~
The Nsear'Ch in this repon was pa:PtiaZ'Ly sponsored by the Nationa1, Science
Foundation GFant No. GU-20S9 and the Sakko-kai Foundation
,1M
and the Institute of statistical Mathemaf;iC8~ To1<yo•
•
ON THE DERIVATION OF ASYMPTOTIC DISTRIBUTION
OF THE GENERALIZED HOTELLING'S T~ ~
by
Takesi Hayakawa
~,.
Department of Statistics
Univel'Sity of NoFth Caz;o'tina at ChapeZ BiZZ
Institute of Statistics Mimeo Series No. 692
July, 1970
On the derivation of asymptotic distribution
2
of the generalized Hotelling's TO
Takes:l. Hayakawa
University of North Carolina at Chapel Hill
Institute of Statistical Mathematics (Tokyo)
This paper considers the derivation of the asymptotic expansion of the prob. 2
ability density function of a generalized Hotelling's TO. To treat this problem,
we make some important formulas for the generalized Laguerre polynomials and for
the univariate Laguerre polynomials.
CL.AsSJFIrATION tbeER:
KEv WORDS
(i)
(ii)
AND ~ES:
generalized Hotelling's
T~
generalized Laguerre polynomial
On the Derivation of Asymptotic Distribution
2
of the Generalized Hotelling's
It
TO
Takesi Hayakawa
Department of Statistics
Univemity of Norrth CCU'oZina at Chapel, HiZl
and
Institute of Statistical
1.
INTROIU:TIOO AND
Let
51' 52
Tokyo
Sf..M.Mv.
be independent
dom, respectively.
Mathematics~
8
2
mxm matrices on
n
l
and
having a Wishart distribution and
51
central Wishart distribution with the same covariance matrix.
eralized
T~ statistic is defined as
When
becomes large, the distribution of
on
n
mn
l
2
degrees of freedom.
n
2
degrees of free-
having a nonHotelling' s gen-
T~ approaches that of x2 based
Ito [6] and 5iotani [9] have independently derived
asymptotic expansions for the cumulative distribution function (c.d.f.) of
in the central case by using the idea of a perturbation as in physics.
T~
The non-
central distribution was treated by 5iotani [10) in the same way and later by
tto [7], who used the integral representation of the characteristic fwction of
2 due to Hsu [5].
-2 • Recently,
2
2
-2
5iotani [11] has derived the non-central distribution of TO up to order n
2
TO
.
But neigher could obtain the term of order n
from the expansion of its characteristic fmction by applying the method of
perturbation.
The research in this repopt !Vas partialZy sponsoNd by the National Science
Foundation Gpant No. GU-2069 and the Sakko-kai, Foundation.
2
The exact distribution of
T
over the range
0
S
T < 1 has been obtained
in the general non-central ease by Constantine [1]. using the method of zonal
polynomials and generalized Laguerre polynomials of matrix argument developed by
James [8] and Constantine [1].
Constantine's solution has the form
~-l
f(T)
(2)
'1~
..
1. e.t't( -S)
where
m
r m(a) .. 'ITb.(m-l) n r (a - ail),
p
..
t(m+l),
a=l
(a)
(a)
K
and
L-tD-P (5)
K
n
..
a(a+l) ••• (a+n-l),
is a generalized Laguerre polynomial of matrix argumentS,
sponding to a partition
K
of
k
into not more than
m parts.
corre-
Davis [2] sug-
2
gested that the asymptotic expansion of the c.d.f. of TO can be obtained from
(2) by changing the argument to
...L2
T = n L
and integrating term by term.
2 O
In
this paper, we give another method of derivation of the asymptotic expansion
(for
n
0
~T
< n )
2
of the probability density function (p .. d.f.) of
T~ up to
-2 by preparing some formulas for the generalized Laguerre polynomials of ma2
trix argument and for univariate Laguerre polynomials..
We think that this method
is simpler than those of previous authors.
2. SorJe USEFUl FORMlA.AS
lAGLERRE
FOR GENERALIZED
UGERRE
POLYr«lMI~ Nm THE WIVARIATE
Pa..YNOMIAlS.
Constantine defined the generalized Laguerre polynomial
Ly (5),
Ie
by the Hankel transform of matrix argument in the· following way ..
(y > -1),
3
Let
LY(S)
Sand
R be positive definite symmetric matrices of order m,
then
1s defined by
K
~(-S)LY(S)
(3)
..
K
where
A (RS)
Y
fR>O
A (RS)(detR)Y
Y
et~(-R)C
(R)dR,
K
is a Bessel fwction of matrix argument and
polynomial corresponding to a partition
K of
k
C (R)
K
is a zonal
into not more than
m parts.
The following lemmas are very important for our argument.
~
1.
(Bayakawa [3])
I
(4)
L~-P(S) .. ~mn-l(~),
p
..
~(m+l),
K
where the left hand side (L.B.S.) is a summation over all partitions
K of
k
into not more than m parts and the right hand side (R.H.S.) is a univariate
Laguerre polynomial.
~
2.
Let
tion
(Sugiura and Fujikoshi [12 J)
C (S)
K
be the zonal polynomial of degree
K" {kl, ••• ,k }
m
definite matrix
S.
of
k
(k
l
~
k
2
~
•••
~
k
m
Put
(5)
m
L
(6)
a= 1
2
ka (4ka -6aka+3a 2 ).
Then the following equalities hold
(1)
~
xk
, L al(K)C (S)
lk
k-O • K
K
..
2
2
x ~ ett(xS),
k
~
0)
corresponding to a partifor an
mxm positive
4
(8)
+ xVtS]eVL(XS).
(9)
We can give a similar set of results for Laguerre polynomial
Let
L1?n-P(S)
I<:
be a generalized Laguerre polynomial corresponding to a
partition 1<:. {kl, ••• ,k } of
m
definite matrixS,
(10)
and
2 al(I<:)L:U-P(S)
a (1<:)
1
•
I<:
k (k
and
l
~
! a2(I<:)L~-P(S)
..
~
•••
k
m
~
for an mxm positive
0)
are given by (5)
a (1<:)
2
and (6).
Then
k(k-l)r(n~1).~1(t1r.S)
_ (n+m+1)tJts
(11)
L:U-P (S).
'L~2(tIlS) + tIlS 2
iK-2
k~ L~l(tItS)
+ 3k(k-l)
- tItS
t{~il
fim{ (m+l)n+m+3}
[4
- {(m+l)n + m + 3} .tItS
+ 1/(tJts)2 +
21
- .tItS'
~+5
(tltS
8
L.%mn+1 (.titS)
1K-2
~+2
+ {(tIlS)2 + tIlS 2 } ~+3 (tltS
+ 4k(k-1) (k-2)
1,
'LJsmn+3 (tltS
1K-2
~
(bLS)
8
~{n2+3(m+l)n+m2+3m+4}'L~2
[
8
(n+m+2)tlr.S 2} L..;mn+4 (btS)
.
(WD.
iK-3
K-3
(tItS)
(12)
~ ai(lC)L~-P(S)
•
k(k-l)
r
5
H m+11n+m+3}
- {(m+1)n + m + 3} tItS
+ {(bts)2 + .tJr.S2}
tt~+l
(.tJr.S)
~~2 (.tits)
~+3
(.tJr.S8
+ 4k(k-1)(k-2) ~{n2+3(m+1)n+m2+3m+4}~+2 (ns)
2
2
- i{n + 3(m+1)n + m + 3m + 4} .tJLS
+
~ (.tItS).
21
2 + (U+m+2).tJts 2}
L.~+4
~-3
~3
(.tJLS)
(.titS)
- W 3 L bn+5 (.:tJtS)l
k-3
~
+
k(k-1)(k-2)(k-3)~{mn3 + 2(m2+m+4)n2
2
2
+ (m3+2m +21m+20)n + 4(2m +,Sm+S>}
2
+ 4{2m +5m+5)} titS
+
~ 2[ (m+n+1) 2
rtr
+ 6](.:tJtS)
4
2
~3
(tILS)
(.tItS)
+ [(mn+20)(n+m+1)
_ {2(n+m+1).tJtstlr.S 2 + 8tJtS3} ~6 (.tJr.S)
+ (tltS 2)2
PRooF:
From the definition of
function of
E al{K)L~-P(S)
lC
K
~+7 (.:tJts~.
L~-P(S)
and Lemma 2, (7), the generating
lC
is given by
6
•
x2
f
A,--_ (RS) (de.tR)~-P e.tJt(-(l-x)R) brB.
R>O 7U P
2
Hence by the definition 6f
L~-P(S),
K
again,
dR
the R.H.S. is
Therefore, we have
r .!.... 1:
k-O k!
00
(13)
k
K
L~-P
a (K)
1
(S)
K
2
= x (l-x)
~-2
r:-~-p
x
e.xp(- 1-x:tltS) t(2)
S
<I-x)
Since from the tables of Constantine [1],
the R.H.S. of (13) becomes
Hence by the use of the generating function of a univariate Laguerre polynomial,
i.e.
(I-x)
-a-I
x
e.xp(- --I-x z)
00
•
k
1: :, x.:<z), <Ixl
k-O
<
I),
7
we can expand the R.H.S. of (13) as a power serJ.es in
efficient of
x
k
x,
and comparing the co-
on both sides of (13), we have (10).
The proofs of (11) and (12) are done completely the same way as the one of
(10) •
In these cases, as we need the explicit fonDS of Laguerre polynomials
corresponding to .tItR.,
VcR2+(tltR)2 ,.t1tR 3 and
(tltR2) 2 ,
we write them here and
will omit the details of the proof.
~-p(5 )
L(1)
(14)
DID
= T
-
"".. '"
"VW
(15)
+ tJcs2 + (tJcs) 2 •
~n-p
(16)
L-n-p
L~-p
L(3) (5) - %L(21)(S) + 4L(13) (5)
mn
2
2
= 11
{n +3(m+l)n+m +3m+4}
2
- ~ {n +3(m+l)n+m2+3m+4} tits
+ ~ {(.tJr.s) 2+ (n+m+2).tJr.s 2 } - .t!l.S 3 •
L~-P(S) - ~~-p(S) + ~~-p(S) - !L~-P (5) + ~~-p(S)
(4)
6 (31)
12 (22)
6 (212)
4 (14)
(17)
Put
h
(x z)
2a+2j'
= h
2a+2j
•
a+j-l
ClCl
(!!.)k
exp(- x)
x
t -,,-;2;;.-_.___
2 2 a +jr(a+j) k~O k! (a+j)k •
8
and put
i.e.,
g2a+2j(x,z)
tribution with
is the probability density function of non-eentral
2a+2j
degrees of freedom and non-centrality parameter
X2
dis-
z,
and,
for convenience
(a) (x)
..
a-l
_x _
a
2 r(a)
•
Then the following equalities hold.
(20)
(21)
(22)
(23)
(24)
(25)
•
- zh 2a+4
+
(z-a)h2a+2
(26)
IX)
(27)
(a) (x)
I
k=l
~)~ (z)
(k l)!()
a k
(-
..
zh 2a+6 - (2z-a-l)h 2a+4 + (z-a-l)h2a+2
9
ClO
(28)
(a) (x)
(_
l
k=1
~)~+I(Z)
(k I)!()
a k
•
- zh 2a+ 8 + (3z-a-2)h 2a+6
- (3z-2a-4)h 2a+4 + (z-a-2)h 2a+2 -
(20) is another type of generating function for
(21), (22), (23) and (24) in the same way_
~ (_;)k~ l{z)
L
k=O
ClO
•
kl(a)k
-JL- I
(a)2 k=O
l.ka-1 (z) _
We can show
We prove here as an example, (22).
(_~)~+l(z)
k!(a+2)k
{k(k-1) + 2(a+l)k + a(a+1)}
Hence the third term is
(29)
By differentiating both sides of (29) with respect to
sides by
x,
x
and multiplying both
then
Thus, the second term is obtained.
By differentiating both sides of (29) twice with respect to
tiplying both sides by
ltr a+1(z)
2~
k-2 (k-2) I (a+2)k
co
t
L
(_ x)
2"
x,
then
~2 co
(xz) it
xx
t
2
.. e.Xp(- -) L
- 2
.
2 22 k=O k!(a+2)k
x
and mu1-
10
Adding these results, we have
Hence we have (22) by multiplying both sides by
a(x).
Formulas (25), (26), (27) and (28) can be obtained in the same way.
example, we prove (27).
By
differen~1at1ng
From (21),
both sides of the above equation with respect to
multiplying both sides by
As an
x,
we have
x
and
11
Since
(k-l)!(a+3)k_l'
(27) follows by simple calculation.
ft>TE:
In Section 3, we sometimes need the following type of summations.
example,
a-I c:o
x
I
2a r(a) k=4
X~+5
(--)
2
-
4(z)
(k-4)!(a)k
,
'
etc.
The first sum is obtained in the. following way.
a+3
(_x)k-4~+5(z)
x
I 2 ~-4
2a+4r(a+4)k=4 (k-4)!(a+4)k_4
CIO
=
(a+4) (x)
c:o
(~2)lt.ra+5(z)
keO
kl(a+4)k
I
~
The second sum is obtained in the following way.
=-
zh2a+12 + (3z-a-4)h2a+lO
For
12
3.
IERlvATION OF nE ASVrtPTOTIC PRJBABIUTY IENSITY MerION OF
In this section, we denote
n
by
1
T~.
n.
Let us write
(30)
Then the p.d.f.
of
= T~
x
is represented by (31),
n+n
.1-
(31)
f(x)
n --pmnr (---1.)
2
m 2
n
r (....£)r(mn)
m 2
2
This series is convergent for
Ixl < n •
2
Using a Stirling-type asymptotic expansion, we have
(32)
r
•
n
(..1.) r (ID11)
m 2
~ + .E!!!..
(n-m-1)
4~
1
2~r(mn)
2
2
+ mn2
96n
{3m3n-2m2 (3n2-3n+4)
2
3 2
+ 3m(n -2n +5n-4) -
(33)
=
L 1{
n2kl~
(T)
+n
2
an2 +
Un + 4} +
O(l/ni~.
}
a 1 (K) + nk
+ 1 2 {3ai(K) - a 2 (K) + 6n{k-1)a1 (K) + 3n2k{k-1) + k}
6n
2
+
O(l/"2~l
Therefore, by inserting (32) and (33) into (31), we obtain the asymptotic expansion of the
p.d.f. of
x
up to order
2.
13
dJt(-S)
(34)
x~-1 00
I
1
(- !.)k
2~r (mn) k=O k I (!!!!)
2
2
2 k
+
4~2 {~{mn(n-m-1)
+
~
96n
2
~I L~-P
K
2
- 8n + 12n + 4}
L:U-P (S)}
+ 4(&1(te)+nk)}
2
3
{I{mn{3m n - 2m (Jn2-3n+4)
te
+ 24mn(n-m-1)
+ 3m(n3-2n 2+5n-4)
{a (te)
1
+ 1613ai(K) - a 2«) + 6n(k-l)a1«) +
+
(i)
O(l/n~~.
(S)
te
+ nk}
3D~(k-l) + k)}L~-P(S)}
.
The liNt 't6rm is obvious from (20) and Lemma 1.
(35)
(ii)
The term. of ordeza 1/""2:
From (10)t (20)t (21).8Qd (22)t
~-1
(36)
~(-S)~
2
rem)
2
00
I
(_
x)k
2
k=O k I (!!!1)
2 K
I
K
a (te) L~-P (S)
1
te
14
•
~
2
hm(n+m+1) - 4(n+m+1).tJts + 4t'tLS } gmn+4
2
+ {(n+m+1).t'Jl.S - 2.tItS } gmn+6 +.tItS
2
gmn+8'
and from Lemma 4, (25),
(37)
= -
.tIlS~+4
+ (.tItS - U;>8mn+2 •
Hence, by combining these results, we have the term of order 1.
(38)
mn(n-m-1)gmn - 2n(mn-2.tItS)8mn+2 + {mn(n+m+1)
1 •
A
2
- 4(2n+m+l)VLs + 4tltS }gmn+4 + 4{(n+m+1).t!tS
2
2
- 2.t1tS }gmn+6. + 4.tItS 8mn+8 •
(iii)
The tem of oI'dsr 2:
From Lemma 3, Lemma 4 and Note, we have the following formulas.
(39)
=
~ [mn{ (m+1)n + m + 3} - 4{ (m+l)n +
+ 4tltS
2
+ 4(.tItS)2] 8 +4 -
mn
m + 3} .tItS
~ [m{n2 + 3(m+1)n + m2 + 3m + 4}
- 2{3n2 + 10(m+l)n + 3m2 + 10m + lS}.t!tS + 4{4(.tItS)2
+ (3m+3n+7)VLs 2 } - 8.tItS 3 ] 8m+6 +
ft (mn{mn3 + 2(m2+m+4)n2
15
2
2
2
+ (m 3+2m +39m+38)n + 2(7m +19m+S2)} .tJr.S + 16{n + 2(m+1)n
+ m2 + 2m + 20}(btS)2 + 8{mn 3 + (m2+m+44)n + 44m + 82} .tItS 2
+
t [{mn3 + 2(m2-f1n+4)n2 + (m3+2m2+2lm+20)n + 4 (2m2+5m+5) }.tItS
- 4{n
2
2
2
2
+ 2(urH)n + m + 2m + 10}(.tItS)2 - 2{mn + (m +m+-23)n
2
+ 23m + 38} W
+ 20m + 32}tltS
2
+ 12{(m+n+l).:tItStltS
- 4{3(n+m+-1).tJtStItS
2
2
2
2
+ 6tJL.S3} - 8(.tItS )2]gmn+10
3
+ 14.t1tS }
2
+ 12(btS )2]gmn+12 + 2[ (n+m+1)tltStJtS + 4btS
3
2 2
- 2(W ) 18mn+14
+ (.tItS2) 28mn+16 •
(40)
· -i {mn -
2t!tS} 8mn+2
+ ~ [3mn{ (m+1)n + m + 3}
- {3(m+1)n + 3m + 10} -tItS
+ 3{(btS)2 + bl.S 2 }]&mn+4
-i [mn{n2 + 3(m+1)n + m2 + 3m + 4} -
6{n
2
+ 6(m+l)n
2
2
- 3[ {n + 3(m+1)n + m + 3m + 4} tits - {S(tItS)2
2
2
3
+ (4m+4n+9).:tItS } + 4tltS ]gmn+8 - 6[ (.tJc.S)2 + (n+m+2)tltS
3
3
- U1tS ]gmn+10 - 4tItS gmn+12 •
16
~-1
(41)
~(-S) ~
2
GO
I
(~)k
2
r (~) k-o kl (T)k
I
k(k-1)
K
t%n-P (S)
Ie
- 4(mn+2)tJts + 4(t'ILS)2}8mn+8
1..-
(42)
~(-S)
7
Zmn
21'
=
U1U
-1
CD
I
x k
(_-)
2
(k-1)
r (~) k=O kl (~)k
I
K
a (1e) L%n-P (S)
1
Ie
1 2 1
4 [mn(n+m+1) - 4(n+m+l).tItS + 4tItS ]8mn+4 - 8 [mn(n+lIrH) (mn+4)
- 2(n+m+1)(3mn+16).tItS +
- 8tltstJts 3 ]8mn+6
4{2(n+m+1)(~)2 + (mn+12).tItS 2 }
- Z [3(n+m+1)(mn+4)t!LS
- 8(n+m+1)(t'It.S)2
221
2
- 4(mn+9)tItS + l.2tJLS.tItS ]~+8 + '2 [2 (n+mH) (t:!lS)
2 2 2
+ (mn+8).tItS - 6t1tStJtS 19mn+10 -.tJtstItS 8mn+12.
Combining (35), (36), (37), (39), (40), (41) and (42) with (34), and
arranging them appropriately, we obtain the term of order 2 in the following
form.
(43)
2 2 2
- 12m (n-m-1)(mn-2tltS)gmn+2 + 6n[mn{3mn + an - (m+1)(m +m-4)}
- 4{4mn
2
2
2
- (m +m-8)n - (m3+2m +5m+4) }.tJtS + 8n(t:!lS)2 + 4{mn
2
2
3
2
2
- (m +m-4)}t:!lS ]8mn+4 - 4[mn{3mn + (3m +3m+16)n + 24(m+1)n
17
+ 12{2mn2 - (m2+m-16)n + 4 (nri-2) }.t!tS2 - 24nbtStltS 2
3 3 2
232
- 32t1ts ]gmn+6 + 3[mn{mn + 2(m +nri-4)n + (m +2m +21m+20)n
2
2
+ 4(2m +Snri-4)} - S{4mn 3 + (Sm2+Sm+-24)n 2 + (m 3+2m +4Snri-44)n
+ 4(3m2+8m+9)}.tItS + l6{6n2 + 6(nri-l)n + m2 + 2m +
S}(~)2
+ l60mn2 + 36n + lSm + 32}.tItS 2 - 32{4n +m +U.tJt5.:tJr.S 2
- 2S6tJts
3
+
16(bts2)2]~+8 +
2
3
2
24({mn + 2(m +m+4)n
+ (m3+2m2+2lmt-20)n + 4 (2m2+Sm+4) }.t!tS - 4{2n2 + 3(m+l)n + m2
+ 2m + 9}(~)2 - 2{2mn2 + (m2+mt-32)n + 8(3m+5)}~S2
+ 12(2n+m+l).tJLs.:tJr.S
+ 96[(n+m+l).tIlStJts
2
2
+ 64.:tJr.S 3 - S(~S2)2]gmn+lO + 8[6{n2
+ 4W
3
2 2
2 2
- 2(W ) 1~+14 + 48(W ) gmn+16"
Hence we have obtained an asymptotic expansion of the probability density
2 as far as the term in n -2 •
function of TO
2
We summarize our results in
THEOREM.
Let 8 1 and 8 2 be independent Wishan matm b1'lth n and n2 degttees
of freedom, and let 8 1 have a non-oentm'Lity parametof' matJ'i,:z: S, then the
asyrrrptotio e:x:pansion of the probabiZity density function of a genera'Lized
.,., "
,
Hote(,.(,'l-ng 6
_2
1'0
= n2<vw182-2
4o.. Cf
• g1.ven
"
by
1.6
18
x2
1JJhere gmn(Z4VrSJ
is a probabiZity dsnBity function of a non-centmZ
variable b1ith mn
degNes of fNedom and non-aentmlity prm:cmetora tIcS.
are given by (38) and (43J~ N8p6ativeZy.
and A
2
rtrrE:
Al
It should be noted that the coefficients of each chi-square probability
density function are in a slightly different form from the ones of Siotani [11.
1.2.44].
However. by setting S.
~
•
~t-~ and
(&)2 •
.tItn in (38) and (43).
and by rearranging the variables of each coefficients, we have the same result
as Siotani's.
:
The author wishes to express his sincere thanks to Professor N.t. Johnson
who read the original version and gave some advice.
19
REfERENces
[1J
CONSTANTlNE~
[2]
DAVIS, A.W. (1968), "A system of linear different1a~ equations for the
distribution of Hote1ling's generalized TO. tI Ann. Math.
Statist. 39, 815-832.
[3]
HAYAKAWA, T. (1970). nOn the distribution of the latent roots of a
complex Wishart matrix (non-central case).
Institute of Statistics Mimeo Series 667, University of North
Carolina at Chapel Bill.
[4]
HOTELLING, H. (1951). nA generalized T-test and measure of multivariate
dispersion." Proc. Second BerkeZey Syrrp. Math. Statist. Probe
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[5]
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[6]
ITO, K. (1956). "Asymptotic formulae for the distribution of Bote1l1ng's
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[7]
ITO, K. (1960). "Asymptotic formulae for the distribution of BotelU.ng' s
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1148-1153.
[8]
JAMES, A.T. (1964). "Distribution of matrix variates and latent roots
derived from normal samples. n Ann. Math. Statist., 35,
475-501.
[9]
SIOTANI, M. (19 S6) •
A.G. (1966). ''The distribution of Botelling's generalized
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"On the dis tribution of Botell1ng' s T2. " Frac.
(In Japanese, with English
Inst. Statist. Math•., 4, 33-42.
abstract.)
2
(10]
SIOTANI, M. (1956). "On the distributions of the Hotelling' s
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[11 ]
SIOTANI, M. (1968). "Some methods for asymptotic distributions in the
multivariate analysis." Inst. of Statist. Mimeo SePies 59S,
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[12]
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T -statis-