Probability distributions with given multivariate marginalsLeon Cohen Citation: Journal of Mathematical Physics 25, 2402 (1984); doi: 10.1063/1.526462 View online: http://dx.doi.org/10.1063/1.526462 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/25/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Constructing multivariate distributions with generalized marginals and t-copulas AIP Conf. Proc. 1621, 435 (2014); 10.1063/1.4898503 Marginal probability distributions for seabed parameter values from simultaneous processing of vertical andhorizontal apertures. J. Acoust. Soc. Am. 126, 2315 (2009); 10.1121/1.3249546 Quantum Teleportation in Spacetime, and Dependent Clones With Given Probability AIP Conf. Proc. 750, 382 (2005); 10.1063/1.1874590 Measures of dependence for multivariate Lévy distributions AIP Conf. Proc. 553, 289 (2001); 10.1063/1.1358198 The probability of exciting an elastic structure with a given resonant amplification factor J. Acoust. Soc. Am. 72, 2035 (1982); 10.1121/1.388636

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Probability distributions with given multivariate marginals Leon Cohen Hunter College o/the City University, New York, New York 10021

(Received 25 October 1983; accepted for publication 23 March 1984)

A method is presented for obtaining joint probability density functions which satisfy given multivariate marginal densities.

PACS numbers: 02.50.Cw, 02.30. + g

I. INTRODUCTION

Suppose we have the marginal probability densities PI and P2 of the two sets of random variables XI = (x I,X2, ... ,xN) and X2 = (x N, + I ,x N, + 2 , ... ,x N, + N,)' respectively. The pur­pose of this paper is to present a method for obtaining a joint distribution P (x 1,x2"",x N, + N,) consistent with the margin­also The case where the marginals are a function of only one variable has been extensively studied 1 and a general method has been given to generate an infinite number of joint distri­butions for that situation. 2

-4 We shall here consider the case where the marginals are in general multivariate, that is, functions of more than one random variable. We assume that the given marginals do not have any variables in common.

What we seek are positive functions such that

(1.1)

(1.2)

II. JOINT DISTRIBUTION

Choose any positive function, h (u l'U2"",U N) of N ( = NI + N 2 ) variables defined in the N-dimensional unit cube and normalize it to 1,

f h (u l ,u2, .. ·,uN)du l du2· .. duN = 1, (2.1)

and let

hl(UI, ... ,UN,) = II h (ul, ... ,UN, + N,)duN, + 1 .. ·duN, +N,'

p(UI, ... ,UN) = h - hI - h2 + l. An example of such a function is

N

h = 2N II U" j= 1

N,

hi = 2N, II u" ;= 1

N

h2 = 2N, II U j ,

i=N1 + 1

(2.2)

(2.4)

P = {2N' .It U j - I} {2N,. IT U j - I}. (2.5)

1=1 I=N1 + 1

An infinite number of joint distributions satisfying the mar­ginals may be generated by

P(X I,X2, .. ·,XN) = PIP2[ 1 + cP(U I,U2, ... ,UN)], (2.6)

where now the u's are defined in terms of the x's

u j = uj(X I,X2, ... ,XN,), i = 1,2, ... ,NI (2.7)

uj = uj(xN, + I ,,,,,XN, + N,), i = NI + 1, ... ,NI + N2, (2.8)

in such a way thta the Jacobian of the transformations satisfy

(2.9)

(2.10)

Also, the transformation must map the infinite NI-dimen­sional space into the NI-dimensional unit cube and similarly for N 2• InSec. III we will give the explicit transfomation.

The constant c is chosen so that P will be positive. Iden­tical arguments to that previously given2 implies that c may be chosen to be any number in the range

(2.11)

where - II and 12 are the absolute minimum and maximum ofp.

To show that the marginals are satisfied consider

f: 00 P2(XN, + I , ... ,xN, + N,)h (ul, ... ,uN)dxN, + I ... dxN, + N,

(2.12)

= II h (ul, ... ,uN)duN, + 1 .. ·duN = hi'

Furthermore

f: 00 P2 h2 dxN, + 1 .. ·dxN = f h dUN, + I .. ·duN = 1, (2.13)

and hence

f P2pdxN,+I···dxN,+N, =0, (2.14)

which proves that the first marginal condition is satisfied. Similarly for the second.

III. TRANSFORMATION AND JACOBIAN

What remains to be shown is that the transformations can be found such that the Jacobians are given by Eqs. (2.9) and (2.10). Define

2402 J. Math. Phys. 25 (8). August 1984 0022-2488/84/082402-02$02.50 @ 1984 American Institute of Physics 2402

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p\N, - \)(Xi + I , ... ,xN,) = J: 00 p\(X\, .. ·,xN, )dxc··dxj>

then a transformation satisfying Eq. (2.9) is

SX; piN, -i+ I)(X~ ... ,x )dx~ - 00 \ I' N, I • / N

IN .) , I"", I' PI,-I(Xi+\, .. ·,xN,)

U i =

(3.1)

(3.2)

Since

aUi

aXj

= 0, j < i, i<N\, (3.3)

The Jacobian is the product of the diagonal elements which are given by

P\N, -i+ \)

p\N,-i) ,

and therefore

N IN.-i+\) J- '(j)\' \-.n pIN,-'1

,~I \

i<N, (3.4)

(3.5)

For the other set of variables the identical procedure is fol­lowed. Also, it s clear that as the x's range over all space the range ofthe u's is from zero to one.

IV. CONCLUSION AND GENERALIZATION

An infinite number of P 's can be generated by choosing different h 's and c's. We point out that sometimes different H's will lead to thesamep and that also for some choices ofh one could get a p identical to zero.

Generalization to more than two marginals is as fol­lows. Suppose we have M marginals each having Ni random variables then the joint density will be a function of MN random variables where N is the sum of the Ni's. For each marginal the u transformations are chosen as above. The joint density is given by

M

P(x\,x2 , .. ·,XMN ) = IT P;lXi ) [ 1 + cp(U\,U2,· .. ,UM )], i=1

(4.1)

2403 J. Math. Phys., Vol. 25, No.8, August 1984

where now h is a function of MN variables and

with

Ui = (UKi + \' ... 'UKi + N)'

Xi = (XKi + \ ,···,xKi + N)'

i

Ki= n~, j~ I

P(UI""'UM ) = h (U\,. .. ,UM )

M

- L hi(Vi) + (M - 1), i~1

(4.2)

(4.3)

Finch and Groblicki5 have shown that for the case where the marginals are a function of only one variable, the procedure given above encompasses all solutions. A similar proof holds for the multivariate case. We note that in general h can be a functional of the marginals.

ACKNOWLEDGMENT

This research was supported in part by a grant from the City University of New York Research Award Program.

'See, for example, the following papers and the references therein: D. Mor­genstern, Mitt. Math. Stat. 8, 234(1956); E. J. Gumbel, Rev. Fac. Ci. Univ. Lisboa Ser. 2A Ci. Mat. 7, 179 (1959); D. J. G. Farlie, Biometrica 50,499 (1963).

2L. Cohen and Y. I. Zaparovanny, J. Math. Phys. 21, 794 (1980). 3L. Cohen and T. Posch, "Positive Time-Frequency Distribution Func­tions," submitted to IEEE Trans. Acoust., Speech Signal Process.

4L. Cohen, Proc. ICASSP 84, 41BI.I (1984). 'P. D. Finch and R. Groblicki, "Bivariate Probability Densities with Given Margins," Found. Phys. (to appear).

Leon Cohen 2403

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