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TABLES OF DISTRIBUTIONS OF (UADRATIC FORMS
IN CENTRAL NORMAL VARIABLES I
By
Samuel Kotz••
and
N. L. Johnson.
University of North Caroline
Temple University-Philadelphia
Chapel Hill, N. C.
Philadelphia, Pa.
Institute of Statistics Mimeo Series No. 543
December 1967
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• Supported by National Institute of General Medical
Sciences under grant 5ROIGM-12363 and by Air Force
Office of Scientific Research under grant AFOSR-681415
•• Supported by Air Force Office of Scientific Research
under grant AFOSR-63-141l.
DEPARTMENT OF STATISTICS
University of North Carolina
Chapel Hill, N.C.
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Introduction
•
Table 1 gives the probability density function (p.d.f.) and cumulative distribution function(c.d.f.) :>f 36 positive definite quadratic forms of rank n
==
4
in illultinorlTIally distributed ranGJD variables with zer:> expected value vector.
In general let X ,X , .•. ,X be a set of such variables, with variancen
l 2
covariance matrix V.
We consid er the distribution of tIle quadratic form
y
==
X'AX
Where A is a positive definite nxn matrix.
C'VC
==
There exists s nxn matrix C such that
I, and C'AC is a diagonal matrix with diagonal elements a ,a , ••. ,a which
l 2
n
are roots of the determinantal equation
IA -
.. 1
o.'V
I =0
(or equivalently the latent roots of AV
).
Applying the linear transformation
X' == U'C'
we see that
n
X'AX
the U.
I
j
S
= U'C'ACU
~. a~u: and also since C'VC
==
j=l
,j
I,
are independent standarGized normal variables.
(i.c.
If V is non-singular
the
joint distribution of the X's is non-
degenerate) then it is positive definite, as is A.
definite
==
J
and the
Then AV
vall also be positive
latent roots a l ,a '."'CX must all be positive.
n
2
We define their
values uniquely by the conditions
... -> exn >
0
nle distribution of Y depends only on the values of C ,C , ••• ,a • Apart from
l 2
n
a multiplying factor,
it depends only on the ratios
n
to use a scale such that
n
tables to cases when
~
j=l
E aj
j=l
f
a~
= n.
a l :a2 , .•. :an • We have chosen
It is, of course, quite easy to apply our
J
n, using the identity
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wi l,l1 c:
n
~
j=1
T~lC:
-,
n
-1
o;~
.1
vaLues of the a's e.:i'C
2l~~:;wn
at the head of each al'IJ1'opriate sectio11
,)1'
Table .:.•
The i'or::mJ.as used ia
:L
-11-
(y!t3)2
( 1)
the cUT:lUl.ative
andy
:':~ir
(2)
Prey :: yJ
==
c8.i.cul.ation 'vTere, f()r tile
1
-2.:-:/'arJ
t~lC
'1
. c
di::"l.l'ib'_:tL~m
'~en8:',- ty
functi.:"m
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functi.:m
r
1
-----J
~·o,
22 r(~)
w~erejJn)(x)
and
the
Cj
III
lF1
(-j;~F~h
P
it
t(~
,t
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~)
_I
's are defined by the equations
j-1
Co
lI:=2
"
n
.E
= 1; C j =0
~: dj_iC i
.,-0
.
(i.-C/f3 ),~ .
l=l
(See ::. :JrcruJ.as (18) anJ. ('If) .)f [lj, "here the i~he:,:;:,y . :'; (:f.:veloped on Hhic;i
tl1esc: 'ca1Jles are based)
taJ:e
J:
I~l ')l'l~e:-'
t'J estimate the
neces~;a}'Y
nu.r:tber of ter(:1s t::-,
1.'.nc series eX"pansix",~ t::e foLLowing u~)per bOlillC:S ::,'~T error in terMinating
tht: ::;c::~s at the (N-~'l)"·t]1 ·~c:;:" llere used
L
F'J:,' (2):
n
!~·:·1
1
2r(~+J.)
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Cslcul~tions
arithmetic.
The
were
c~rried
v~lues
out on
shown to 5
~n
=5
will be provided in
~rec1sion
places were based on original calcu-
decim~l
1 at ions which shou].d be correct to about
TQbles for n
IBM 360 machine using double
pl8.ces.
(l
2
later report.
(Tables for n
= 2,
3
r.re alre~dy available in [2J and [3J.)
As the value of n increases, provision of an adequate grid of values of the
Ct's repidly incre2ses in difficulty.
wOrk to give
25
We have attempted, within the scale of our
complete coverage as possible, sUbject to the limitation that
a Jan should not be so large as to require too great a number of terms in the
series expansion we
~re
rising.
(In fact this means that, usually, we have
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Interpolation is likely to be arducus, if considerable accuracy is required.
Lagrangian multivariate interpolation, using tabulated sets of values of the
eI's
in the neighborhood of the required value, may be expected to give useful results,
but will not in general be possible to use precalculated sets of coefficients,
oWing to the irregularity of the tabwlated sets• of
.J ~
cr ~
For many purposes,
good approximations may be obtained using methods of the following type, or
variants thereon.
n
The expected value of Y is E a.,
which is eQual
to n for all tabulated
J
.
n
2 j=l
sets. The variance is 2 r. OJ' The distribution of
j=l
(3)
(n·· l
n
E
j=l
c?)J
n
E
j=l
O~ degrees of freedom)
i
tije"
8.2
1".~'
',;11e same first
:.~:.;s:~ble
approximatL.n
n
2
j:::l
,J
.• i
(
Pr[Y <
'·.:Jments as that )f Y
ApproXi.r:le.ting i.r.. '-
,:lC::::'C'::'):
~i,;ld.
''light be conciderec:
J.
to the distrj.l),,:t'!.)!:.
d'
Z. ex.)
n"
n:..:r"a:~
U is a standardized
(4 )
,:;'.:C ,m(~
C,,[l
'~
yJ
.11;
(·F2tr) '.
+
IJIIJ
,
.2
Z
.j:::l
du
Z
c~.,
,;
..
'~ \
.-
I
I
l
1
rl;
I
,j =1
Sin:i:',ar ,Ly if the fractile Y
Prey
e
2
Jl~2
;J2~y
0
1.
S C1Cf""
E:
-,
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variable Vie obtain
-2'u
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that
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el
n
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;2.)
C.
1::
j=l
(J2IIr
1
i
{,U
I
€
2'
- 2'u .
e
U1.:
J -tIS
A::'Pl' .:,xi\l1ate values
S803
an(;,
~:;E.-::r De
:):.=' va.l..ues of the o:'s.
.~',
c2.J.culated for both
Y'~t
tab1.;,1,3:~:8C·
is possible to find sJ.:e iunction of the exact
'roXililate val,ues "\vl'r:,cL vcw:'ies only slowly wi tIl tllc ex' ',' then an exact non-
tabu:.ated 'ralue lnay be est "need \lith reasonable accUl'ac:/
t,o
7.,:1e
and non-tabulatee
tabulated value.
tile ":-1"':"
(Of
(')U1'S8,
!':COlIl
other approximatiJl1E :;ay be used, th.:::mgl'l
0" 'rcr, f1Cre seems t:J be ;:mc>f tile simplest,
pr[~
+
4~
I
the approximation
:Jnce chosen),:ms'::; be adhered to thr'Jughout t.l:e c:1J,culations.
As an exa.'Jlple) s'lrppoce'ti'2 need
I
t;)
Tile
f;)rl:~
though by n:' .'eans the ::lost accurate)
calcuLate
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Where Ul ,U ,U3 ,U4 ~re independent stnndnrdized normal variables.
2
restate in stand~rd form, giving
pr[1.75U~
Hence 0: 1
The
= 1.75,
+
0:
approxim~tion
2
U~
+
0.75U~
0.5U~ ~
+
= Ip 3 = 0.75,C~
described above
2.51
4
= 0.5
4
We first
c1:J = 4.1375
end E
j=l
(4), in particular) gives
(~t
.,
= (d160'·
- 1
-
J2i7)/-!39
= 0.33336
2nd, as the c';.J:)roxim::-,te value for the probnbility
~
J -0.33336
-1
('"12,,0 -
1
e-~ u
2
du
= 0.36945
_W
For neighboring sets of values of cJ.. 's we have the following results
2
Prof: O:jU j :- 2.5J
0:
0:
0:
Exact
Jl.pprox.
Exe.ct - A,pprox.
01
3
4
2
I.G
LO
0.9
0.3
0.39239
0.37699
0.01540
LG
1.0
0.3
0.4
a.3CeOO
0.37419
0.L138l
1.5
I.e
1
_.IJr_
1.0
0.5
0.37214
0.35790
0.01416
0.7
0.5
0.30522
0.37156
0.01366
LO
(The set of values in the last line is
~uite
close to the
re~uired
values).
The
values shown suggest that a better estimate of the required probsbility will be
obtained bY,increasing the approximate value.
Linear interpolation between the last two lines suggests an addition of
about 0.01374 leading to an estimated value of 0.3832 (to four decimal places).
vi
ACKNO~EMErrr's,
The authors vrish t:., thank Dr. D. W. Boyd for
formulati)L
to Mrs. C.
01' U11';
Ne~'nan
his valuabl<- aS8i.stance in
c-:Jnputatim pr:.:>cC(;-,lre· for these tables.
who has written
~h~
Than}:s are also due
Kotz, Sa,,:ucl, John S (ill, N. L. an6 B()yd, D. W. (1967) Series ;"cpl"esentatbn
.
;.;.
of distributions :)f qua-drat:.c :C
~. ~. Statist.
[2j
J
38,.
Marsa~a) G. (:~9~O)
)rlS
in normal variables
j
1.
Central case,
823-J:j~
Tables :):'~l'lc'; '~istribution :)f qua(::.'aci.~ ~nor:r.1S of
and three; ~~er. Ny. Dl-82-0015-1, Boeing Sc >cntific
ran.:".:3 'e\{.
Reseal"c11 Labol'atories.
[3J
S:d.om~)n> H.
-,
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cJmputer program.
REFERENCES
,
;-1]
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(l~16o) Distribut2_,n of' cluadratic forms '. tab~Les
Tech. Rep. Ii,:). 45 Appl. Math. La...,., Stanford UnJversity.
,1n(' applications;
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...
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,.
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eX
y
0.1
0.2
0.3
O.l"
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.l~
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
. !j·.0
4.1
q.• 2
4~3
4·.11·
Q·.5
4.6
4'.7
4.8
l~. 9
5.0
p(y)
.01l·3393
.0797(,1.1·
.11003
.13501
.15541
.17185
.18487
.19495
.20250
.20790
.2114'~·
.21342
.21406
.21359
.21218
.20998
.20714
.20377
.19998
.19585
.19146
.18687
.18214·
.17732
.17244
.16755
.16266
.15781
.15300
.14827
.tl~362
.13906
.13460
.13025
.12601
.12189
.11788
•11!j·OO
.11023
.10658
.10305
.099638
.096339
.093152
.090074
.0871O!t·
.084238
.081474
.078808
.076237
.
(2.5, 0.7, O.ll·, 0.4)
Pr(y(y]
.0022324
•008q·ll-ll·8
.017982
.030275
.0411·832
.061225
.079087
.098101
.11799
.13853
.15951
.18077
.20215
•223511·
.24481.1·
.26595
.28681
.30736
.32755
.34735
.36671
.38563
.40408
.42206
.43954.45654.If·7305
.q-8098
.50462
.51968
.53427
.54841
.56209
.57533
.58814
.60054
.61252
.62412
.63533
.64617
.65665
.66678
.67658
.68605
.69521
.70!j·07
.71264
.72092
.72894
.73669
y
5.1
5.2
5.3
5.4
S.5
5.6
5.7
5.8
5.9
6.0
6.1
6.2
6.3
6.4·
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
10.0
p(y)
.073758
.071369
.069066
.066811·6
.064,706
.062644
.060656
.058739
.056892
.055111
.053393
.051737
.0501q·0
.048600
.01.1-7114
.045681
.044297
.042963
.041674
.040lfJO
.039229
.038069
.0)694-9
.GSI61
.0y.,821
.033810
.032833
.031889
.030976
.030093
.029239
.028412
.027613
.026839
.026090
.025364
.024662
.023982
.023323
.022685
.022066
.021467
.020886
.020323
.019777
.019247
.018734
.018236
.017752
.017283
1
PrlY{y
.744019
•751411·
• 758l~6
.76526
.77184
.77820
•784J7
•790311·
.79612
.80172
.80715
•812q·O
.81750
.82243
.82721
.83185
.83635
.811·071
.84494·
.84905
.85303
.85690
.86065
.86429
.86782
.87125
.87458
.87782
.88096
.88402
.88698
.88987
.89267
.89539
.89803
.90061
.90311
.905511·
.90791
.91021
•912!j4
.91462
.91674
.91880
.92080
.92275
.92465
.92650
.92830
.93005
I
-20/:
y
10.1
10.2
10.4·
10.6
10.8
11.0
11.2
11.l~
11.6
11.8
12.0
12.2
12.1.1·
12.6
12.8
13.0
13.2
13.4
13.6
13.8
14.0
14.2
14.4
14·.6
14.8
15.0
15.2
15.4·
15.6
15.8
16.0
16.2
16.4
16.6
16.8
17.0
17.2
17.11·
17.6
17.8
18.0
16.2
18.1118.6
18.8
19.0
19.2
(2.5. 0.7. 0.4, 0.4)
p(y)
Pr(y(y]
.016828
.016386
.93176
.93342
.93661
.93964
.01g541
.01 ,744·
.013992
.013283
.012613
.011980
.011382
.010816
.010281
.00977l l8
.0092955
•ooe "VI·1i)
•ooe':115
.00800110
.00761.77
.007?513
.006903g
.006.57'-/0
.0062611
.00;")639
.00;6~18
.00;/r.138
.0051591
.00lf?!7!
.00lf·6371
.00411685
.00lf·2605
.001l·o627
•00 387116
.00369.56
.0035252
.00336)1
.0032087
.0030618
.0029218
.0027885
.0026616
.002jll·06
.0021.1·2511·
.0023155
.0022109
.0021111
.00201~0
.0019253
.0018)88
.9l l·251
• 9452'~
.91.1·783
.95028
.95262
.95484
.95695
.95895
.96086
.96267
.961.140
.96601,-
.96760
.96909
.97050
.97185
.97313
.974-36
.97552
.97663
.97769
.97869
.97966
.98057
.98144
.98228
.98307
.98383
.98455
•9852lf-
.98589
.98652
.98712
.98769
.98823
.98875
.98925
.98973
.99018
.99061
.99102
.99142
.99179
Y
19.4
19.6
19.8
20.0
20.2
20.4
20.6
20.8
21.0
21.2
21.1.1-
21.6
21.8
22.0
22.2
22.4·
22.6
22.8
23.0
23.2
23.4
23.6
23.8
24.0
24.2
24.4
24.6
24.8
25.0
25.2
25.4
25.6
25.8
26.0
26.2
26.4
26.6
26.8
27.0
27.2
27.4
27.6
27.8
28.0
28.2
28.4
28.6
28.8
p(y)
.0017564
.0016778
.0016028
.0015312
.0014629
.0013978
.0013357
.0012763
.0012197
.0011657
.001111.1-2
.0010649
.0010179
.00097307
.00093022
.00088931
.00085024
.00081293
.00077729
.00074325
.00071073
.00067967
.00064,999
.00062164
.00059454
.000)6866
.00054392
.00052028
.00049768
.00047609
.00045545
.00043572
.00041686
.00039883
.00031860
.00036512
.00034937
.00033430
.00031990
.00030613
.00029296
.00028036
.00026832
.0002)680
.00024578
.00023525
.00022.517
.00021553
Pr[~yl
.99215
.99250
.99282
.99314
.99344.99372
.99400
.99426
•99Q·51
.99474
.99497
.99519
.99540
.99)60
.99579
.99597
.99614
.99631
.99647
.99662
.99677
.99691
•99701.J·
.99717
.99729
•997lf.Q
.99752
.99762
.99772
.99782
.99791
.99800
.99809
.99817
.99825
.99832
.99839
.99846
.99853
.99859
.99865
.99871
.99876
.99881
.99887
.99891
.99896
.99900
I
e.
I
I
I
I
I
I
el
I
I
I
I
I
I
I
e.
I
I
,e
-30(:
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
y
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1. 1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
(2.5. 0.6. 0.5. 0.4)
p(y)
Pr(yly]
.042057
.0775!f·5
.10729
.13201
.15237
.16891
.18215
.19252
.20042
.20617
.21009
•2121.J4
.21344
.21330
.21219
.21027
.20768
.20453
.20093
.19696
.19270·
.18822
.18357
.17881
.17397
.0021613
.0081926
.017479
.029483
.043736
.059829
.077408
.096164
.11583
.13618
.15700
.17814
.19945
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6.0
6.1
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6.4
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
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8.2
8.3
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8.7
8.8
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1.9
2.0
2.1
2.2
2.3
2.4
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2.6
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2.9
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3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
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4.2
4.3
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4.6
4.7
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5.2
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6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
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7.8
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8.0
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8.2
8.3
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8.5
8.6
8.7
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9.5
9.6
9.7
9.8
9.9
10.0
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.082026
.079529
.077104
.074750
.072466
.070249
.068099
.066013
.063990
.062028
.060126
.058282
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.054761
.053081
.051453
.049876
.048347
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.044039
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10.2
10.4
10.6
10.8
11.0
11.2
11.4
11.6
11.8
12.0
12.2
12.4
12.6
12.8
13.0
13.2
13.4
13.6
13.8
14.0
14.2
14·.4
14·.6
14·.8
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9.3
9.4
9.5
9.6
9.7
9.8
9.9
10.0
p(y)
.085800
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.08~40
.077874
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.04304-8
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.0)4167
.033061
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.030958
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.028059
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.025441
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.019631
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Prlj~y]
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10.1
10.2
10.4
10.6
10.8
11.0
11.2
11.4
11.6
11.8
12.0
12.2
12.4
12.6
12.8
13.0
13.2
13.4
13.6
13.8
14.0
14.2
14.4
14.6
14.8
15.0
15.2
15.4
15.6
15.8
16.0
16.2
16.4
16.6
16.8
17.0
p(y)
.016721
.016120
.015196
.014261
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17.2
17.4
17.6
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18.0
18.2
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18.6
18.8
19.0
19.2
19.4
19.6
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20.2
20.4
20.6
20.8
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21.2
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22.4
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23.4
23.6
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24.2
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.0012318
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9.8
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p(y)
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2.6
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3.2
3.3
3.4
3.5
3.6
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.030621
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.1)462
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.16791
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Pr(~y]
.0015635
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5.1
5.2
5.3
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6.0
6.1
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6.4
6.5
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7.0
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7.4
7.5
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8.5
8.6
8.7
8.8
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9.0
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9.3
9.4
9.5
9.6
9.7
9.8
9.9
10.0
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.08701
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Pr[Y'y]
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14.8
15.0
15.2
15.4
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15.8
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16.8
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p(y)
.0164·00
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19.6
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Pr[Y!y]
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1.6
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17.2
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17.6
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18. 11'
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19.8
20.0
20.2
20.4
20.6
20.8
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21.6
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22.0
22.2
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22.6
22.8
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.0014619
.0013732
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0.2
0.3
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0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.~.
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.42.5
2.6
2.7
2.8
2.9
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3.3
3.4
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3.6
3.7
3.8
3.9
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4.2
4.3
4.4
4.5
4.6
4.7
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.13715
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(1.8. 1.2. 0.5. 0.5)
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.69394
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5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
6.1
6.2
6.3
6.4
6•.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7•.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
10.0
p(y)
.0871211,
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.081792
.079235
.0767~9
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.026129
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.018829
.018223
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Pr (Y'y]
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.77135
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Pr[Yly)
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11.4
11.6
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17.2
17.4
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18.4
18.6
18.8
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19.2
19.4
19.6
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13.0
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13.6
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14.0
14.2
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5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
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8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
10.0
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.087e07
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.72905
.73769
.74608
.75420
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.76972
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15.6
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16.2
16.4
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(1.8, 1.0, 0.9, 0.3)
Pr[Y{y]
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16.8
17.0
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20.6
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21.2
21. 11.
21.6
21.8
22.0
22.2
22.4
22.6
22.8
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.0019262
.0018072
.0016957
.0015912
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.000118282
.0004.5380
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0.7
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1.1
1.2
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2.1
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2.3
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2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.1
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
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p(y)
.030833
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.081148
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Pr[Y$yJ.
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.0060)61
.013008
.022161
.033196
.045847
.059875
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.091239
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.16255
.18140
.20045
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.23883
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.29615
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.38800
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.47237
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.56182
.57544
.58869
.60159
.61413
.62633
.63817
.64968
.66085
.67170
.68223
.69244
.70235
.71195
.72126
y
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
10.0
p(y)
.088862
.086110
.08)428
.080827
.078275
.075803
.073399
.071062
.068792
.066587
.064446
.062368
.060351
.058395
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.054659
.052877
.051149
.049475
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.046282
.044761
.043288
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.036602
.035391
.0)4220
.033087
.031991
.030931
.029906
.028915
.027956
.027029
.026133
.025266
.024428
.023618
.022835
.022078
.021)46
.020639
.019955
.019295
.018656
.018038
.017442
Prlrty]
.73029
.73904
.74752
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