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This PDF is a selection from an out-of-print volume from the National Bureau of Economic Research
Volume Title: Annals of Economic and Social Measurement, Volume 5, number 4
Volume Author/Editor: Sanford V. Berg, editor
Volume Publisher: NBER
Volume URL: http://www.nber.org/books/aesm76-4
Publication Date: October 1976
Chapter Title: Numerical Aspects of Multivariate Normal Probabilities in Econometric Models
Chapter Author: J. E. Dutt
Chapter URL: http://www.nber.org/chapters/c10495
Chapter pages in book: (p. 547 - 561)
Annals of Economic and Social Measurement 5/4, 1976
NUMERICAL ASPECTS OF MULTIVARIATE NORMALPROBABILITIES IN ECONOMETRIC MODELS*
BY J. E. Durr
The role of Multivariate Normal Probabilities in Econometric Models has in the past been somewhatrestrictive because of the unavailability of useful computational formulas.
Using the author's recent integral representations for the Multivariate Normal Probability Integral,ThaI (1973) and (1975), highly accurate and efficient computational formulas are now available forcomputing normal probabilities of dimension up to 6. These formulas have direct application to theMaximum Likelihood procedures which are of interest in econometric modelling.
1. INTRODUCTORY SUMMARY
Prior to 1972 and after years of considerable effort, the only known generalrepresentation for multivariate normal upper and lower probabilities consisted ofPearson's tetrachoric series (Kendall, 1941) which is well-known to be computa-tionally unattractive for dimension K >2. A reasonably complete bibliographyrelating to rnultivariate normal probabilities up to 1972 can be found in Johnsonand Kotz (1972). Milton (1972) applied a method based on a multidimensionaliterated Simpson's quadrature to the customary iterated form for either an upperor lower probability integral. Milton's computerized procedure, however, appeaisto be at least one order of magnitude in running time slower than what is nowavailable.
In the recent paper DuIt (1973), this author obtained an integral transformrepresentation over (0, co) for upper and lower multivariate normal probabilitiesusing Pearson's tetrachoric or orthogonal series, Kendall (1941), as a startingpoint. A simplified representation for the normal and an extension to themultivariate t are given in Dutt (1975). Tne representations are for arbitrarynormal and t probabilities of arbitrary dimension and correlation matrix.
The integral transform representation for multivariate normal probabilities isvery useful when numerical evaluation is by the GaussHermite quadraturemethod. A short table based on the integral transform representation for thequadravariate normal orthant probability P4 which, except for nearly singularcorrelation matrices, is accurate to 7+ significant digits, is found in Dutt and Lin(1975). A more extensive table for P4, Dutt and Lin (1975a) and a short table forthe trivariate normal, Dutt, Lin and Desai (1976) will be available shortly.Accurate computational formulas arc also derived for the exponential, error andarcsin functions, Duu, Lin and Tao (1973). Integral transform representationsover (0, cx)) for arbitrary upper and lower multivariate probabilities with applica-tion for computing bivariate and equicorrelated trivariate x2 probabilities isdiscussed in Dutt and Soms (1976). A table of the trivariate t for unequalcorrelations is found in Dutt, Mattes, and Tao (1975).
* Presented at the NBER-NSF Conference on 1)ecision Making Under Uncertainty, University
of Chicago, 16-17 May. 1975.
547
Attention here is focused on properties of the integral transforni represcnta/ tion over (0, ) for mullivariate normal probabilities which might he of interest in/ econometric niodels. Numerical results are discussed for several correlation/ structures and dimensions up to six.
2. INTEGRAL TRANSFORM REPRESENTATiONS oVER (0, aD) FOiL UPPER ANDLOWER MULTI VARIATE PR0BABILrnESIntegral transform representations over (0, co) are here summarized for anarbitrary continuous multivariate distribution and in particular for the mul-tivaria(e normal. The integral representation follows in the general case from aslight modification of a theorem of Gurland, Gurland (1948), Dull and Soms(1976). That such a modification was possible in general was only realized afterthe integral representation for the multivariatc normal was derived from thetetrachoric series, Dint (1973, 1975). Iloth approaches however, follow eitherdirectly or indirectly from the Inversion theorem.Let X.,.... X, have the K dimensional cdl FK(x) and correspondingcharacteristic function /(). For k K, let j(f) be the characteristicfunction corresponding to the marginal distribution of X,, . . . , X,,, whereI1,..-,Jk isasubsetoftheintegers I,.. .,K.Now define 'kj,..... as the integral transform
k(2,1) 'kj1,.. = (1/2 j J {Real ,kj, fl fri dt,
where IXk;ji.......k[ek;f,,.. ,jk(b] and &[f(t .....1k)] is the kth centraldifference about 0 of f(t, .. . , t)
k{f(tl .....')]=f(t,... tk) f(-', t2.....4)- 1(t, 1 ......- I, t)+f(t, 2, :4.....4)+ +(I)kf(:1,.. .
and is a continuity point of the distribution of x,......,Then, from Dutt and Sonis (1976, equation 2.3), if a (a1.....ak) is acontinuity point of FK, the integral transform representation over (0, co) for anarbitrary Continuous multivariate lower probability is(2.2) FK(a)= (I)k (I)Kl I(a,)
(4)2I2(a1, a)
IK-3- () 13(a1, a, ak)
+...+JK(aI,...,aK).with 'k;j,.....A Ik(aj.....,
548
For upper probabilities, the negative signs are simply changed to positive
ones. In Gurland's work, 'k;ji 1krelates to the real part of
(2.3)(21)k r\ 4 dt,
where 4 is a continuity point of the marginal distribution of X,, .. , X and for
any function gW, using the notation of Outland (1948).
..gQ)dt=iimJ... J gQ)th.
l;
As they stand, the Cauchy principal value integrals in equation (2.3) are
divergent. This can be seen in the case of the bivariate normal. One of the integrals
in equation (2.3) is of the form
(2.4) J cos a(11 + t2)2(, t2; p) dt1 dt2/t1t2
which as E-O, T-co is divergent. On the other hand, using equation (2.1) the
integrand is bounded at the origin and the integrals can be used in numerical
integration.The integral representation over (0, x) for the multivariate normal may be
either treated as a special case of (2.2) or obtained from thc tetrachoric series in
the following way.The K dimensional normal probability integral is defined as
LK(xl,...,xK;R)=I ...j nK(I0,R)dyXi Xl.
for any real numbers x1, . . . , x,. The integrand nK(yj0, RK) denotes the K
dimensional standardized normal density with correlation matrix R.
Consider the representation of L by the tetrachoric series (Kendall. 1941)
(2.5)
where
LK(x!,...,xK;R) -..fl2O )-t,i
K
1K J)K [1 (r'/j,!) II (nk!)rflk(xk),kI
rn <n
KK
1K; (iK), n,2ñ,1=1
549
I
I
say, and rm(x) is the rnth tetrachoric function (Ahiamowitz & Stegun, 1964, .934).
(2.6) T,(x) Z(x)He11(x)J(rn !)2, in 1, 2,...,with
Z(x) = (l/(27r)112) exp (/2)and He(x) is the nth degree Hermite polynomial
He(x) = [(_-1y/Z(x)](-)Z(x) it = 0, 1,.From the integral representation of the Hermite polynomial (Abramowitz &Stegun, 1964, p. 786), an integral representation of the tetrachoric function Tm(X)is obtained as
(2.7) Tm(X) = (1/(rn i)2)J
exp (s2/2)s' cos (rs - (in -- 1)42) ds,rn = 1,2.....The direct substitution oi Tm (x) given by (2.7) into the tetrachoric series (2.5)yields after considerable manipulation Dutt (1973, 1975)
(2.8) L(x1,. ..,xK;R)=
(!)X_(1)KLD'.1+()2 D1+()"3ii i<j. 1ri*
The D functions are defined by
where, for the first few, K,
d'=sin1=sin(x1s1),d=e_12cos1_2e12cos12,d
elZ.f13+23 sin1,23e121323Sifl,243 e
-12+13-23 sin1_23e12323 sin123,= e1 2+13+23+14+2+34 COSI+2f3+4 +12-Li- 23- 14--2434 COS_1 --2-f 34-4
+1241323_14+24_34 COS2_34+e2_13231424_34 cos1234
e_12_1323_14f2434 Co5
+2+3+4eI?13_,l+1424f34 COS123f4e1213-23+14+24-34 C0S1234
e1213+2314_2434 cos14234;andD;j11k = D(x11
,...,xjk). For notation,epq,+.p= { (rpjqispjsq +
. +rpqspsq,)},sin1++=sin(x1s+ . . .
COs,+cos(xs+ ...where R = ((z,))
550
Ix(2.9) D(x;R)=2(2ir)_KJ
0ds1
.. .J ds,e2d(s;x; R)/ fl S,I) kl
I
A negative sign on the index q1Corresponds to +rq,sp,sq, and p1
corresponds to x,,,s1,.
The k dimensional normal cumulative probability is defined by
xI
(2.10)nk(y0,R)dy.
In terms of the upper tail probability Lk,
(2.11) 'lk(xl,...,xk;R)=Lk(xl,Xk;R)
3. PROPERTIES OF D FUNCtiONS
Consider again the integral representation over (0, co) for Lk
(2.8) Lk (_(1)k D.;+(2 ...
-,;-, .,. .-.
It is noted that in particular
(3.1)D(x)=4erf(x/'J)
with
(3.2)D(0)=0
and
(3.3)Dx) -
Since D(x) for k odd, is a sine transform with argument x's then equation (3.2)
generalizes to
(3.4)D(0) 0, fork odd.
Moreover, it can be shown easily with equation (2.8) that by mathematical
induction on k, equation (3.3) generalizes to
(3.5)
1.
From a practical point of view, equations (3.5) means that to at least 3 digits of
accuracyD(4)
(_)k, for any k > 1.
In general, D(x), for k > I is non-negative and monotonicallyincreasing for
k even and, non-positive and monotonicallydecreasing fork odd. There is aslight
inconsistency for k = I in that D1'(x) is defined in equation (3.1) as a positive
function for positive x.For theorthant case (i.e. x =0) in addition to identity (3.4), it was previously
noted, Dutt (1975) that
(3.6)D(0. 0; r) = (arcsin r)/2ir.
551
;-,-c,,.-_,r-- ,--a,c
If P Lk(O,. . , 0) and D°= D(O,..., 0) then from equation (2.)
(arcsin,)/2rI+ ( D,1114I
3.1 Transfer of Sign changes from x, toFor Ic I it is clear that
(3.1.1)
For k> I, attention need be focused on only d. For k 2, considerd' = e+'I2S cos (x1s1 x2s2) e l22 cos (x1s1 +XiS2)for a single sign change (i.e. either x1 orx2) and then for a double sign change (i.e.
both x1 andx2).For a single sign change
(3.1.2) D(x1, x2; r12) D'(x1, X2; r12) = D(x1, x2; r12)while for the double sign change
(3.1.3) D(x1, x,; p) = D(x1, x2; p12).Therefore, for a single sign change a negative sign is transferred from either x, orx2 to r12 with a negative sign in front of D. A double sign change of x1 and x2leaves D unchanged.Fork 3 and considering a sign change ofx to x1,
(3.1.4) D(----x1, x2. x3; r12, r13, r23) = D'(x1, x2, x3; r12, rfl, r23).The sign change is transferred to the correlation in which one of the subscripts1=1.For the double sign change x - x1, x2 -*
(3.1.5) D'(x,, x2, x3; r,2, r13, r23) = D(x , x2, x3; r17, r13, r23)and the triple sign change(3.1.6) D(-x1, -- x2, x3; r12, r13, r23)=z D(x1, x2, x; r12, r13, r23).From equation (3.1.3) and (3.1.6) it should be clear that in generalD(x; (l)kD(_x; rq).Moreover, for any r and k if x (xh,.. . , xJ) then
D*1k . . . , X1., + X1 ......+Xjk,= (-1 )rDc(X; _r, . . . , r,, +r1,,, , . . . , +r)provided 1ik, ij, and 1int. In other words, a change of sign occurs
only among the r, for which one subscript is from the set 11.....j,.552
then
(3.2.1)
L(--x,,,. . . ,-x1,, x1,,. . - ,
-r,,,, . . . ,
provided 1isk,ij,,, and irnt.The results of Sections 3.1 and 3.2 would apply also to the mixed case.
3.2 Symmetry Considerations
For k = 2. observe thatx2; t)r D(x2, x; r12)
which implies that for fixed x1 and x2, there are two equivalent probabilities
x2;P12) = L2(x2, x1; p12).
For k = 3, there are the following six equivalent Di's:
x2, x3; r12, r13, r23)
=D(x1, x3, x2; r13, r12, r23)
D(x2, X1, x; r12, 23 r13)
= D(x2, X3, X1; p23, p12, r13)
x; r13, r23, r12)
x1; r23, r13, r12).
The six equivalent Dr's lead to six equivalent L3's.
Fork = 4, the general pattern is readily apparent. For a fixed x1, x2, x3 and x4
and fixed {rq}, there are 4! 24 equivalentDi's relating to the permutations of (1,
2, 3, 4). However, for a given set of the six correlation coefficients there are
6! 720 corresponding Di's which taken together with the above mentioned 24
yields a total of 30 distinct Dr's with each occurring in 24 equivalent ways. The
permutations of {p,j yield, therefore, the total number of probabilities and the
permutations of {x1}, the subset of equivalent probabilities.
3.3 Mixed (Upper and Lower) Probability Integral
If the mixed probability integral is defined as
fx, (X1 f.fk =J . . . J ' J .. . J
{r,}) dy
-c A
4. SUMMARY OF COMPUTINGFORMS FOR L,
There are a variety of different options in computing the D7 functions
depending on the primary needs of the user. There may be, forexample, interest
in high accuracy (of the order of 7-8 digits) for a relatively small list of prob-
abilities (less than 100), or moderate accuracy (of the order of 3-4 digits) for a
large number of iterations (1,000+), or interest in dimensions greater than five
where overflow can be a problem. Clearly, the more specific the case of interest
553
the greater would be the advantages in computer running time and accuracy.Moreover, it would not be prudent to use the geneal form of Lk lot- theequcorrelated case or for one of the orthant cases.A summary of the computing formulas for L1 are therefore presented in theitmost general form with the integration coefficients specified SO as to anticipateoverflow in the higher dimensions.For k I in equation (2.8)
(4.1) L1(x1)D1where
DT1 (1/v) f sin (x1s1) ds/s1.Application of the Gaussian quadrature formula (Abraniowitz and Stegun,1964, p. 924) yields the computing formula for D'.1 uS
(/) Y sin (Ax1)where y. w1/A, A1 = {w1} are the Christoflel weight factors, and {z1} are thezeros of the Mth degree i-termite polynomial. With M even, N = MJ2 so that Ndenotes only the positive zeros. The {yJ and {Aj are used for all D. The { z1) and{w1} are found in Stroud and Secrest (1966).For k = 2, equation (2.8) yields
(4.2)
where
For k 4(4.4)
L2(x1,x2;p,2)=H-[D.1 +D;2}+D;17
D';i2=(1/ir2) y1y,d'i=1 j-1d=
exp[p12A1A2]cos(x1A2x2A2)
exp[--p12AA2cos(x1A1 +x2A2).For k =3(4.3) L(x1, x2. x1; P12, P13, P23) 1/8{D1 +D;2+D:1]
+
1where
YY,Y d.i I j I k
L4(x1, x2, x3; P12, . . . ,f234)= 1/16[D1+D'2+D'3+D4]
±1[D'12 + D;13 + D';23+ D4 + D;24 + D'.14]ir,-* r* r-i* r*21'-'3;123 T '-'3;124 LI3;134 ' '3;234r*TL4;1234
554
4
where
where
D1234 = (1/2ir4) f YIYfYkY,d.I I j = I k I t I
Fork'S,(4.5)
L5(x1, x2, x3, x4, x; P12, . . . P4s)
1/32 [D'.1 t + D;3 + D't4 +DJ
+ LF5.12345
YiYjY&YrYsd'i1 j1 k1 rl s1
d(A1, A,, Ak, 'r A5 x1, x,, x3, x4, x5)
Jsin(-f+±++)
exp[++-- +-- + ]sin(-- ++++)
+ }sin[± --I- + +)
exp[+ + --+-- +Jsin(++--- +4-)
exp[-- ++ + ----+]sin(-f-+++)
exp[+exp[ + ++ + ± + ---}sin(
+exp[+++++--+--]sin(+-4-+)+exp{.+ + ++ ++ ]sin(+ ---++)
+exp[+--++--+---++Jsin(-f-++)
+exp{ -f ++ + +]sin(+ + --4-)
+exp[+ +-4- -----++ +Jsin(+++)+exp[+ -4-- + + + ±}sin(+ ++)+exp[_- 4--f +4-4--- ]sin(+++)+exp[ ++ + ++ + ]sin(+ ++ )}
555
10 terms
1r* im* r* ir*4tL 3;123 '-'3;I24 -'3;I34 '-'3;234 L3;J25
D*+ 313S*3;235
* j * *3; 145 3;245 3;345
Jr r * * * im*
21-'4;I234 -'4;1235 -'4;I245 '-'4;I345 -'4;2345
where the order of correlalion isP12' P13, P23 Pl4 P24 P34, P15' P25' P35' P4
5. NUMERICAL. RtsursAs was mentioned in the Introduction, a variety of tables and numericalresults are now available based on integral transform representations of mul-tivariate probabilities. In the normal case, these evaluations cover primarily L3,L4, P4 and P5 and, are probably far more accurate than necessary for moststatistical applications. An accuracy of four significant digits seems reasonableparticularly for higher dimensions. It is also clear that for higher dimensionsexcessive computer running time becomes a serious problem and some alternateapproach is needed.
The value of N, the number of positive Hermite zeros, needed for a specifiedaccuracy depends to a large degree on the determinant IR! of the correlationmatrix R and the limits (x1} of the probability integral. A guide for choosing Nbased on jR and max x, so as to achieve 4 digit accuracy (i.e., four correct digitsafter the decimal point) in D is available in Table 5. 1. Any rule based solely onRj would not be completely adequate. However, for a given accuracy, therequired N does not seem to vary significantly for Di,. . . , D although larger x1require higher N. Somewhat larger N are needed for D while generally smaller Nwould be more satisfactory for D than for D, and more satisfactory forD thanforD'.In the four variateorthant case, a table of P4 is available for the 21 correlationsets of Bacon (1963) as a function of N and IRI, Dutt (1973). An enlargement ofthat table to include 4)4 for x, = 1, 2 and 3, as a function of IRI and the J%T which aresufficient for fourdigit accuracy appears in Table 5.2. The required N appears inparentheses. It is also noted that a given N is satisfactory for both +x, and x1 sothat the 4)47s could be replaced by a corresponding set of upper tail probabilities,L4's.
TABLE 5.1A GUIDE Bisnn ON AN!) max X FOR CHoosING N IND'(k >2)FOR FouR DIGITACCURACY
* For the orthant case (x = 0), D =D 0556
I
maxx1IR - 0
1 2 3
0.7<IR 1 1-2 2-3 3-4 5-60.5<IRI0.7 2-3 2-3 3-4 5-6O.3<IRIsO.5 2-4 3-4 3-4 5-b0.2<IRIs0.3 3-4 3-4 4-6 5-ti0.1<IRI0.2 3-6 4-6 4-8 6-5
O.05<jRaQ.1 4-6 4-6 4-8 6-80<fRc0.05 4-12 4-12 4-12 6-12
MW
V
TAB
LE 5
.2IN
BA
CO
NA
S A
FU
NC
TiO
No
NFO
R T
HE
21SE
__-=
===-
----
-C
ase
r12
r23
r14
.
180.
500.
500.
309.
.-0
00
0.56
57(4
)0.
9201
(4)
200.
500.
500.
500.
500
0.59
30(4
)0.
9241
(4'!
140.
500.
500.
500
00
0.57
73(4
)0.
9219
(4)
210.
612.
.' 0.
666.
..0.
612.
..0.
408.
.0.
250.
408.
..0,
6322
(4)
0.93
02(4
)
170.
309.
-. 0,
809.
..0.
309.
..0
00.
5792
(4)
0.92
39(4
)
90.
500.
309.
..0,
809.
.0
00.
5938
(4)
0.92
59(4
)
8(1
,50
0.50
0.70
7..
00
0.59
39(4
)0.
9251
(4)
160.
500.
809.
.0.
135.
.0
00.
5826
(6)
0,92
49(6
)
130.
500,
790.
..0.
250
0.58
60(4
)0.
9248
(6)
30.
790-
0.25
0.79
0...
00
00.
6150
(6)
(1,9
303(
4)
ii0.
809.
..0.
500.
309.
..0
00
0.59
25(4
)0.
9258
(6)
100.
500.
707.
.-0.
500
00
0.59
29(4
)0.
9251
(4)
10.
925.
-0.
135.
--0.
809.
.-0
00.
6324
(4)
0.93
51(8
)
70.
500.
500.
709.
..0
00.
6029
(6)
0.92
71(6
)
150.
309.
-.0.
925
-.
0.13
5...
00
00.
5873
(8)
0.92
73(6
)
120.
500.
809.
..0.
309
..0
00
0.59
15(6
)0.
9257
(6)
20.
809.
.0.
309.
-0.
809.
.-0
00
0.62
24(4
)0.
93 1
6(6)
50.
500.
309.
..0.
925.
00
00.
6113
(10)
0.93
02(8
)
60.
500.
500.
809.
--0
00.
6052
(8)
0.92
76(6
)
4o.
707.
.. 0.
500.
707
00.
61)5
(8)
0.92
85(6
)
190.
500.
500.
500.
500
00.
6028
(4)
0.92
50(4
)
digi
Ts.
0.99
48(6
) 0.4
2133
8137
0.99
49(6
) 0.3
1250
000
0.99
48(6
) 0.3
1250
000
0,99
51(6
) 0.2
1701
389
0.99
50(6
) 0,1
6362
712
0,99
51(6
) 0,1
6362
712
0,99
50(6
) 0.1
2500
000
0.99
51(6
) O.0
8181
350.
9951
(6) 0
.078
1250
')Q
9953
(6) 0
.078
1250
00.
9951
(6) 0
.062
5000
)0.
9950
(6) 0
.062
5000
00,
9957
(8) 0
.031
2500
00.
9951
(6) 0
.031
2500
00.
9953
(10)
0,03
1250
000.
9951
(6) 0
.023
8728
70.
9954
'6) 0
.023
8728
70.
9954
(10)
0,01
1936
430.
9951
(8) 0
0091
181)
20.
9952
(6)
00
0
* in
a n
umbe
r of c
ases
the
liste
d N
in (
) is s
atis
faC
tol"
Yfo
r fiv
e si
gnifi
cant
F
(
Work is now in progress to Complete the Computer routines for D and Dwhich would then permit computation of the general normal probability Lk for allk 8 and for the orthant probability Pk for all k 9. The approach then would beto look for sinipic approximations for Di,..., D which would he accurate toabout three or four digits and which would he generally useful in iterativemaximum likelihood procedures Numerical properties of D',... D' areexamined graphically in an appendix available on request from the author for fourcorrelation matrices which are identified as EquicorreJae Markov, Toeplitz andNested. The corresponding probabilities L2,. . . , L for the first three matrices areavailable in that paper.
6. APPLICATrONS
Two applications where computational formulas of multiyariate normalprobabilities would be useful are briefly discussed. The first application relates to amodel of contraception discussed by Heckman and Willis (1973) in which themathematical details are here presented in a slightly more general way. Thesecond application pertains to the mtiltivariate probit probleni Ashford andSowden (1970). Other applications might be inferred from McFadden (1974).The maximum likelihood method is used for illustration purposes although otherestimation methods are available, Amemiya (1972). See also Tobin (1955, 1958)for an application in economics
6.1 Application # 1A Model ofContraceptionConsider a set of Continuous dependent random variables S1, S2, .. wherethe index refers to time. Specifically, let S1 denote a woman's "level of contracep-tion" at month j and consider M relevant economic variables E1,. . ., E1. E1 mayrelate to education level, E2 to income level, etc. The event that a woman becomespregnant in thejth month and leaves the sample is defined under this model by<A, where A =ao±MiaE and {a1J are unknown parameters relating tofor example, first pregnancies only. The a1} would presumably change for second,third, etc. pregnancies The inequality IS reversed when shc does not becomepregnant, and hence remains in the sample.The Probability of a Woman becoming pregnant in the kth month is
(6.1.1) Pf{Si>A .Skl>A;Sk<A]_(A)If now there is an independent sample of such Women with different birthintervals the method of maximum likelihood in principle may be used to estimatea, a1,. . by choosing those parameter values which maximize the joint probabil-ity of observing the sample distribtition of birth intervals. To carry out the MLmethod, it is necessary to specify a Probability dstribution fo S1, . . . , S.To put this in a somewhat more general framework, let 5 be represented asthe sum of two independent random variables S = U1 + E, where (U1,..., Uk) isdistributed as the mulfivariate normal k (/LIO {o..}) and e as n1(e O, o). It thenfollows that (S1, .. , S) is distribrited as the multivariate
normal ilk(SIO, {o +558
o}). In particular the correlation coefficient between S and S5 is given by
Ph = (o +u)/'Jo1 + o)(o-, +o)=ôI'94
In this context then the probability in equation (6.1. 1) would he conditionalon e and interest would be in the probability
JOD
-,which in terms of the Lk notation takes the appearance
(6.1.2)J
L(x1,.. .,XkI,Xk)flj(E)dE
withx=(A--e)/o-1fori=1,...,k.
6.2 Application #2Multivariate Probit ModelConsider k response systems S1,.. . , S,, in which the reaction of system S1 is
defined to be of the form
y1x1(z)/i1 fori=1,...,kwhere x-(z) is a suitable response and i/i is referred to as the tolerance for systemS1. In other words, if x, (z) > a toxic effect occurs.
It is reasonable to assume that the tolerance vector !/' = (i .....1/1k)' isdistributed as multivariate normal. The response functions x.(z) are so chosenthat all univariate inarginals associated with 1/i are standardized normals.
Let lFk(xl,. . . , xk)Lk(---xI, ... , b) where the subscript is dropped fork = 1. Then the probabilities of quantal response (+) and non-response (--) forsystem 5, are
p(z) =pT(z)F[xa(z)] 1t'[x1(z)J.
The probabilities that systems S1 and S1 both have positive responses is
p(z) = 42[x1(z), x.(z)].
In the bivariate case the other three probabilities of interest are
p(z) I2[xl(z), x(z)] p'(z)p(z)p(z) = F2[x1(z), x(z)] = p7(z) - p(z)
andI (p+p+p).
For the k dimensional case, interest is in an expression of the form
pr,.....
559
PkOtlE)flI(e) de
6
aIi groups) all sets of
CONCLUDING REMARKS
The computational formulas based on integral transform representationsover (0, ) fo multivariate normal probabilities have been summarized withspecific emphasis given to properties of the D* functions in the representations. Inan appendix available on request from the author, numetical aspects of
are examined for four important correlation matrices identified asEquicorrelated, Markov, Toeplitz and Nested. Thegeneral curve shapes of D inthese four cases suggest the possibility of obtaining simple approximations inmore general cases.
A variety of numerical results, most of whichwere previously unavailable aregiven for the multivariate normal probabilities L2.....L4, in an appendix avail-able on request from the author. Two specific applications to econometric modelsare noted.
ACKNOWLEDGEMENT
Acknowledgement is given to Dr. Peter Cheng for helpful numericalimprovements in the computer routines and to Li Chun Tao for the originalcomputer programs.
E. M. Laboratories, Inc.Elmsford, N. Y. 10523
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560
Now, let each response function x(z) he of the form
x1(z)=c fori=l,...,k
1
where is a k-dimensional vector of known constants and f3 for i fixed, is ak-dimensional vector of unknown parameters. Let denote the number oforganisms in which the systems S1,... , S exhibit the responses (±,..., ±), Theparameter vectors 3......Pk can then be estimated by the log likelihood functionof a given set of independent samples,
k ogp1 .k+conttant-
C .kfl*4fl
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