'Whose probability clement is given by
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The null-distribution of the variate
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(1.14)
before the randomization is, independently of the hypotheses, the non-central
chi-square distributions of degrees of freedom n-b-vl-l = b(k-l)-( v-l) with
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the non-centrality parameter
K
l
I
= 1t' [In -!( u=l
~ cu--U
A."6~' ]1t/ a2
••
•11
•-I
,
Hence the probability element of the variate ~
is given by
X 2 n-b-v+l
2
+ v (2)- . ~
,
1
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I~
1
Since
~
and
~
are mutually independent in the stochastic sense,
the null-distribution of the F statistic given by
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e,
I~
4
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J
J.
1
1
]
before the randomization is the non-central F-clistribution, whose probability
element is given by
exp (
J
2cr2
t :
)
.'
~ (~ +~)r(n-b~v+l +v)
n~-v+l
(1 +
a
n-b-v+l
+~ -1
/
I
n-b-a
F)
- (-
2
+~+v
)
d'(
a
n-b-v+l
F}
Hhere we have put
m
a =u=l1+l
i:
a = v-l-Ci,
u
,
]
/
#= re' (
Wl1en
h~,
~ V#)re/b and
u=l u
e = re (~ viu!> re/AI
u=h+l
the-null-distribution of the F-statistic
]
F
= n-b-v+l
v-l
before the randomization is the non-central F-distribution, whose probability
element is given by
1
I·
J
'~a-
(' a
)
.
]
J
1
-b
(~2;;"-';"
n-b- a ;.~+V
r
•
J
]
82
e
A
11
S~(h)
= n-b-v+l -..
F
(1..18)
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rf(l-e) v
(1.22)
r( n-b +.t)
(
'-r-(-V";::;:'::-l-+-li-)r-(-n~-b:-~-V-+":"l-+-V-)I
_(E.:E...
(
1 +
~
v-l F
n-b-v+l
V-l
- + Ii - 1
n-b~;~l
F ) 2
f
+.t )
2
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d ( v-l
F)
n-b-v+l
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vThere
e = e + '9
(1.23)
In the previous paper [
= 1t' (
I
~ v'u!J 4 f).
u=l
values and variances of both distributions in a very tedious way:,. that the
pernmtat.ion distribution of (
-
5] the authors showed, by compairing the mean
e, '9 ) due to randomization
can be approximated
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by the Dirichlet distribution, whose probability element being given by
r(n;b)
(1.24)
'r
(_i2'/'
1)
h, pt_CL)p(n-b-v+
\ 2
2
(o:::;e,'9,e
+~)
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for sufficiently large b, under the conditions
= ro' p = 1, ••• , b
where
1tip)
stands for the unit error of the i-th unit of the p-th block.
Then, they integrated out the factor e~v(l_e_~)r in (1.19) with respect
to
(1.24) to show that the null-distribution of the F-statistic given by (1.18)
after the randomization is approximated, for large b, by a familiar central
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F-distribution with degrees of freedom
In the special case ,.;rhen h=m
(a, n-b-v+l).
(see [ 4 ], for m=2) , it was shown, by
the same method, that the permutation distribution of
e
due to randomization
is approximated by the Beta-distribution whose probability element being
r(n
(1.26)
2b)
v-l
e2
-
1
n-b-v+l
- 1
2
de, (0
(l-e)
rev ~ l)r(n-b~V+l)
for large b, under the conditions (1.25).
Then, the factor
.:s e .:s
1)
el-1(l_ e) v in
(1.22) was integrated out with respect to (1.26) to get the central F-distribution of ,-,rees of freedom
(v-l, n-b-v+l) as an approximation to the null-
distribution of thaF-statistic given by (1.21) after the randomization.
From the probability theoretical paint of view, the above argument presents
us two questions:
(e,
~) or
(a)
e approximated
In What sense is the permutation distribution of
by (1.25) or (1.26), and (b) in what sense is the
null-distribution of the F-statistic given by (1.18) or (1.21) after the
randomization approximated by the central F-distribution of aegrees of
freedom
(a, n-b-v+l) or (v-l, n-b-v+l)?
Furthermore, the condisions given by (1.25) under which our argument carried
out seem too severe from the viewpoint of practical applications.
This article treats these problems and gives satisfactory answers to
them, where the uniformity conditions (1.25) are
2.
As;yrn.ptotic Permutation Distributions of
we&~ened.
(e, e)
and
e.
In this section we shall derive asymptotic distributions of
and
of
e
(8, e)
Which are asymptotically equivalent to the permutation distributions
(8, e) and e in the sense of type (8) as
We shall
t~e
b-+~, under certain conditions.
a special numbering of the whole set of units suCh that
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the i-th unit of the p-th block is numbered as
p=1,2, ••• , b.
Then it is clear that
Gk
(2.1)
B =
G
k
designate the
.tt
becomes
0
G
k
•
0
'Where
B=t
f = (p-l) k+i, i=1,2, ••• ,k;
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,
·G
k
k x k
matrix whose elements are all unity.
us take an arbitrary but fixed incidence matrix of the treatments
~,
Let
and
let
s
(2.2)
u(J
=
o
(Jls
0'2
o
be the permutation matrix corresponding to the set of permutations
2.
cr (2)
p
• •••• k
)
b
p
Then, for any incidence matrix of the treatments,
a permutation matrix U such that
cr
1 2
cr (k) , p= , , ... ,
Ucr',*
1*,
= 3:':tor
n
_I
•
there can be found
~
x =
u+'I
cr.
m + 3 matrices
Now, since the
1
m
1.
1
u=l, 2, ••• , m, (In - - k B - LI
~ V #), - k B - - n
u=l u
.
Gn ,
...L G
n
n
with respective ranks
ex , u=1,2, ••• , m, n-b-v+l, b-l,
u
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are mutually orthogonal idempotent matrices [ 8], there exists an orthogonal
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matrix of order n
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~.
•
•
•
•
•
e (1). • • e(l) •• ·c (2). • • e (2) • •••• • e (p )•• • e (pJ. • • •• • e (b) .•• e (bJ
v~l
vu .v.;Ll \Llk
v.:Ll
v:-ll
v-ll
~
9
•
•
•
A
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
d12• • • • • • • • • • • •.• • • • • • • • • • • • • • • • • • • • • • ••' • • • • • • • • • ~n
j
d11
:I
1
J
dn_v +11' • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • .' • • • • • • • dn _v +a
satisfying the conditions
r
p
0
o
•
V#
p' =
u
Ol~.
1 a:u
•
, u=l, ••• ,
m
·1 .
O'
]
•
o
••
,
Je
p(r -
J
1,
.0 i
III
.l:... B
1~
o
0
m
-
r. VJk) p'
u=l ti
•
J
J
1
1
1
p(- B - k
n
·0
1
n
and p(- G )p' =
G ) pI ==
n
1
•
~t
o
n
·0
J
I·
]
simultaneously.
fhe
submatrix
e
1
consisting of the first v:-l rows of P is
divided into two lubmatrices
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C_ and C_, consisting of the first
ex
for Wlich we have
a
.
~a
(2.6)
•
ex rows and of the last CXrows
of C,
h
:/I:.. ~
m
(E V" J C- = 1- and Ca' (I: V#)
u=l u
ex
ex
u=h+l u
ca = lei •
It is also noted that any row of the matriB: C satisfies the following conditions
k
E C(~f = 0, p=1,2, ••• ,b
i=l
Now, since
1C
is orthogonal to the matrix B and hence to G, and
r
the sum of m+" matrices in (2.:3) is equaJ. to
A =
n'U
(~ vi)
(}' u=l u
= 1C'U _'
p'p(
f:f
U n +n'U ,(
In' we have
~
v#:l
(}' lIl=ll+l u
(}'
~ V#) P'p U 1C +
u=l u
f:f
1C
..L B
U n+1tU' (I •
(}'
U' pIpe
0"
(}'
n
k
-
~ ~ II
V )
u=l u
~ V'!J P'p U 1C
u=h+1 u
0"
+ 1C'U ' P' P (I n
f:f
..1:...
B
k
~ V#)P'PU
1C
u
f:f
-
u=l
= 1C'11'
P'.
- 0"
o
PUf:f
o
1C
·0
0"
o • 'l)
o
o·
•
·0
o•
+re' U ' P'
f:f
0
'Oo
o
10
Ip U re
1~E2
'0
•
~
o.
+ 1C'U0" ' P'
•
PUre
f:f
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'0
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and therefore
We shall demonstrate the asymptotic normality of the vector variates
c ucr~
and
under certain conditions.
For this, we
prepare the following
LEMMA 1.
a.s b
-+
eo,
~
All the elements of the matrix C are of order $(
keeping v, k, nl , ••• , nm and P~k fixed.
Let (:' be any row vector of the matrix C.
follows that
e must
Then, form
)
(2.6) it
be a. linear cOl1bination of v-:llinearly independent
column vectors of the matrix
(2.10)
1
'.Jb
e
(~=l V~),
ie.,
v-l
= !: A.V.
. 1
J..
J. J.
is a set of linearly independent column vectora of
~ V#.
u
u=l
(2.11)
W= r
\I vTJ.J
• • ~(."-l
._')
~J-, ••• ,
V":I.
we have
(2.]2)
e It
= -
1
r
,
X W). = 1-
11
Ij
Since the
Vi ••
~J
I
S
#
m
LV, it follows from (1.5)
are elements of the matrix
u=l u
that each element of the matrix W given by (2.il) is of order 0(1) under
the limiting conditrhon stated in the lemma.
~laracteristic
simgular and sjrnmetric, its
Furthermore since
roots
W is non-
0l""'ova are all positive
and of order 0(1) as b.-+- and there exists an orthogonal matrix K of order
w.., '..h ose
elements being of order O( 1) as b.-+ 10, such that
0
1
°2
=
KWK'
Hence, putting ~
= K~,
0
0
'0
v:a.
~' = (~l"' •• ~~)' we obtain by (2.12)
vl
and
L o.~
i=l ~ ~
=r,
from which it is easy to see that all components of 1\ must be of order O(.Jb)
as b.-+ 10,
s1Dce r and b are of the same order of magnitude as b.-+ 4'
•
j
Hence it follows from (2.10) that all the elements of the matrix Care
of order G(l/./b) as b
ivhich proves the lemma.
UBing this result,
LEMMA 2
v~
can show the following
Suppose that the following conditions are satisfied:
b
(2.14)
E == ; ~
6
p=l P
for some 6 0 > 0 and 0 ~ 1/2.
Then , the permutation distributions of the vector
variates C U 1r, C. U 1r and C Ucr1r converge to the multi-normal distributions
ex cr
a cr
6
tn4 B(O, k-~ I~l) respectively, in the sense of
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b.
PROOF
We shall shovr tl1at CU :rc converges to N(O, , 0 I ~ ) in the
cr
l~- 1
v:..L
sense of type (8) as b ~ ro
•
Let us put
f c
.
(2.15)
b
-. -
C Ucr" -
~ ~(p)
"-'
p=l
.
,
,(p)-
1
]e
J
k
.t
i=l
k
.t
j
~
~p)
i=l l~
i=l
rC
(p)
cr(i)
c(p):rc (p)
2i cr (i)
C (p)
v;a.i
:rc
,
(p)
cr (i)
Then, since
and,
~;) ••• , ~(b)
are mutually independent in the stochastic sense, we have
J
]
]
J
(2.18)
I
= -L
]
I·
J
k=l
Since
I
v;,l
+
---L
~ (b.p - L5:»).~
C~p)
II ~=l Ji
k-lp=l
dP)11
J~
1
1
(- a
and, by lemma 1,
(.~
p=l
·1
= 1).
+ - a'
a
~
C(p)C(p)(
i=l si 'i
we obtain, putting
a'iT
)
&=1+0,
b
(2.19)
1: (.6 p=1 p
M = 0(1)
k·
6)
(b -+00),
.
1: c(P)C(~)/l+0
. 1 8i 'J.
J.=
b
<..l...
=
b
1:/6 _
P
p=1
A/. 1 +O
Hence, we have
(2.20)
Now, let 1(t)
'Where
®:;: 0(1)
be the characteristic function of
cucr1[. Then, by (2.17),
as b -+ co. Since
"re get
k
1: C ~
i=l SJ.
(2.22)
From lemma 1 it follows that
14
3
()2-
)2
and hence
~
(2.23)
p=l
E<[t ' g(p)
Hhere M = 0(1) as b~to.
Since
~E <It'
(2.24)
3
/3) ~ M (tt
1
T
g(P)/3)
t)2
r
b
3
1: D.
p=l
2
P
3
b 2
2
1: Ii. ~ Ii.
p=l p
0
~O
~n
2-
b
(b
(b ~oo), vie have
~1lO).
p=l
Therefore, it follows from (2.20), (2.21) and (2.24) that
D.
(,.,(t)
r
(2.25)
~-
1
2
--£..
t' -t'
k-l
(b~to
)
v,hich proves the lennna.
From this lennna, it is straightforward to prove the following lennna.
LEMMA 3
(a)
Under the conditions (2.14),
the permutation distribution of b(k-l)e, b(k-l)~ and b(k-l) e converge
to
xi'
~and ~-l in the sense of type (JJ) as b~1lO , where ~
designates a random variable distributed according to the chi-square
distribution with degrees of freedom f, and
(b)
(b(k-l)e, b(k-l)
e} and hence (e,
t}
are both asymptotically independent
set of random variables in the sense of type (8) as b ~fl.
It should be noted that the limiting chi-square,', distribution in(a),
~
and ~ , can be asswned to be independent in the ctochastic sense.
Applying Theorem 4.1 of [ 7 ], we have
THEOREM 1
Under the condition (2.14) of Lennna 2, (a) the tvro random
vectors
(2.26)
_==
1
2
1
(e, e) and (b(k-l) XCi ' b(k-l)
2
xa )
are asymptottcally equivalent in the sense of type (S) as b ~ 00, where vie
15
2
1
1
~
b(k-l) XCi and b(k-l)
can assume that
are mutually independently
distributed for each fixed b, and (b) two random variables
~SJ as~ptotically equivalent
in the sense of type (S) as b ~ 10.
3. Alternative Asymptotic Permutation D.istributions of ee,~) and
In this section, we
de~ive
as~ptotically equivalent to
other types of distribution which are
('e, e)
e in
and
the sense of type ($) as b~
In the first place, vTe consider the latter variable
LEMMA
4
e.
IX)
•
e.
Two random variables
(3.1)
1
b(k-l)
2
Xv-I
and
®
v-l
«(
are as~ptotically equivalent in the sense of type (J)
7]) as b
~IO
,
'ithere @w. stands for a random varia.ble whose probability element is given
by (1.26).
PROOF
Since type (8) and type (I) a.symptotic equivalence are mutually
equivalent in this case, Theorem 1.4.2 of (6) is applicable.
The probability density functions of the variables in (3.1) are given
by
v-l
2
(3 •.1)
Y
-
1
-
e
and
b(k-l)
2
Y
, (0< Y < GO)
v-l
y
""'2 -1
n-b-v+l
(l-y)
2
,
(0 < Y < 1)
rp.spectively~
Hence, the Kullback-Leibler mean information in the Theorem
1.4.2 of [ 6 ] is given in the present case by
16
J
-}
e
1
1
J
J
1
(3.4)
l(g:
f)=
Jl
g (y) log
a
This is calculated as follows:
.. ~
log -rr:YY
(n-b-v+l). y-1
2
+ ~
r
2
B;{ using the _tirling f0rnD.11a,
!log
r(b(k-l») _
- 2
log
=log
v-l
X log 2 -
~l
'tIe
r ( b(~-l»
have
_ (b(~-l)
]
1J
~ dy
+ log
I
.J2rc ~
n-b-v+l
2
-
(
~) log b(~-l)
_
1 )
2
/ b(k-l)
2
0
log
n-b-v+1
2
n-b-v+l
2
+ log .}2; (~ 0
as
b~ co.
Furthermore,
]
J
J
o
r
g(y)b~k-l)
ydy =
v-l
2
and again using the Stirling formula, we have
]
( n-b-v+l
2
- 1 )
1(
) dy = - (n-b;v+l -1)
g Y ) log (
l-y
[.
f
J
x[
.......;:;~-y-
(log
r
(p+q) - log
]
,e
J
r(p) ) ]
b
( p:::ln-b-v+l)
2
v-l
)
( q=
2
/'1
I
= lim [ - n-b2-v+l {lOg (1+ n_bv-_vl+l) + ~~~v~-l~
~_ } ]
b(k-l)(n-b-v+l)
b~oo
(:5.8)
.-I
I
v-l
= ---r
It follows from (3.4)
f)
I (g:
I
I
I
I
(3.5) (3.6) (3.7) and (3.8) that
~O,
(b~
00),
which proves the lemma.
Next, we shall show the following
5 Two variates
LEMNA
1
X ~ , b(k-l)
(::5.10)
2
~ ) and
(@a , ot
)
are asymptotically equivalent in the sense of type (8) as b~ 00, where
(<»ex
,~
)
is a random vector whose probability element is identical to
that of(1.24)
EBQQ[ Analogously to the the proof of the preceding lemma, it suffices
to prove that the Kullback-Leibler mean information of both variates of (3.10)
tends to zero as b
~
00 •
I
-I
I
I
The probability density function of the first variate of (3.10) is
I
given by
v-l
f(x, y) = .l.bk-l .....,.:::--_
2 v-I r( ti ) r( (12-)
2
~ -1
x
b(~-I)()
a
9
) J2
.r (
x+y
-2 - 1-
y
e
,
2
(0
< x,
Y
< 00)
and that of the second variate of (3.10) is
(3.12)
S! -1
r (b(~-l»
x
2
g(x, y) = - - - - - - - - r(;~ ) r( ~ ) r(n-b-~+1 )
g
y
2
-1
n-b-v+1
(l-x-y) 2
(0 < x,y,x+y<l)
18
•
1
I
I
I
I
I
,
I'
I
I.
I
I
I
cl
Hence the Kullback-Leibler mean information is given by
(3. 13)
I (g: . r) =
J
1[
1 g(x, ,., log
Thus, fP. (3.11) and (3.12) we have
r(b(k-l»
(3.14)
I(
g :)
f = E( x,
y
) [ log (
~1~: ';l
dxdy
v-l
22
(n-b-v+l
2
- 1 ) log «1- X+Ji»
v- 1 J +
r(n-b-v+}(b(k_l»2
2
2
,I
b(k-l) (
. x+y
2
'I
I
I_
I
I
.nlere E(x,y) stands for the expectation with respect to (3.12).
x + y is distributed according to the Beta-distribution (3.3).
l(g:
Under (3.12)
TheTefore
and hence
l(g:
f)
-4),
(b~oo)
vlhich proves the lemma.
From these results and Theorem 1 in the preceding section, we have the
following
·1
distribution of (8, e) are asymptotically equivalent to
-I
probability element being given by (1.24), in the sense of type
I
,
f) given by (3.14) has the same value as that in tl1e preceding lemma,
I
·1
I
I·
)]
THEOREM 2 Under the conditions (2.14) of Lemma 2, (a) the permutation
b ~oo, and (b) the permutation distribution of
to~,
as
e are
(Q.t, <»&),
whose
(~)
as
asyrnptotically equivalent
'whose probability element being (1.26), in the sense of type (8)
b~eo.
19
_
4. Asymptotic Null-distribution of the F-statistic after the Randomization
In the present section, we show that the null-distribution of the
F-statistic given by (1.18) or (1.21) after
the randomizatiOJl;can be asymptoti.
'~.
cally approximated in the sense of type
($), by a familiar central F-distribution
with degrees of freedom (a)n - b - v + 1) or (v;,l, n-b-v+l),
which is obtaine4
by integrating out the conditional variates in (1.19) or (1.'22) by the approximatll distribution (1.24) or (1.26) of the permutation distribution of
(e, e)
I
I
I
I
or
e•
First we shall prove the following
THEOREM:;
Let
p (zJx) d z d F (x)
(4.1)
s
s
S
be a probability element of the two dimensional real random variable (zs, X ),
,nlere F (x) stands for the cumulative distribution function of X S, S=1,2, •••
s
Let yS be another probability distribution Whic4~1t~asymPtoticallyequivalent
s
to X in the sense of type
(~)
as
s~oo,
Let us assume that tile following conditions are satisfied:
(i) ps(zjx) is continuous vdth respect to (z, x) jointly, for every s.
(ii) !For yS,_ere exist constants c > 0 and ds, s=1,2, ••• , such that the
s
variable
~ (yS _ d s )' s=1,2, •••
s
tends to a certain fixed distribution of the continuous type, y, in the sense of
type ($)
Then, two distributions whose probability density function
being·givep by
(4.3)
f
-00
OOp (zlx)
S
d'F (x) andJ'; (zlx) d G (x)
s
s
-00
20
_I
and let the cumulative distribution function
of yS be Gs(Y)' s=1,2, •••
(4.2)
s
I
I
I
I
I
I
I
I
I
I
I
I
e
I
I
~ co
are asymptotically equivalent in the sense of' type (J) as s
•
PROOF To prove this, it is sufficient to show that
(4.4)
_
:~ < ~ j
IS
liz
-J~P • (zlx)
elF .(x)
~!Sdz _!~ P • (Z)x)
~co).
(s
Let us put
(4.5)
CPs (alx) =
fa ps(z/x) dZ,
s=1,2, ••• ,
- co
and
(4.6)
-00
-co
s • 1,2, •••
Then,
(4.7)
(J.~.4)
can be written as
sup !Hs(a) - Ks(a)1 -.+0, (s ~co)
-co < a<co
Now since
H (- 00) = 0, H (+00 ) = 1 and K (- 00) = 0, K (+
s
a s s
for every s, there eXitts
a point a==as such that
.
IHs (as ) - Ks (as )/" _= supa I<H (a) - Ks (a)1
00<
~
'
-00<
00) = 1,
a
s
< 00
for every s, and therefore (4.7) turns out to be
~ 0, (s~oo).
s s
s s
Since the cumulative distribution function of the variable given by (4.2)
(!~. 8)
" H (a ) - K (a ) J
'
is G (c x + d ), the condition (ii) of
s s
s
21
thet~eoremassures us
that
dG.(x)
1"",0,
I
sup
-00
< x < 10
I G (c x + d
S
S
S
.-I
) - G(x)/ ~ 0, (s ~co)
where G(x) stands for the cumulative distribution function of Y, and therefore
1re
I
have
(4.10)
sup.
_10
< x<
10
IF s (c sx + d
S
) - G(x)/~O, (s~1O )
I
I
I
Taking a transformation of the variable, we have
00
(4.11)
-00
-1
10
I
cP s (a s lc s x + d s )dG s (c s x + d s ) / .
The function CPs (alx) given by
(4.5)
has the properties that
I
O:s cps(a/x) :s 1
eI
I
for all (a,x), and cps(a/x) is continuous in (a,x).
a convergent subsequence
(0=1,2, ••• ) and a
{cp' (a
'I c
Henee, we can always find
' x + d ')) s' ~
funotion~;x) ':Oh :hat
o
00
.
)
of{"tp (a
•
Jc x
+ d
• ••
))
II
•
I
CPs '(as' Ic s ' x + d s ') ~cpo(x) (s '~IO)
q.early, CPo (x) is continuous and 0 SCPo (x) :s 1 for all x.
Now, for any given e
I G(M)
(4.13 )
- G (-M)
I < e,
and hence, by (4.9) and (4.10), we can find a integer s such that, if s'
e
then
F ,(0 ,M + d ,)
I
{ Gs ,(c s ,M + d s ,)
s
Is s
(4.14)
- F ,(- 0 ,M + d ,)/
s
s
< e,
s-
- G ,( -
s
(4.12) can be assumed to be uniform, i.e., there exists
&;
'POSitive integer solt••
e'
L
s'
> s* implies that
-
> se
cs ,M + ds
,)/ < e,
is uniformly continuous on the interval 1M' and the convergence
Note that CPo ex)
such that
I'
> 0, there ensts an interval IM=(-M, M) such that
e
22
I
I
I
I
••
I~
I
I.
i
I
-J
(4.15)
for all x belonging to
1M.
Hence, if s' ~ max (se' s:), then
J
(4.16)
Mcp S ,(aS ,Ie S tx+d S ,)
J
-M
S
S
S
-M
<e
-1
Mcp S ,(aS ,Ies ,x+ds ,)
IJ.
-M
~l
and,
by
_JM cp tea. ,Ie
S
(4.17)
J
-M
-
e
J
dG ICC ,x+d ,)
s
s
S
_JMcp 0 (x)dG S ,(c S ,X+dS ,)!<e
-M
(4.14)
1
-1
JM cp 0 (x)elF S ICC S ,X+d,~
S 1
elF ,(c ,X+d ,) -
S
M
J
S
,x +
d ,)
S
elF S ,(c S ,x + ds
,) I<e
,
cP S ,(aS ,Ie S ,x + d s ,) dGS ,(c S ,x + d S ,)
I<
-
e~
-M
TI1US,
(4.18)
jIHS'(SS) -Ks,(Ss)j -
tiM
'l'o(X) dFs.(cs.x + ds')
-J'1p (x)dGs •(cs ,x
J
0
'!' d...)
II~
4e
-M
J
J
Since, by
(4.9) and (4.10),
1
and
]
(4.20)
-1
there exisits, from
I
J
-M
Mcp (x) dG
o
S
,Cc S ,x+dS ,)
0
(X)dG(x)/~
0,
(S'~IlO),
-M
(4.18), a positive integer
I·
1
_JMcp
23
S
such that s' > s
0 - 0
implies that
(~~%)
I Hs,(as ')
I
eI
1<
- Ks,(a s ')
6e
This shows that (4.8) is true for the subsequence {s'},
frOJTl
which it is easy
to see that the theorem 110J.ds.
This result can easily be extended to bivariate case, which will be
stated without proof in the followiag.
THEOREM 4
Let
:p (zfx, y) dz d F (x, y)
s
s
be the probability element of a three-dimensional real random variable
(4.22)
s
(3 ,
of
x1,
x~), 'Where Fs(.X,y) dmsignates the cumulative distributionf'unction
(x;., ~)!:',
I,et
S=1,2, ...
(~, Y~)be
equivalent to
another probability distributior which is aJY1l1.Ptotica1ly
O{,
~)in the sense of type (IJ) as s ~ 1I1J, 'whose cumulative
distribution is denoted by G (x, y), s=l, 2, ...
s
Suppose that the following conditions are satisfied:
(i)
:p s (z Ix, y) is continuous with respect to (z, x, y ) jointly, for every s.
(ii)
For (~, ~) there exist contants
c1, c~(>
0) and
d~ d~, s=1,2, ••• ,
such that the variable
(
tends to a certain fixed distribution (Yl' Y.) of the continu~ type in the
sense of type (JJ) as s~ co.
Then, two distributions whose probability density functions being
given by
(4.24)
-10
-10
24
I'
I
I
I
I
I.
I
_I
I'
I
I
I,
I
I
I
e.
II
I
I.
'I
I
~
J
1
J
I
1J
are "syrqptotically equivalent in the sense of type (I) as s~eo.
Applying these results to the problem stated in the section
1, we have
the following
THEOREM5 Let us
and the conditions
null-hypothesis
given by
con~der
(2.14)
the limiting process stated in lemma 1
given in Lemma
HO(h) given by
(1.6),
Then, under the tJ!~:r$.~,l'
the distribution of the F-statistic
(1.18) after the randomization is asymptotically eqQi.alent to the
F-distribution of degrees of freedom
as b
2.
(a,
n-b-v+l) in the sense of type ($)
~t:l.
Likewise, in the special case when h=m., under the 'total' null-hypothesis
H ('1' =0), the distribution of the F-statistic given by (1.21) after the
o
randomization is asymptotically equivalent to the F-distribution of degrees of
freedom (v:i, n-b-v+l) in the sense of type
(i) as b ~ 10.
.
~
This gives a justification and a probability t11eoretical meaning to all
the results
~hepto
obtained, concerning the asymptotic null-distribution of
the F-statistic after the randomization ~,2,f,4,5.).
1
]
I
1
ACKNOWLEDGE.MENTS
The authors wish to e}Q?ress their gratitude to Professor
L. Johnson
for his helpful discussions through which the accomplisl'llnent of this work
was accelerated.
Thanks are also due
skilful typewriting of this manuscript.
J
I
1·
1
~r.
25
to Mrs. Kay Herring for her quick and
REFERENCES
[lJ
J. Ogawa, "The effect of randomization on the analysis of randomized
block design", Ann. Inst. Stat. Math., Vol. 1.3, pp. 105-117 (1961)
[2J
J. Ogawa, "On the randomization 10 Latin-square design under the Neyman
model·, Proc. Inst. Stat. Math., Vol. 10, PP. 1-16 (1962)
[3J
J. Ogawa, "On the null-distribution of the F-statisticin a randomized
balance. incomplete block design under the Neyman.model.
Math. Stat., Vol• .34, pp. 1558-1568 (1963)
Ann.
[4J
J. Ogawa, S. Ikeda
[5 J
J, Ogawa, S. Ikeda and M. Ogasawara, "On the null-distribution of the
[6J
(7J
and M. Ogasawara, "On the null-distribution of tl1e
F-statistic in a randomized partially balanced incomplete bloclc
desigll with tvTO associate classes under the Neyman model", Inst. of
Stat., Univ. of N. C" Mimeo Series No. 414 (1964)
F-statistic for testing a 'partial' nUll-hypothesis in a randomized
partially balanced incomplete blo~k design with m associate classes
under the Neyman model" (35th session of the Interna.tional Stat.
lnst. 1965)
S. Ikeda, "Asymptotic equivalence of probability distributions with
applications to some problems of asymptotic independence", Ann.
Inst. Stat. Math., Vol. 15, pp. 87-116 (196.3)
S. Ikeda, "On certain types of asymptotic equivalence of real probability
distributions, I. Definitions and some of their properties,"
(SUbmitted to the Inst. of Stat., University of N. C. ~ Series)
. "''''''''.#
[8J
J. OgavTa and G. Ishii, "On the analysis of partially balanced incomplete
vlock designs in the regular caselli, Inst. of Stat., Univ. of N. C.,
Mimeo Series No. 412 (1964)
I
,I
e
I
,I
I
I
I
I
I
_I
I
I
I
I
I
I
I
-.
>1