Gertrude M. Cox
ASlMProTIC FORMULAE FOR THE DISTRIBt1l'ION OF
HOTELLING I S GENERALIZED
T~
STATISTIC
by
Kotchi Ito
University ot North Carolina
Special report to the Office of Naval Research
of work at Chapel Hill under Contract NR 042 031,
Project N7-onr.284(02), for research in probability and statistics. Reproduction in whole or
in part is permitted for any purpose of the
United States Government •
Institute of Statistics
Mtmeograph Series No. 133
June 21, 1955
•
•
ASYMPTOTIC FORMULAE FOR THE DISTRIBt1rION OF
HOTELLING 'S GENERALIZED T~ STATISTIC
By Koich! Ito
University of North Carolina
1.
Summary.
In this paper the asymptotic expansion of a percentage point of Hotelling's
generalized T~ distribution is derived in terms of the corresponding percentage
2
point ofaX distribution. Our result generalizes Hotelling's and Frankel's asymptotic expansion for the generalized Student T
L-2-', L-3-7.
The technique used in
2
this paper for obtaining the asymptotic expansion of T is an extension of the previoua methods of Welch
L-7-i and of
o
James
L-4-7, L-5_7,
who used them to solve the
distribution problem of various statistics in connection
w~h
the Behrens-Fisher pro-
blem. An asymptotic formula for the cumulative distribution function (c .d.f.) of
2
TO is also given together with an upper bound for the error committed when all but
~
the first few terms are omitted in the series.
This formula is a sort of multivari-
ate analogue of Hartley's formula of "Studentization" L-l.1.
2.
Introduction.
In the multivariate analysis of variance we use the folloWing canonical proba-
bility law:
(2.1)
. where Xl and Xo are pxm and pxn matrices respectively, and iii1 XlXl'
=Sl
is the sample
1 XOX ' = So 1s the sample "Within" dispersion matriX,
"between" dispersion matrix and 'ii
O
the prime denoting the transpose of a matrix.
~
1
m
1s a p x m matriX, -
~ ~
,
being the
population "between dispersion matriX, and ;1 is a p x p symmetric positive defilf
nite matrix.
1
It is assumed that m may be
2:
p or
< p, but n 2: p. To test the null
Work at Chapel Hill sponsored by the Office of Naval Research under Contract
NR 042 031. Reproduction in whole or in part is permitted for any purpose of
the United States government.
- '2 -
hypothesis
H:
O
t • 0, H\)telling
L-2.J
proposed a test based on the statistic:
(2.2)
and derived the exact distribution ot this statistic when p = '2 and t
eral values ot
in the null case
p the exact distribution ot TO2
~
•
0:
O.
For gen-
is not available at present, even
O.
'2
3. Derivation ot asymptotic formula of TO.
For general values ot
p
it 1s known that the statistic
X
'2
=
m tr 51
A
2
has a X distribution with m p degrees ot freedom.
pr{mtrS1A
where 29 denotes the tabled value of
p
=mp/2,
II:
x2
•
G (9) ,
p
tor a particular level of significance,
and
Gp (8)
That is to say, we have
[r(p)J-l
f
p
t - 1 .-t dt.
o
2
Hence, if 1\ is known, the statistic X given by (3.1) may be used to test HO exactly, and it 1\ is unknown but it So is based on a large number ot degrees ot
freedom, i.e., i t n is large, we may use as an approximation the result:
This suggests that in the general case we try to find a function h(SO) of the elements of So such that
- ~
When
n
2h(SO)
is large,
2h(SO)
will approach
2
as a series with X
2e ::
2
X,
and we now expect to write
as its first term and successive terms ot decreasing
order ot magnitude.
Now
Pr
~
d So\
)
\
,
where the first expression on the right denotes the conditional probability of the
relation indicated for fixed values of the elements of SO' and the second denotes
the probability element of SO' which has a Wishart distribution
with n degrees of freedom, and the domain of integration R is
values of the elements of SO.
Now we may expand Pr(m tr
over all possible
SlS~l ~ 2h(50 ) Iso)
about
an origin (°11 , 022' ••• , 0pp' 012' ••• , 0p_l,P) in a Taylor series, where
!
°11
°12
...
alp
°21
°22
• ••
02p
•
·..
•
°p2
• ••
°pp
I
/\-1
=
Thus,
(3.7)
Pr (m tr
51S~1 ~
2h(SO) ISo)
= {expL-
i
i~j=l
(sOiJ-Oij)
~ .J}
iJ
Pr {m tr 51 A
~ 2h( I\-l)~
~
where Soij is the i-th row, j-th vcolumn element of SO' and d denotes the matriX of
derivative operators:
40
0
(3.8)
1
I 3011
0
0
30
22
~302l
=
°
\ 1
1
• ••
• ••
°
\
1
230
lp
1 0
230
·..
·..
d
230
p2
230
pl
its typical element being 0iJ
0
2(50
12
2p
•
0
d'(J
pp
= !(l + 5 iJ ) a~ij ,where 5 1j is the Kronecker delta.
Whether uniformly convergent or not, the right hand side of (3.7) is an asymptotic
representation of Pr (m tr
Sls~l ~
2h(So)!So)
,for sufficiently large values of n.
Hence, substitution of (3.7) into (3.6) and term by term integration which 'may be
done legitimately, yields:
(3.9) Gp (9) =
J
ex:p[tr(So-/\-l)oJ Pr
(m tr
Sl/\
~
2h( ;\"1)} Pr
!d So)
R
=
8
l
~
Pr m tr Sl /\
2h( I\-l)}
,
where
8 = )f
expL-tr(so - /\-1)0.1
Pr {d
so} .
R
Since So has a Wishart distribution with n degrees of freedom, we have
\21
n-p-l
n
8 ::
exp
L- -tr /\-lo.1.const.l/\
Isol
expL-tr (SO 0 -
2
~
/\ So).1 d So
R
n
= exp
L--tr A-1....oJ·
const. \/\
\2
f
n-p-1
IsOI
2
R
n
n
= exp L-. tr /\ -1 0_7. 1/\12 1/\- ~ 01 2
n
= exp
L-·tr 1\-1 0_7
• II
_
~ 1\-1 01- 2 ,
expL--
~ tr( /\ - ~ 0)S0-7
d So
- 5 where I is the
(3.10)
;,ie
p
x
- log
identity matrix.
p
II
- YI
= tr
Now using
~tr
Y +
2
y
+
L-5-7,
3
3
tr y +
... ,
obtain
(3.1l)
8 ...
exp
L--
tr A -1~0
==
exp
L--
tr
= exp
-
'2n log I
2 A-l~0
I - Ii
I .J
2 -1::-..) 1 tr (2 -1~)2 1tr (2 _1()3
1\-1d + '2n lJ tr(ii!\
0 +'2
IiA 0 +'3 Ii" + ...
L- -n1 tr (-1::-..)2 + 3n4
~ tr
1\'0
(A-l~)3
I \
0
+ •••
.J
It is to be noted ~~r~ that in (3.11) the operator d does not act on
E1
J.J
/\-1
present in
itself, and it 1s more useru1 ror our purpose to write (3.11) in surfix form:
Q
C7
== 1 +
1:n
L.
0'
0'
rs
+.!...f~r.O'
2
n
13
tu d st () ur
0'
0'
0 () ()
rs tu vw at Uv wr
+.!I.O' 0' CJ a () 0 d 0 1
2 rs tu vw xy st ur wx yv)
+ O(n -3) ,
where L. denotes the summation over all surfixes r, s, ••• each of which ranges rrom
1 to p.
Now we represent h(SO) as
s
hS(SO) being of order n- , i.e., we write h(SO) as an asymptotic series such that
is made arbitrarily small for sufficiently large values of n.
t
substituted into Pr m tr 8 1\
1
~
Then (3.13) may be
2h( 1\ -I)}, and by Taylor's expansion we have:
- 6(3.14)
Pr
tm tr 61 /\ =s 2h( 1\.""1)}
= expL- { h e,,-I) + h 2
l
=L-l+
J D_7
...
Pr
tm tr 8 1 "
=s 2Q}
2
{h (I\-I) + h (/\-I) + ••• JD +
2
l
x Pr
where D ~
(0 -1) +
t
m tr 8
1
1\ ~
~ {hl (/\-I)+h2 (J\-I)+ •••JD2+ •••J
2Q} ,
o By substituting (3.12) and (3.14) into (3.9) we obtain
OQ.
G,,(Q)
=
....
i-I + .!n 1:.
4
a
a
ara at u s tour +.!..2 t.;I.3 ars at u avw 0a touv wr
n
+ l:2I'. ars at u avw axy0 6 tour
°
WX
dyv Jl +
°(n- 3)- 7
~ 2Q } •
x Pr { m tr 8 1 /\
By equating terms of successive order in (3.15) we obtain
0.16)
(3.17)
+
~
3nc::
a a a 0 d + ~ r. a a C1 a 0 a 0 0 )
rs tu vw at uv wr
2nc::
rs tu vw xy at ur wx yv
i. a
!
x Pr m tr 6 1 /\
S
29
J = 0,
and so on, where hlat ) { 1\-1) = d h ( A-I) and
st l
hl
at , ur) (1\-1) = 0urOst h (i\-I).
l
- 7 ~
It now remains to carry out the operations
and D indicated in (,.16) and
0.17)1n order to obtain h (,/\ -1) I h 2 ( 1\-1) and hence h (sJ h (SO)'
l
l
2
:5
torsw1l1 operate on Prtm tr Sl/\
These opera-
2Q} which is a p x m fold integral, and the
operations may be thought of as differentiations, with respect to the boundary only,
of the integral of the probability density function of the Xl throughout a region in
the space of Xl'
t
The method used to evaluate ~st~ur Pr {m tr Sl/\
~st~uv~wr Pr m tr
8 /\ :::; 2Q}
1
I
••• ,
:s 29} ,
is to change the boundary slightly, expand the
integral in powers of the quantities specifying this change, and obtain the derivatives by comparison with Taylor's expansion.
0.18)
where
J =
€
Pr lf m tr Sl ( A1 +
is a p x p symmetric matrix.
J =
We consider
lf 1
+ L
,l
€
+ 1;.. L
+ •••
J
€
E
:::;
'(
2Q) ,
Then by Taylor expansion we have
rs ~rs + 21,• L
E
)-1
€.
rsE t u~rs ~t u
~ ~ ~
€
rs tu vw rs tu vw
Pr { m
tr 8 1 1\
~
+
~
~!
L
€
E
€
€
~ d ~ 0
rs tu vw xy rs t u vw xy
2Q} •
On the other hand, J is, by definition, written as
m
0.20)
J
=
J.6t.
E!:
(211:)2
where X1X~
=
J
,
R'
ms , and domain of integration R' ranges over all possible values of
l
the elements of Xl such that
m tr Sl (!\
-1
+ €)
-1
<
2Q.
It is now easy to show
that integration of (,.20) yields
m
II-D
0.21)
J ..
E
I -2
<1f.:ij )
Tl
G (6),
p
- 8 whereD~
is a diagonal matrix which satisfies
~r' A r =
and
I(p) -
D~,
'r being a non-singular matrix, and E is an operator such that
Now letting
A = E - 1 and using
/I-D ~EI
II-D~I
=
=
(3.22)
we have
II-D1)-D1)AI
=
/I-D ~ I
I~r'" .r - {lr' (A -l+E) -lr. ~r' 1\ r} A I
I~r'!\ rl
II -
X
AI
m
J
= II - X AI- 2 Gp (G)
Now using (3.l0)again, we rewrite (3.23) as
= exp { 2m tr
XA +
33m
4 4
'4m tr X22m
"A + b tr X A + 'S tr X A
+
... } G
p
(e)
- 9 -
2
2
!m
4 m (
)
.3
m (
2 2
+t~ tr X +12 tr X (tr X ) + 32 tr X )
4
3
m
m
7
+ J2(tr
X) 2 (tr X2 ) + 3S4(tr
X) 4} 6 4+ ....vGp(e).
X can be represented as
where
A rs
-1
is a p x p matrix obtained by operating 0 onA, Le.,
rs
1
row, J-th column element, -2(8 r 18 a j + 8 s~.8 rJ. ).
Ars
-1
has its i-tb
Writing
tr(!\ -lA) = (rs),
rs
tr(!\ -1 A)( 1\ -1/\)
rs
tu
= (ra Itu) '.
tr('I\;;;\)(I\~~!\)(I\;;A) =
(rsltulvw),
tr(A -1 A )( A telA HA -11\)(!\ -1 A) = (ra Itu Ivw Ixy),
rs
u
vw
xy
and substituting (3.25) into (}.24), we obtain
2
0.26)
J = L-l+'£'E
+~~'£€ra€tu€vw
rs t -~(rs~} +~~ r€rs6tu [ (r8Itu)(~2)4(rs)(tu)62}
{(ra Itu Ivw)
(_3~_3m2 _m3 )+ (ra) (tu Ivw) (_~262~263)4(rs) (tu)(vw )63)
+hr€r8€tu~€xy { (rs Itu Ivw IXY)(1~18:1Jl62+1~3+3mA4)
I I
222324
·2223324
~(rs)(tu vw xy)(6mA+6mA+2mA) + (rsltu)(vw/XY)(3mA.+?m6+4:mA)
- 10 -
4 4 4
+(rs)(tu)(vw IXY)(~'6'+~'6 ) + (rs)(tu)(vw)(xy) ~ 6 } + ...J Gp(e).
Then term by term comparison between two expansions for J , (,.19) and (,.26), gives
t
~rs Pr m tr Sl/\ ~ 2Q} , ~rs~tu Pr Lm tr 61 /\ ~ 2Q j
,
etc., but in doing so we must
take such a care that, for example,
are completely symmetrical in their sufimplies a ijk = b ijk if both a ijk and b
iJk
fices. With this in mind and using the relation
~ G
p
(3.28)
~rsdtu
(,.29)
~rs~tudvw
(e)
Pr {m tr Sl"
Pr
t
m
=-
E gp (e),
~ 2Q J
tr Sl/\ ~
2
=
26}
-l ~(rs I tu)(E 2 +E)+ T(rs)(tu)(E2-E)}
gp(Q)
2
3 2
= {m(rs ltu IVW)(E +E +E)+!ir L-(rs)(tulvw)
+ (tu)(rslvw)+(vw)(rsltu)-l (E 3.E)+
~(rs)(tu)(VW)(E'-2E2+E)}
• gp (Q) 1
().30)
drsdtudvwdxy Pr
{m
tr Sll\ ~ 2Q) = - [mL-(rsltulvwlxy)+(rslvwlxYltu)
2
+(rslxYltulvw)-l(E4+E3+E2+E)+ irL-(rs) (tu Ivw Ixy)+(xy) (tulvw Irs)
2
+(vw)(tulxylrs)+(tu)(vwlxylrs)~ (E 4_E) + irL-(rsltu)(vwlxy)
3
m
+(r s lvw)( tu lxy)+(rslxy)(tulvw)-7(E 4+E3
-E2
-E)+
-g L- (rs)(tu)(vwlxy)
+(rs)(vw)(tulxy)+(rs)(xy)(tulvw)+(tu)(vw)(rslxy)+(tu)(xy)(rslvw)
4
m
4-3E 3+3E 2-E)} gp(Q).
+(vw)(xy)(rs Itu)_7(E 4
-E, -E 2+E)+ro(rs)(tu)(vw)(xy)(E
.. 11 Upon substituting (3.28) into (3.16) we obtain
and also,
Hence we have
rs at u (stlur) =;D(ptl)
~
Ea
and
~
ars atu (st)(ur) = p.
We also note that 26 =
x2, P = mp/2. Therefore we finally obtain, after some simpli-
fication,
In a similar way we substitute (3.29), (3.30) and (3.31) into (3.17) to evaluate
h ( 1\-1). We note here that since h (A -1) given by (3.31) is independent of A-1,
2
1
the terms involving hi st )(I\-l) and hist ,ur)(/\-l) in (3.17) do not appear. As before;
it can be easily shown that
r. ars atu avw (stluvlwr)
= r.
=
ars at u (Jvw (uv)(stlwr)
~0 p(p2+ 3P+4), r. ars at u avw (st)(uvlwr)
= r.
ars at u avw (wrHstluv)
=~D(pt-l),
~
r. ars at u avw (st)(uv)(wr) = p, r. ars at u avw xy (stlurlwxlyv)
(J
=
r: ars at u avw axy (stlwxlyvlur) = ~Lf- P(P+l)2,
- 12 L a
a a a (stlyvlurlwx)
rs tu vw xy
=~p(P+3),L
'+
= I:
ars at u avw axy (yv)(ur Iwx 1st)
= l:
= l:
ars at u avw axy (ur)(wxIYVlst) =
a a a a (st)(urlwxlyv)
rs tu V\'1 xy
ars at u avw axy (wx)(ur Iyv 1st)
~'O(P+1),
~
L ars at u avw axy (stlur)(wxlyv) = :;-o2(pf-1)2,
l: a at a a (stlwx)(urIYV)
~
rs u vw xy
= l:
ars at u avw axy (stlyv)(urlwx)
= ';'O(pf-l),
c:
L a
rs at u avw CJxy (st)(ur)(wxIYV)
= r.
ars atu avw axy (wx)(yv)(stlur) =
L a at a a (st)(wx)(urIYV)
rs u vw xy
= r.
a at a a (st)(yv)(urlwx)
rs u vw xy
= 1:.
ars at u (1vw axy (ur)(wx)(stIYV) = 1:. ars at u avw axy (ur)(yv)(stlwx)
~-o2(pf-l),
~
= p.
and
r.
a at a a (st)(ur)(wx)(yv)
rs u vw xy
= p2.
Using these results we obtain trom (3.17), after some simplification,
(3.32)
h ( /\ -1)
2
=
1
2
48n
L- 6 (p-1)(f 2 )(m-l)(ml-2) x8
(mpf-2) (mv·4) (mpf-6)
2
2
2
+ •4mp3+2(3m +3u*10)p2+ 2 (2m 3+3m +11m+ 18)pt4 (5m +9m+ 2)
(mpt2)2 (mpt4)
x6
which is independent of /\ -1 just as h ( 1\ -1) •
l
Now we substitute (3.31) and (3.32) into (3.13) to obtain
T~ = 2h(SO) = 29 +
k
J
2
2h1 (So)+2h (So)+O(n- 3 ) = x +
{~1 X4+(p_mr1)· x2
2
2
2
2
+ 1.- (6(P-1)(I*2)(m-1)(mt2) )(8+ 4mp 3+2(3m + 3mt lOh?+2(2m3+ 3m + 17m+ 18)I*4(5m +9mt 2):)(6
2
24n
(mI*2)2(mpt4)(mpt6)
(mpt2)2 (nipt4)
(3.33)
2
+ 13p
2
+;~;llm +7 )(4
+L- 7p2+ (-12mt-12)p + (7m2-12m+1 t7x2 )
+ O(n- 3 ),
- 13 which is the asymptotic expression of a percentage point of the TO2 distribution in
2
terms of the corresponding percentage point of the X distribution with mp degrees
of freedom.
If we put m • 1 in (3.33), we have
which is the asymptotic expression of a percentage point of the generalized Student
T distribution.
This result (3.34) was previously obtained by Rotelling and Fran.
2
4. Asymptotic formula for the c.d.r. of TO.
2
Let F(2Ql) be the c.d.f. of TO' i.e.,
(4.1)
•
Then, as (3.6), we can write
(~.2)
Pr {m tr
818~1 $ 291 }
s
f t
Pr m tr 8 18
0 :: 2<11 1 So l Pr (elS O)
1
R
= e Pr {m
tr Sl 1\
where ~ is given by (3.12).
,
Upon substituting (3.28), (3.29) and (3.30) into
(4.2) we obtain, after some simplification,
1
.24 { mp3+2(m2+mt4 )p2+ (m3+2m2+2lm+20)pt-8m2+2Qrrt-20) Qi
- 48n2 L
(mpt-2)(mpt-4)(mpt-6)
4 { 3mp3_2(3m 2_3m_4)p2_3(3m'+2m2+llm_4)p-4cm2-,an.4j ~i
+
(mpt-2}(mpt4)
- 14 2 {3mp3+ 2 (3m2+3m_4)p2_3(3m3_2m2_5mt4)p_&n2+12mt4 } 9~
mp+2
- {3mp3_2(3n?_3m+4)p2+3(m3_2m2+5m_4)p..8m2+12mt4; 9 _7g (e ) + 0(n-3 ) ,
p l
1
where Gp (9 )
1
f
= ["r (p ).J-l
9
1
0 Gp (0"'1 ) ' and p = mp/2 •
t P• l e-t dt., gp (0)
"'1 = de:"
1
o
(4.3) is a sort of multivariate analogue of Hartley's formula of "Studentization."
In fact it can be shown that when p
for
th~
= 1,
(4.3) coincides with Hartley's formula
c.d.f. of the univariate analysis of variance F statistic. (See equation
5. Discuss ion of the error end rema:':'ks.
In view of the methods used in sections 3 and 4, it is rather difficult to set
a bound for the error committed by omitting all terms after the first few terms in
2
the asymptotic formula for TO
2
of TO (4.3).
(3.33) or in the asymptotic formula for the c.d.f.
There is, however, a method to find lower and upper bounds to the
2
c.d.f. of TO which is fairly good for large values of
n, and they can be used to
set a bound for 0(n- 3 ), say, in the asymptotic expansion of the c.d.f. of T~.
2
We shall first obtain lower and upper bounds for the c.d.f. of TO.
known (e.g. see
L-6.J)
It is well
that the joint probability law of the characteristic roots
e l , e 2 , ••• , e s of m tr 818 -1 under the null hypothesis H is given by
O
0
where 0
~
e s $ e s _l
~
...
S e l < 00,
s
= min
(p,m), t =
max
(p,m) and
s
'2
C(s,t,p,n)
s
= ..2L
st 11"
"2
n
i=l
•
- 15 2
The statistic TO is expressed as
2
and the c.d.f. of TO is given by
s
J••• J "IT
R
1
1=1
t-;-l
e1
where R is the domain of integration such that 0
l
o<
s
S 2Ql.
.t e
i
i-l
S e s S e s _l S ... S e l < 00
Now for any non-negative values of e
i
and
and n , the following
inequality holds:.
e
i
log (1 + - )
n
for i
= 1,
e
< -n1
-
••• , s" where equality holds when e
mtn
s
1T
1=1
(1 +
e
-!)
n
- -
2
mt-n
>
i
=0
or n
->
00.
Hence we have
8
- 2n i~l e 1
e
.
Therefore the probability law (5.1) is bounded from below as follows:
where
It must be noted here that P (el, ... ,e ) is not a probability law although it is
l
s
non-negative for all e i such that 0
sides of
(5.4) in Rl we obtain
S e s S ••• S e l < 00.
Now integrating both
- 16 where
s
J.•• J TI e
i=l
R
l
t-s-1
. 2
de i
i
and also integra.ting both sides of (5. 4) in R2 where 0
s
2Q ~ E e < 00 and subtracting each from 1, we have
l
i=l i
~
es
~
...
~
el <
00
and
00,
2
TO
where
In order to evaluate Fl(2Ql) and F2(2Ql)' we observe that as
•
n tends to
2
s
E e has a X distribution with st degrees of freedom in the Itmit, i.e., we have
i
i=l
(5.7) K(s,t,p)
J J
•••
R
l
s
t-s,.l
1T e i
2
1=1
where
s
K(s,t,p)
=
'2
Itm
e(s,t,p,n)
n->Oo
= ~st
-2
2
and P l = st/2.
1
s
1( r t~(t-eti)} r(~)
1=1
Hence integratLon of (5.5) yields
where
L( 8, t ,p,n )
= Kc~s,t,p,nl
s,t,pY--
Stmilarly we obtain from (5.6)
at
(E-)2
m+n
•
- 17 Now i f we write (4.3) as
,
where R is the error committed by omitting all terms except the first three terms
3
in the asymptotic series of F(2i ), the absolute value of R has the following
l
3
upper bound:
The actual manner in which (3.33)
2
converges to the true value TO or (4.3) to
the true value F(2e ), is not known, but it is hoped that the use of the first few
l
2
corrective terms may result in a test which is more accurate than the X approxtmation, at any rate for moderately large values of n.
In the case of the asymptotic
2
formula for the c.d.f. of TO (4.3) we may judge the magnitude of the error involved
in using the first few terms of the series
small numerically when
n
by
(5.11), which
~urns
out to be rather
is sufficiently large.
The author wishes to express his indebtedness to Professor Harold Rotel1ing for
suggesting this problem and for his guidance in the preparation of this paper.
REFERENCES
L-1J
R. O. Hartley, IIStudentization, or the elimination of the standard deviation
of the parent population from the random sample-distribution
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L-2J
R. Hotelling,
L-3_7
R. Rote11ing and L. R. Frankel, "The transformation af statistics ta simplify
their distribution," Annals of Mathematical Statistics, Vol.
9 (19}8), pp. 87-96.
IIA generalized T test and measure of multivariate dispersion,lI
Proc. Second Berkeley Symposium on Math. Stat. and Probability,
1950, pp. 23-41.
- 18 "The comparison of several groups of observations when the
ratios of the population variances are unknown," Biometrika,
Vol. 38 (1951), pp. 324-329.
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analysis when the ratios of the population variances are
unknown," Biometrika, Vol. 41 (1954 ), pp. 19-43.
lip-statistics and some generalizations in analysis of variance
appropriate to multivariate problems," SankhY!, Vol. 4 (1939),
pp. 381-396.
"The generalization of 'Student's' problem when several different population variances are involved," Biometrika, Vol. 34
(1941), pp. 28-35-
·e