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R O C ZN IK I P O LS K IE G O T O W A R Z Y S T W A M ATEM ATYCZNEGO Séria I : P R A C E M A TEM ATYCZN E X V I I I (1974)

J erzy B artos ( h ô dz)

The determination of the two-dimensional distribution by means oî conditional distributions

1. The setting of the problem. In investigations of the two-dimensional random variables it is sometimes convenient to represent the two-dimen­

sional distribution by means of those of one dimension. I t is known, however, that for instance the boundary distributions determine uniquely the two-dimensional distributions only in case of independence of random variables. In case when the random variables are dependent the two- dimensional distribution can be represented by means of boundary distribution and a corresponding conditional distributions.

H. Bumsey J r . and E . C. Posner dealt in [2] with the problem of how to determine the density of two-dimensional distribution of the random variable with respect to the boundary distributions and some additional conditions. These additional conditions they imposed were certain equa­

tions of the form

holding for given functions rj (x, y ) and for given constants ; to assure the uniqueness of f{ x , y) it is assumed, moreover, that this function attains its maximal entropy

The alternative approach to this problem can also be considered:

since the boundary distributions do not determine the two-dimensional distributions whether then the knowledge of both conditional distributions suffices 4 In the sequel we shall show that the answer to the above question is in general negative. We shall show that for a large class of distributions this is, however, sufficient.

We shall now investigate these questions separately for continuous - and jump-functions random variables.

— CO — o o

в {f) = J f In/ ldxdy.

— OO — OO

9 — R oczniki PTM — P r a c e M atem atyczn e X V III.

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2. The representation of density of the two-dimensional distribution by means of densities of conditional distributions. Let (X, Y) be the two- dimensional continuous variable with density f{oc,y). Let cp{x),ip{x) срг{x/y), щ { у 1х) denote respectively densities of boundary distributions of X , 7 and conditional of X / Y and T [X . To represent the density f(x , y) with respect to (pi(x/y) and yi(ylx) we shall first prove the following

lemma.

L emma . Let D be a measurable subset in the plane and let density f(x , y) o f the two-dimensional random variable (X , Y) be given. I f f ( x , y) > 0 fo r (x, y )e D , f( x , у) = 0 fo r (x, y) 4 D, then

Vi (У/a?) = Ч>{У)

<Pi{x/y) <p{x) holds almost everwhere in D.

P r o o f . Denote by D x and D y the projections of D*, where D* denotes a set of density points of D, onto the coordinate axes Ox and Oy. Clearly then D* a Dx x D y. B y the Fubini theorem there exist, for almost every x a function <p (x) defined as <p{x) = j f ( x , y)dy and, for almost every y, a function y)(y) — J f ( x , y)dx. Bv

Those functions have values greater than 0 in Dx and Dy with exception of a measure zero set in which their values can be zero. Hence the condi­

tional densities <px and грх are well defined, have positive values and satisfy the equations

<Pi№ly) f№ , У )

У>(У) Vi (УМ

=

/(^ У)

<p(x)

almost everywhere in D. Dividing these relations by sides we accomplish the proof.

W ith the help of this lemma we shall prove the following theorem.

T heorem 1. L et D be a measurable plan e set, Dvx° — a projection o f its intersection by the straight line у = y0 onto Oy axis, Dx — a projection o f D onto Ox axis. I f a density f( x , y) is greater than zero fo r (x, y)e D and i f Щ 0 = L>x, then

f { v , y ) yi№)Pi(a?/yo) / Г Г У>Л у 1 я )<Р1( я о ) У>ЛУо1®) W y>i{y 0 læ)

holds almost everwhere in D.

P r o o f . We have for almost all (x, y )eD

f ( x , У)

=

<p{x)fi(y[x).

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Applying lemma, since (x , y0)eD , we get

' <p{®) У’ (У о)< рЛ х1У о) ч у М what substituted to the previous relation yields (3) /О», У) = у у 1 я )< р Л®1У о )

ч у М <Р(Уо) for (X, y) e D .

Let us, moreover, use the condition that J J f(æ , y) dxdy = 1 to express

D

f ( y 0) and substitute to (3) and we shall get the thesis of Theorem 1.

E e m a r k . If Dy° = D y, where T)y and D*° denote respectively the projections of JD and the projection of its intersection with 00 — OC q onto the axis Oy, then the thesi-s of the theorem can be written in the form

Я » , У) <pi(®ly)v>i(ylx o) I г г <Pi(æly)y,i(y læo) dxd '

х Ы у ) V J Vifaoly)

From the above proven theorem it follows that the density of the two-dimensional random variable (X , Y) can be represented by all condi­

tional distributions of the random variable Y/X and one conditional distribution of X / Y == y0.

This problem can also be viewed at in a certain geometrical fashion.

We know that the two-dimensional density represents in general some surface, whereas the conditional densities its normed intersections by the planes parallel to the Oxz and Oyz planes. The theorem says therefore that the surface f( x , y) may be represented with respect to all normed intersections by the planes parallel to Oxz and one of the normed inter­

sections by Oyz.

To fully illustrate the theorem let us explain yet the following question.

Of what a nature must be a set D and what intersection of this set by a straight line у = y0 should be taken ?

We know th at the projection of intersection by the straight line у = y 0 must be identical to D x. Of such a nature is, for example a set which is a product of its projections onto the axes Ox and Oy : В = Dx x x B y. This set can also be of a different form but then its intersection Ъу У = y 0 should not be taken arbitrarily but so as to DV J> = Dx hold.

An example of such a set is shown on Fig. 1 .

I t means that this intersection is such that it connects all intersections

by straight lines parallel to Oy axis. In the case when such an intersection

does not exist (Fig. 3) the representation of the two-dimensional density

by all distributions of Y /X variable and one of the distributions X fY = y 0

is not feasible.

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In the sequel we shall deal with a problem which can be regarded as a converse to the above, namely:

1. Does the knowledge of the functions f x{x, y) and qpx{x) treated as densities of conditional distributions Y fX and X j Y — y 0 suffice to anticipate the existence of some two-dimensional distribution, of which densities of respective conditional distributions are given functions %

2 . Does the knowledge of the functions ipx(x, y) and qpx(x) suffice to express uniquely the density of two-dimensional distribution f ( x , y) and if not, what are the additional conditions to this purpose. In what follows we shall study these problems.

3. Conditions for existence of two-dimensional distribution at given conditional distributions. Let us, firstly, introduce the notations. В denotes an arbitrary measurable plane set, B x and B y as well as B vf> and B y° have the same meaning as before.

T heorem 2 ( existen c e ). Given the functions <px(x, у) and q?x(x) satisfying conditions

1° чрх{х, у) > 0 fo r (x, у)й В and equal 0 fo r (x, y) 4 D, qpx(x) > 0 fo r x e B v x« and equal 0 fo r x 4 _D^°;

2 ° f i p 1( x , y ) d y = 1 , f cpx (x) dx — 1 ;

D x

V

D y0

X

/ ipi(x, y0) dec •— - Tc 00 .

Then the function

c<Pi(x)y>i{x, y) Уо) е {х )щ {х ;у )

О

for {X, у ) е В п В у х В У х о, for (x, у ) е В \ В у X В у°, for {х,у) 4В ,

(4) у ) =

(5)

where e(x) is any continuous positive function satisfying condition f e(x)dx = l < 1 , and c is a positive constant such that ck + l = 1 , is Dx\Dvf>

a density o f some two-dimensional distribution (X , Y) fo r which the densities o f conditional distributions are Y/X and X /Y = y 0 are

f x ( x ,

y) and ц

ox { x ) .

P ro o f. (Fig. 4). Since щ { x ,y 0) differs from zero on DHf hence the

w A x , у ) ® - , ( x )

term --- --- -— is well-defined in D n D y x B y°. Also the integral У о)

f f f { x , y)dxdy exists since f( x , y) is non-negative nad there exists iterated D

integrals namely

j j f(oo,y)dxdy = JJ f( x ,y ) d x d y + JJ f(x ,y )d x d y

/[/ &Pi{œ)V>1(x, У)

n\(Dyxnvx 0 )

V i{x,y)

dVx° D y n x \ V yx ° D y

+ ^ ^ e{x)'ip1{x, y)dy]^dx

r cwDxjax . r

— --- - + e(x)dx = c h f- l = 1 . J ipi(x) у о) J

Dl° Dx\Dyf

Hence the function f( x , y) of the theorem as non-negative in D and satisfying condition f f f ( x , y)dxdy = 1 is density.

D

I t remains to show that yAx, у) and (рг{х) are the densities of distri­

butions.

Denote by у (x) and y{y) boundary densities of variables X and Y respectively and by yA y/x) and <Pi(x[y) densities of conditional distributions

Y/X and X/Y. Then

Vi № ) = №> У)

cp{x)

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and 9 )(x) equals

cp(x) =

I y) л dy — ----—---- r ctp^x) for X e B l0,

D уЛ®,Уо) n i® , У о )

f s(x)yj1(x, y)dy = s{x) for X€D x \D%>,

d : Thus

n i y l ® ) =

ccpi(x)yi(x,y) cy^x) Wii®, Уo) Vii®, Уo) е(х )щ (х , y)

e(x)

for x 4B x.

П (®, У) for i®, У ) е В п В у х B vx\

= n i ® , y) f o r i®, У)€ B \ B y x Щ 0, and finally у>^у/х) = y) for (x, y)* B .

То express (p1ix /y Q) we shall use the relation ni®Iyo) fi®, y)

wiyo) W e know that

vi y) = f f i ® j y ) à ® ,

D *

hence

and

Wiyo) = / fi®, Уо)Я® = / c e p ^ d x = c

D v 0 D VJ

Vi ( * Ы = = Vl (æ). Q.E.D.

Viyo)

We deduce from this theorem that knowing the functions regarded as densities of conditional distributions we can find such a two-dimensional distribution, whose densities of conditional distributions are these functions.

Because density of this two-dimensional distribution depends upon choice of both the constant c and function e (x), it is not uniquely determined, i. e. taking various c and e(x) satisfying condition clc-\-l = 1 (see (4)) we shall get different two-dimensional densities f ( x , y) which will satisfy the required conditions.

Next chapter will be devoted to studying the following problems:

1. Under what assumptions the functions y^ x, y) and (pxix) determine uniquely the density f(x , y) ?

2. If uniqueness is lacking what additional conditions should be

assumed in order to get the uniqueness ?

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4 . Sufficient conditions for unique determination of the two-dimensional distribution by conditional densities.

T h e o r e m 3 . Given some plan e set D with a point y0 such that — I)x.

I f there exist two functions <px{x) and y)x{x ,y ) satisfying the hypotheses o f Theorem 2 , then the function

Я®» У)

yjx{x, y)(pt {x)

D

n ( x , y)<Pi{oc)

Vi(x, Vo) dxdy fo r (x ,y ) € B , Vi(x,yo)

0 fo r (x, y )4 D ,

is the only function which is a density o f a two-dimensional distribution with the densities o f its conditional distributions Y fX and X J Y — y 0 equal to cpx{x) and y)x(x, y).

P r o o f. From the existence theorem (Theorem 2) we know that there exists a density f( x , y) of which densities of conditional distributions X / Y = y0 and Y IX are given functions. Therefore as the condition

= Dx holds, we can apply Theorem 1 thus accomplishing the proof.

The formula for density given by this theorem is identical to that of Theorem 2 because if D ^ ° = D x, then D \ (D y x D%>) is a void set and hence

С(рх{х)щ{х, у) Я®> У) =

and the constant c equals c =

D

ipx (Х)'У0) 0

<рх{х)у>г{х, y) У a)

for ( x ,y ) e D , for { x ,y ) 4D ,

dxdy

To illustrate the way of forming of f(x , y) as well as to show the necessity of Assumption 3 in Theorem 2 let us consider an example.

E x a m p l e . Take the functions cpx{x) 1

7Г(1+Ж2) ’ y x{x, y) V l + x 2 тт (1 + я 2 + у2) I t is clear that

+ o o + o o

J <px(x)dx = 1 and J ' y x{x, y)dy = 1

— 00 —00

and so we may regard them as densities of conditional distributions of some two-dimensional distribution. Assume y0 = 0. Then

V i +<

y)

7 г ( 1 + a?2 + y 2) 7 r ( l + a?2)

1 j/l + æ 2 ТС 1 + х г + У 2

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However, this function as non-integrable in the plane — oo < x, y < -f oo cannot be a density. I t happens so because Assumption 3 is not

cpx{x) + oo

satisfied, i. e. J is not integrable.

Уо)

In what follows we shall consider the following question. Since to determine uniquely the density it does not suffice the knowledge of (p{x) and ip(æ, у ) when Dx\T)y° is of a positive measure, what else conditions must be additionaly assumed ? In order to answer this question we shall prove the following:

T h e o r e m 4. Let D be a measurable plan e set and let 1)xl and BH2 be projections onto x axis o f its intersections by the straight lines у = y1 and у = у 2 (уг Ф у 2). Let L>x = B yïu B x2 and let D ^ n h J 2 be o f positive measure.

I f given are functions ipx(x, у), (рг(х) and <p2{x) satisfying conditions:

1 .

2 .

Vi(®> У) > 0 f or У)

<Px

(x) > 0 fo r

Х е Щ 1

<p2(x) > 0 fo r

X e B vx 2

f y>i(æ, y)dy = 1 ,

and equals zero outside this set ;

D~

3.

4.

J q>x{x)dx = j cp2(x)dx

B VJ

cc

D v

cc

2

<px{x) Уг)

= i ; (p2(x) V i fa y i ) 9>i(»0)

the integrals

Vii®, У a)

/ YbO»? Ух)

Yhfoo» Уa) 9*2 (®o)

<Pt(x)

Dy\ ' ' BV f 2

a : x

are finite. Then the function

s W i ( æ oj У

2

) < P i ( x ) v > i ( œ , y )

^2(^7 У 2 )

fo r x 0e Dvxin D l2-,

dx

(б) /(®, 2 /)

9h (#?0) Sip1(x01 уг)

<p2M о

Ч р х { х , У х )

ср2{х)у)х{х, у)

fo r (x, у ) € В п В у х В р ,

V,i(**b У2) where

s = / У1) Г

\ <Pi(æo) J

У 2 И 2 /i) ■dx

fo r {x, y)e D

Г\

Dy x B x2, fo r (x, y) 4 D ,

Vi(x o, 2/2) Г

<P*{xQ) V ifo У2) dx

is the only function which is a density o f the two-dimensional distribution

and whose densities o f conditional distributions Y jX , X j Y — y 1} X j Y = y 2

are respectively the given functions y > i (&, y), <Pi(æ )j «M®)-

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л

P ro o f. Knowing -уг{х, y) and срг(х) we can by the existence theorem construct the two-dimensional density which on Dr\ByX B vxl takes the form

№ , У ) Сх<Рх(х)у>х{®, У) Wifai Ух)

Analogously, for хрг{х, у) and <p2{x) on B n B y x B v x2 this density equals

Write

/ ( « » У) c2(p2{co)ip1{x, y) Vi(x, У г)

€iCp1{x0) _ C2(p2(x о) Ух&отУх)1 2 Y>2(®o ,y%Ÿ

where » 0 fD"inD|J2. Substitution of sx and s 2 from these relations to former equalities yields

У) =

SiViiXp, Ух) yi(a?)yx{æ, y)

<Pi(x0) щ{х,Ух) S2Vl(®0, У2) <P2(æ) n ( x , У) У

У

2

(®о) У х ^ У ъ )

for (х, у) с В п В у х B vxl ,

for (х, у)е D n D y X B vx2.

Taking into account our third hypothesis we get that == s2 = s.

I t remains only to take s such that f( x , y) becomes a density. To this end we shall make use of the condition f(x , y)dxdy — 1 .

Hence

J f f(x ,y )d x d y = J J eVi(®o? yi)<Pi{x)Vi{v, У)

dxdy -f Dr,ByXZ?n

+

? i ( ® o ) y i ( ® , Ух)

8Vx{Xf»y*) УЛх)у>х(я,У)

D\DyXDVl и У2Ы Vxfa У2) dxdy

- f

8V>x(®0> Ух)Ух(х) , . --- dx +

Vi ( ®o) V i( ® j

Ух)

Г SVlfaoi У 2) ’ 4 ) 2 {х ) J <р2Ы - ъ ( х , У 2)

Dx\dVx1

=

1

and

Ух(®о, Ух) Г У х И ^ Vhfoo> Уа) Г Уя(®)

Ух(^о) ~ Ух( ^} Ух) У

2

{®о) J Wxi^) У2)

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To accomplish the proof we need only to show that the densities of conditional distributions X / Y = y l i X j Y — y 2, Y jX are equal respect­

ively to ср1{х)срг{х), 1рг(х, у). This can easily be verified similarly as in Theorem 1.

The above proven theorem allows us to determine the density when the projections onto Ox axis of two intersections of D by lines у = yx and у = y 2 cover Dx and their common part is of positive measure. This theorem could be generalized to the case when a finite number of inter­

sections cover JDX and each of this intersections has a common part of positive measure with at most one of the remaining intersections.

Finally we conclude that the density f(x , у ) can be determined if all conditional distributions of Y [X are known and if given is a finite number of distributions X fY = yi such that projections of B yJ cover

Dx. The number of these intersections could be reduced if instead of taking intersections by planes parallel to the coordinate planes intersections by other planes were taken as illustrated on Figs. 5, 6 .

Fig. 5 shows the case of intersections by straight lines parallel to the axes. I t is seen that then to connect all intersections by lines parallel to Oy axis at least 2 intersections by lines parallel to Ox axis are needed.

Fig. 6 shows, however, that taking intersections by lines not neces­

sarily parallel to the axes but parallel to the eclipse main axis — one intersection suffices. The number of intersections can also be reduced whenever a type of a required distribution is known. Knownig, for instance, that the two-dimensional distribution {X, Y) is normal to establish its density it suffices to give the densities of distributions of X ] Y = yx, X / Y = y 2 and Y jX = х г {уг Ф y 2).

Therefore in this case only three normed intersections of the surface f ( x , y) are sufficient.

We shall now give a simple example illustrating applications of

Theorem 3 to determining densities of the two-dimensional distributions

at given densities of conditional distributions.

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E x a m p l e . Determine the density of the two-dimensional distribution of the variables (X, Y) assuming that the densities of distributions of X fY and Y jX = 1 are the functions

< р Л®1 у ) =

ye xy 0

Wiivl1) =

ja+l

Г ( а ) y e

for x, у > 0 , for others, Py for у > 0 ,

for others.

Since all hypotheses of Theorem 3 are fulfilled for x, у > 0 we have

№ , y)

00 00

^ <Pi(xly)yi{yll) / Г r (Pi(xly)Vi(yl1 )

<PiQly) \J J V ii 1 ly)

= ( ^ - 1) ° r c-xvc-v ^ -if

dxdy - i

5. Representation of the probability function of the two-dimensional jump distribution by means of conditional distributions. Similarly as for continuous distributions we can deal with a two-dimensional jump variable the only difference upon approach being the fact that instead of integrals in corresponding formulae the sequences and series will appear.

L et x 1} x 2, ... and 2 /i, ? ••• denote jump points in boundary distri­

butions respectively of X and Y.

Let further p (X j,yk) —P (X j = x j f y = yk) designates a joint probability distribution of variables ( X , Y) whereas y{Xj), (p(yk), (pi(oojlyk) y\{yklxi) denote probability function of boundary distribution of X , Y and condi­

tional random variables X [Y , Y jX respectively.

T h e o r e m 5 . I f D is a set o f points (xj, yk) fo r which p {x j: yk) > 0 ,

then fo r all points o f В there exist y i(y kIXj) and ip2{Xj[yk) and the following equality holds :

V i(yklxi) = wiVk) Ч>Лхэ1Ук)

Since £ p jk and fy p jk both differ from zero, the functions

j к

v i ( y k h ) V (Щ, У к)

Ï P j k ’ <рЛщ1Ук) = P ( xj , У к)

2 р »

exist in В . Dividing these last equalities by sides we accomplish the

proof.

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T heorem 6. I f B is a set o f points (Xj, yk) fo r which p{Xj, yk) > 0, B V J° the projection o f its intersection by y = yl(j onto Ox and Dx is the projection o f В onto Ox axis and i f B V = Dx holds in every point o f B , then the relation ( 8 ) P(®j, У к) = v(ykl®j)<Pi(Xjlyi0) ( x ' 'Ы У к М г Л Щ у

ViiViofa ( s

3,k Vi (V iJX j)

is satisfied.

Proof of this theorem is completely identical to this for a continuous case with a replacement of an integral by a sum.

Similarly as for continuous random variables the conditions and methods of unique determinations of the two-dimensional jump distri­

bution by means of the conditional distributions will be given.

T heorem 7. Let В denote a set o f points (Xj, yk) o f the plan e (j , h =

= 1, 2, ...) . Given are functions (px{Xj) and y^Xj[yk) satisfying the conditions 1 ° (Xj, yk) > 0 fo r (Xj, уk)e В and = 0 fo r others, <рх {xf) > 0 fo r xj e B v 4 and = 0 fo r Xj 4 B%.

2 ° (xjj yk) = 1 , 2 <рх {Xj) = 1 .

A

j

3 ° у —(f }Sx^ _ ^ < oo and holds the condition B yJ = B„.

j Vi)

Then the function

Р(Ц ,Ук)

<Pi{æj,yk)<Pi(æj) I V V i (xi , У к ) ?Л хз) \ 1

Vx{®j,yi) \^jJ у Л я ц У г ) I fo r {Xj, yk) e В ,

0 fo r ( X j , y k ) 4 B

is the only function representing some two-dimensional jum p probability distribution and whose conditional distributions Y /X and X jY — уг are given functions.

The proof of this theorem is analogous to that for continuous case.

C orollary . I f a set o f points (Xj, yk) is finite, then fo r the theorem to hold Condition 3 can be dropped.

References

[1] H. C ra m e r, Melody matematyczne w statystyce, Warszawa 1958.

[2] H. E u m se y Jr. and E . C. P o s n e r, Joint distributions with prescribed moments, Ann. Math. Stat. 36 (1965), p. 286-298.

[3] R. S ik o rsk i, FunTccje rzeezywiste I , Waraszawa 1958.

T EC H N IC A L U N IV E R S IT Y , IN S T IT U T E O F M ATHEM ATICS, L Ô D 2

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