t , ,~- '-S-z.T r ~

E L S E V I E R Fuzzy Sets and Systems 92 (1997) 83-93

FUZZY sets and systems

On the variance of fuzzy random variables Ralf Krrner

Faculty of Mathematics and Computer Science, Freiberg University of Mining and Technology, 09596 Freiberg, Germany

Abstract

This paper deals with an expectation and a real-valued variance of fuzzy random variables. The expectation and the variance of a fuzzy random variable is characterized by Frrchet's principle in a metric space. We study properties of the variance of a fuzzy random variable and compare it with the common variance of real-valued random variables. Using the expectation and the variance of fuzzy random variables, we consider a linear regression problem and limit theorems. (~) 1997 Elsevier Science B.V.

Keywords: Probability theory and statistics; Random fuzzy variables; Expectation and variance; Linear regression; Limit theorems

I. Introduction

In many real situations the variability is given by two kinds of uncertainty: randomness (stochastic vari- ability) and imprecision (vagueness). The concept of fuzzy random variables, introduced by Puri and Ralescu (1986) as a generalization of compact random sets, combines both randomness and imprecision. The stochastic variability is described by use of probability theory and the vagueness by use of fuzzy sets introduced by Zadeh [11]. This concept has been found very convenient in studying linear statistical inference, limit theorems and so on. Indeed, many results can be regarded as generalizations of results of real-valued random variables. The notion of expectation and the notion of variance are the relevant notions for a linear statistical inference with fuzzy random data (see Section 5). By Frrchet's principle (see [3, 12, 7]) we can define an expected element and a variance of a random variable in a metric space. Here, we use an L2-metric on the space of convex sets, defined by their support functions. Lyashenko [6] observed that the variance, defined by this metric, leads to an appropriate additive variance. Many other properties of the common variance of real-valued random vari- ables are also preserved (see Section 3). Furthermore, the expectation defined by the L2-metric of Frrchet's principle is equal to an extension of the well-known expectation of Aumann (see [5]). Laws of large numbers for sums of independent fuzzy random variables based on this variance are also obtained (see Section 6).

2. Prdiminaries

Following Kruse and Meyer [5] the set of all normal compact convex fuzzy subsets of R" is denoted by ~ c , i.e. any fuzzy set AE~c with the membership function mA : R" -* [0, 1] satisfies

0165-0114/97/$17.00 ~) 1997 Elsevier Science B.V. All rights reserved PII S0165-0114(96)001 69-8

84 R. K6rnerlFuzzy Sets and Systems 92 (1997) 83-93

(i) A is normal, i.e. A 1 = {xE~" : mA(x) = 1} is non-empty, (ii) the a-cuts of A

A~ = {xERn:raA(x)>>.a} , 0 < a ~ l

are convex and compact and (iii) the support of A

"4°= U A" (I) ~E(0,1]

is compact. A linear structure of a cone in : c is defined as extension of Minkowski operations of addition and scalar

multiplication by Zadeh's extension principle. In terms of level sets the operations for A , B E : c and ). E R

result in

( A + B ) ~ : = A a + B ~ and (L4)~:=2A a, aE[0,1].

Moreover, each fuzzy set A E ~ c corresponds uniquely to its support function

sA(ot, u) = sup{(u,a): a E A ~ } , uES "-1, ~E[0,1],

where S "- l is the (n - 1 )-dimensional unit sphere of R n and (., .) is the inner product of the Euclidean space R n. It follows that sA( . ,u ) represents a fuzzy set for any fixed u E S " - l and sA(~,-) is the support function of the convex 0t-cut of A for any fixed ~E[0, 1].

Now a metric on ~ c is defined by the L2-metric on the space of Lebesgue integrable functions (use sA instead of A E ~-c)

d=(a,B) = lisa - so l12 = n . . - , IsA(~,u)- sn(~,u)lZt~(du)da (2)

for all A, B E ~ c and a norm

11,4115 = IIsAII2 = n. , - , Is,4(~,u)121~(du)d~t (3)

Diamond and Kloeden [2] have shown that the space (~rc, d2) is a complete separable metric space. Another helpful notion is the Steiner point of a fuzzy set A E ~-c, defined by

l'/s ¢r,4 = n . u . sA(u )kff du ) da, (4) n - - t

where/~ is the normed Lebesgue measure on the unit sphere S n - I ( p ( S n- I ) = 1) (see [9]). For any A , B E ~ c

and any two real numbers 2, ~ the Steiner point ira satisfies

aA E A and tr~A+~B = A . trA + 7 " trB.

The set A0 = A - aA is centered, i.e.

a A - ~ = 0. (5)

Note that the Steiner point of a fuzzy set A E ~-c can be written as the average of the Steiner points of the a-cuts (4) or as the Steiner point of the average of the or-cuts A ~ by a simple change of the order of integration.

R. Krrner l Fuzzy Sets and Systems 92 (1997) 83-93 85

Let ( f2 ,~ , P) be a probability space. Now, a fuzzy (-valued) random variable X is a Borel measurable mapping X : f2 ~ ~ c . It follows that for each ~ E [0, 1] the ~-cuts X ~ are non-empty compact convex random sets (see [2]).

Assumption 1. In the following we restrict ourselves to square integrable fuzzy random variables, i.e. EI[XH~

In the next section we give characteristics of fuzzy random variables.

3. Expectation, variance and covariance

The approach of Frrchet [3] handles with an expectation and a variance in a metric space (./-t',d). The expectation of a random element X in ~¢l is built by the set of all elements A E J// with

Ed2(X,A) = inf Ede(X,B), (6) BE.g/

provided that there is an A EJ/¢ with ~_d2(X,A) < oc. The infimum of (6) is called variance of the random element X (Vat(X)). This approach has been shown to be reasonable in other fields of application. In particular, the least-squares property of real-valued random variables x is generalized by this principle, i.e. E(x - Ex) 2 = infcen E(x - c) 2.

Hence, the expectation and the variance of a fuzzy random variable X can be defined in the metric space of normal compact convex fuzzy sets equipped with the metric (2), i.e.

and

OgFX = { A E ~ c : Ed2(X,A)= inf ~_d2~(X,B)~ B E a~c J

(7)

Var(X) = inf IEd~(X,B), (8) BE~arc

The assumption 1 ensures that the expectation as well as the variance always exist. Usually, the expectation of a fuzzy random variable is defined by the generalized Aumann expectation EX (see [5])

(EX) ~ = sup{~: ~ is a selector of X a, FII~II2 < ~} , ~e(0,1] (9)

or alternatively by the Bochner expectation of the corresponding support function of X (see [10])

s~x(~,u) = ~-sx(~,u), uES "-1, ~E[0, 1]. (10)

But, the theorem below shows that both, (9) and (10), coincide with the expectation (7).

Theorem 1. Let X be a fuzzy random variable. Then the only solution of the Eq. (7) /s oiven by the expectation of Bochner (10).

Proof. The infimum in (7)

inf Ed~(X,B) = inf En (sx(o~,u)- ss(e,u))2~(du)da

86 R. K6rner/Fuzzy Sets and Systems 92 (1997) 83-93

is reached if for each u E S "-~ and a E [0, 1] the term E(sx(a,u) - sB(~,u)) 2 is minimized. But, the only solution is given by

sB(~,u) = Esx(~,u), u~ S "-~, a~[0,1]

by the use of the results of real-valued random variables. []

Many properties of the expectation and the variance of a fuzzy random variable are given directly by use of (7)-(10). Some interesting properties of the expectation and of the variance of fuzzy random variables are put together in the following theorem without proofs.

Theorem 2. Let X, Y be fuzzy random variables, x a positive square integrable random variable, A E ~c and 2,7E R.

(i) E(2X + ~,Y) = 2EX + ~,IFY, (ii) Var(X) = ~llXll~ -II~[[~,

(iii) Var(2X) = 22Var(X), (iv) Var(xA) = IIAII " Var(x), (v) Var(A + X ) = Var(X),

(vi) Var(xX) = I/=[[XII ~ • Var(x)+ Fx 2- Vat(X) i f x and X are stochastically independent. (vii) Var(X + Y) = Vat(X) + Var(Y) i f X and Y are independent fuzzy random variables.

Furthermore, for the L2-metric d2 we have the following property:

Proposition 1. For any A, B E ~'c it is

d,2(A,B) = d~(,4o, So) + d,2(aA, aB),

where aA resp. aB is the Steiner point o f A resp. B and Ao = A - a~ and Bo = B - as.

Proof. By use of sA(=,u) = SAo(a,u)+ a~u for each uES "-1 and :cE[0, 1], it follows:

d~(A,B) =d~(Ao, B o ) + ( a A - - a B ) T ( n f s , _ u u T g ( d u ) ) ( a a - - a B )

+2n(aA--aB)Tfolfs,_,(S,~o(CC, u)--SBo(~,u))ulz(du)d ~.

Since n fs.-~ uurl~(du) is the n-dimensional identity matrix the second term is reduced to (aa--O'B)T(o'a--aB) =

d~(aa, aB) and since f~ fsn_j u. sAo(a,u)#(du)dec is the Steiner point of the centered set (5) aA0 = aa-~A = 0 (for B analogous) the third term vanishes. []

The above proposition shows that the expectation and the variance can be splitted up into two parts, in a part of location and a part of shape (vagueness).

Theorem 3. Let Xo = X - ax and ax be the Steiner point o f X then

EX = EXo + Ear,

Vat(X) = Vat(X0) + Var(ax). (11)

Proof. The first line of (11) is given by the linearity of the expectation (see (i) of Theorem 3)) and the second by the definition of the variance (8) and the Proposition 1. []

R. K6rner l Fuzzy Sets and Systems 92 (1997) 83-93 87

Moreover, a covariance between two fuzzy random variables X and Y can be introduced by the scalar product of support functions (inner product in the Hilbert space of square integrable functions)

Coy(X, Y) = ~: ( s x - s ~ x , s y - s ~ y )

and the correlation between X and Y by

Cov(X, Y) Cor(X, Y) =

~/Var(X). Var(Y) '

where

/o'Is (SA,SB) = (A,B) = n . sA(~,u)" sB(e ,u)#(du)d~, A, B E :~c. (12) n - I

In the space of normal compact convex fuzzy sets ~ c the operation (12) fulfills properties of a scalar product except the multiplication with negative scalars, i.e., in general, is ().A,B) ¢ ).(A,B) for 2 < 0; A , B E ~ c .

By a straightforward proof the following theorem is valid.

Theorem 4. Let X, Y be two f u z z y random variables. Then (i) Cov(X, Y) = Cov(X0, Y0) + Cov(ax,~ry),

(ii) Var(X) = Cov(X ,X) , (iii) Coy(X, Y ) = 0 i f X and Y are independent, (iv) ICor(X, Y)I ~< 1, (v) Cov(X, Y) = F (X, Y) - (EX, ~:Y).

In the next section we will calculate the expectation, the variance and the covariance of the so-called LR-fuzzy sets of the real line lt~.

4. Example of application: LR-fuzzy variables

Let L,R be fixed left-continuous and non-increasing functions L,R : [0, 1] ~ [0, 1] with R(0) = L(0) = 1 and R(1 ) = L(1 ) = 0. Then, an LR-fuzzy set ALR = (m,s, l, r)LR is a fuzzy set with the membership function I Oi'm x'

(x-s-m)

i f x < m - s - l ,

i f m - s - l < ~ x < m - s ,

i f m - s < ~ x < r e + s ,

i f m + s < ~ x < m + s + r ,

i f x > m + s + r ,

xER,

where mER, s,l,r>~O.

A LR-fuzzy random variable is defined by XLR = (m,s,l,r)LR, for a square integrable random variable m and three positive square integrable random variables s, l, r.

Clearly, the LR-fuzzy set XLR can be represented by XLR = m + s ' A I +r.AR-- l .At . , where mA,(x) = l[_l,ll(X) and AR resp. AL is the fuzzy set with the membership function R-110,1] resp. L. l[o,l]. The linearity of the expectation leads to

EXtR = IFm + At • ~-s + AR • F_r - AL • IEl.

88 IL K6rnerlFuzzy Sets and Systems 92 (1997) 83-93

Again the expectation n:XzR has a set representation of an LR-fuzzy set: ~_XLn = (E_rn, ~.s,g-l, g-r)~8. If m , s , r , 1

are independent then from Theorem 2 follows

Var(XzR) = Var(m) + Var(s) + IIARII2~, War(r) + IIALII2 2. Var(/),

1 where IIALII = ~ 2 d~ and 11.4R][ 2 = ½ fol(R(-')(ct)) 2 d~. Since the Steiner point of the fuzzy set XLR is given by

/o ) ~x = m + ~ r R ~ - l ) ( ~ ) d ~ - I L(-1)(~)d~ = m + r . aAR -- l . aAL (13)

the expectation of the Steiner point is

~-ax = Em + aAR " ~_r -- trA,• El.

As special case the triangular fuzzy number is received by L ( x ) = R ( x ) = 1 - x and P ( s = O ) = 1. The variance is simply Var(XLR) = Var(m) + ~Var(r) + ~Var(l).

Kruse and Meyer [5] used fuzzy sets which violates the demand of compactness of the support (1),

mx = g(a,b) with g(a'b)(x) = exp - , x E N ,

where a, b are real-valued random variables and P(b > 0) = 1. But, the expectation as well as the variance of such type of fuzzy random variables exists:

mEx = gtEa,~) and Var(X) = Var(a) + Vat(b),

if n=la[ 2, n=lb[ 2 < oo (independence is not necessary). For the existence of expectation and of variance we only have to assume that H=IIXI[ 2 < ~ (the compactness of the support is not necessary). Therefore, the class of LR-fuzzy random variables can be extended to fuzzy random variables of the form XLR = m + r" AR -- l • AL, whereby the membership functions of the fuzzy sets AL,AR are in the class of left-continuous and

non-increasing functions f : [ 0 , ~ ) ~ [0, 1] with f ( 0 ) = 1 and f2(f<-l)(at))2d~ < 00, where fC-O = inf{y E R : f ( y ) >~x} is the pseudo-inverse of f .

Of course, the expectation (10) may exist although the variance does not exist as well as in the real-valued case.

Now, we will describe the covariance be tween two L R - f u z z y random variables. IfA = (mA,sA,/A,rA)LR and B = (mB,sB, la, rB)LR are two LR-fuzzy sets, then

(A ,B) = mAmB + SASB + IA/BIIALI[2 2 + rArBIIAR[I~

+(salB - malB + sBIA -- m~la)IIALI[I + (sAm + mArn + snrA + mnrA)llAnll~

and the covariance of two LR-fuzzy random variables X, Y is given by

Coy(X, Y) --- C o v ( m x , m r ) + C o v ( s x , s r ) + Cov(lx, Iv). I[AL[I 2 + C o v ( r x , r r ) . IIARII

+ ( C o v ( s x , li") + Cov(sr, I x ) - C o v ( m x , l y ) - Cov(mr,/x))l]Adll

+ (Cov(sx, r r ) + Cov(sy, rx ) - Cov(mx, rv ) - Cov(m r, r x ))I[A,~ Ill,

R. K6rnerl Fuzzy Sets and Systems 92 (1997) 83-93 89

where t[At~lll = ½ f01 Z(-l)(~)d~ and [IARI[1 = ½ f01 g(-~(~)d~. The form is more convenient under additional assumptions:

(i) if s = 0 (fuzzy number) then

Coy(X, Y) = Cov(mx, m y ) + Cov(lx, Iv )" IIAL Ilg + Cov(rx, rr )- lIAR II g - (Cov(mx, l r ) + Cov(m r, /x ))ll A/; 111 - - (Co¥(mx, rr ) + Cov(m r, rx ))[IAR I11,

(ii) if L = R, Ix = rx, Iv = rr (symmetric fuzzy set) then

Cov(X, Y) = Cov(mx, m r ) + Cov(sx, sr ) + Cov(lx, Ir ). IIA, I1~ + 2 . (Cov(sx, l r ) + Coy(st, Ix))IIAL[I~,

- 2. (Cov(mx, I t ) + Coy(mr, Ix))l[Al_ II~,

(iii) if L = R, Ix = rx, Iv = rr ,s = 0 (symmetric fuzzy number) then

Coy(X, Y) = Cov(mx, my) + Cov(Ix, l r )" [IALI[29 -- 2- (Cov(mx,/r) + Coy(mr, lx))llALIl~.

Now we will discuss a simple linear model.

5. Example of application: A linear regression model

Consider the linear regression problem

X ( z ) = f ( z ) r O + ¢(Z), z E D C R d,

where q~ is a fuzzy random variable, f : R a ~ W is a fixed function and 0 is an r-dimensional vector of parameters. With the assumptions,

~_a~(z) = 0, ~:4i(z) = B and Var4i(z) = a~(z),

we want to estimate the parameters ~ and B by a best linear unbiased estimator based on the observation of the process X at n design points z~ . . . . . zn. Denote

X = ( X ( z l ) . . . . . X(z.)) T, q,=(q,(zj) . . . . . q,(z.)) r and

Then it is

X = Ftg + ~ with EX = Ftg + B, B = (B . . . . . B) T.

Since the covariance

CovX = ( (Cov(X(z i ) ,X ( z~) ) ) )~ j= j

and the expectation (14) can be split up into the two terms

C o v X = C o v X 0 + C o v a x = Z 0 + Z o and ~ _ X = F O + B ,

we can split the model (14) into the two models

ox = FO + o ~ a n d X o = ~ o ,

F = ( f ( z l ) . . . . , f ( z n ) ) T.

(14)

90 R. KrrnerlFuzzy Sets and Systems 92 (1997) 83-93

where trx is the vector of Steiner points ax(zi) of X(zi) and Xo(zi) is the vector of the centered fuzzy set X ( z i ) - ax(zi) (a# and ~0 analogous). Moreover, the best linear unbiased estimator of v9 is given by the classical BLUE-problem (in the regular case):

= (FTS~1F)- IFTZ~Iax .

A linear estimator of B is n n

/~ = #rXo with EB = E p i ~ X o ( z i ) = E ( / a i B ) . i = 1 i-----I

If B is a symmetric fuzzy set then B is unbiased, if ~ i~l gi = 1. But if B is asymmetric then we have to assume additionally that/t~>~0, i = 1 . . . . . n. The estimator B is 'best' if the variance of/~

Var(B) = ~ IpilzjlCov(sign(giPj)Xo(zi),Xo(gj)) i = 1 j = l

is minimized. The variance of B in general cannot be expressed only by Z0. We have to distinguish three cases.

(i) If the variables Xo(zt ) . . . . . Xo(zn) are independent and identically distributed then the best linear esti- mation is given by/~i = 1/n, i = 1 . . . . . n.

(ii) If the variables Xo(Zl ) . . . . . Xo(zn) are symmetric then the best linear estimation is given by minimizing n of Var(/}) = It~lT~:,l~l with ~i=1 kti = 1, where I/tl = (1~11,..., Iml)L

n (iii) For asymmetric X0(zt) . . . . . Xo(z~) f.s. we have to minimize Var(/~) = pxSap with ~i=1/~i = 1 and under the additional constraint #i ~> 0, i = 1 . . . . . n.

The considered model can be used in cases if the additional uncertainty, the fuzziness, is independent on the design points z.

Let us illustrate the model by a simple example (see [2]). Consider for the linear model

Y(x) = ax + 4~ with EdP = (roB, IB, rB)LR

the observations

Xi Yi -~- (mi , li, ri)LR

21.0 (4.0, 0.6, 0.8)LR

15.0 (3.0, 0.3, 0.3)LR

15.0 (3.5, 0.35, 0.35)LR

9.0 (2.0, 0.4, 0.4)L R

12.0 (3.0, 0.3, 0.45)LR

18.0 (3.5, 0.53, 0.7)LR

6.0 (2.5, 0.25, 0.38)LR

12.0 (2.5, 0.5, 0.5)LR.

Then the splitting of the model and formula (13) leads to the equations,

fig, = axi + 64~, : mi + rirAR -- litYA L = axi + m B + rBtrA R -- IBtrAL + ~i

and to

Y,9=4'°: l i = l , + 6 [ and r , = r z + O ; ,

R. K6rnerl Fuzzy Sets and Systems 92 (1997) 83-93 91

5

4,5

4

3.5

3

2.5

2

1.5

Lineare l:l~ssion: Y=ax+B

s~ s

O-lear"

- ' " 1-1

' ' 'o '2 ' 6 8 1 1 14

I ).-

16 18

Fig. 1. Example of linear regression.

where e,, 6~ and fir are independent random variables with Eei = Ef~ = E6 r = 0, lB + ~ >>- 0 and rs + fir >i 0 a .s .

The assumption of independence of ei, f: and f~ leads to the case (i) above. With the values

1~=0.4037, PB=0.485, f i B = l . 3 9 and 6=0.1193,

the model

Y(x) = 0 .1193x + ( 1 . 3 9 , 0 . 4 0 3 7 , 0 . 4 8 5 ) L S

is obtained. In Fig. 1 the triangles represent the data, the solid line the I-level-cut and the dashed lines the 0-level-cut of the estimated fuzzy line.

Note that these calculations are obtained by the assumption that the left and the right fuzzy parts are equal (L = R). Futhermore, the calculation only depends on Steiner point of L and is independent on the other concrete settings of L.

6. Example of application: A linear prediction problem

The problem of linear prediction is solved by similar reasoning. Suppose there are given n observations X(zl ) . . . . . X(z , ) of a weak stationary random process X(z) with EX(z) = B (aB = 0) at the points zl . . . . . z,. Our purpose is to study the unbiased linear predictor of the form

J'l

X(zo) = ~_, 2iX(zi), i=1

(15)

92 R. KfrnerlFuzzy Sets and Systems 92 (1997) 83-93

which minimizes the predictor variance

Var(X(z0)) = Z C o v ( 2 i X ( z i ) , 2 j X ( z j ) ) . i=1 j = l

Since the expectation of (15) results to

EX(zo) = Z 2i EX(zi) = 2iB i=1 i=1

it is necessary to assume that ~i~1 2i = 1 and 2i~>0 for ~-X(zo)= B. The problem is now simplified by

rain 2TI;2 21>0

and the solution is given by the classical linear predictor solution

~. = _r-~, + (1 - UTr~-~U)~-iU/(Ur~-tU),

where u = (I . . . . . 1) T, ~ = (rx(zl) . . . . . rx(zn)) T and 2; is the covariance matrix with the entries rx(zi - z j ) ; i , j = 1 . . . . . n of the covariance function

rx(z) = Cov(X(O),X(z)).

7. Laws of large numbers

A first limit theorem for fuzzy random variables was given by Puri and Ralescu (1986). Lyashenko [6] proved a strong law of large number (SLLN) and a central limit theorem (CLT) for sums of compact convex random sets, based on the three-series theorem of Kolmogorov, by use of an L2-based variance. Other proofs of limit theorems were given by help of results from Banach-space-valued random variables. Our definition of variance enables us to prove WLLN and SLLN by a direct application of classical methods (e.g. used in [81).

Let X be a fuzzy random variable with n:llX[l~ < oo. The WLLN can be proved by the inequality of Tchebyshev for fuzzy random variables

Var(X) P(d2(X, EX)>>.~)<~ ~2 E > O.

Theorem 5. Let {X/} be a series o f independent identically distributed fuzzy random varhTbles with El[X11122 < oo. Then by the inequality o f Tchebyshev we obtain

limp d2 1 X/,I:X ~>¢ =0 e>O. n--co0 k n i-----]

Now, let Xt . . . . . X. be independent fuzzy random variables with EI[Xk[[ ~ < oo. For the proof of a SLLN we use the inequality of Kolmogorov for fuzzy random variables

P max d2 Xj, Z IFXj i> e ~< e2 e > O. l<~k<~n \ j = l j = l ,]

P,. KrrnerlFuzzy Sets and Systems 92 (1997) 83-93 93

Theorem 6. Let X1,X2 . . . . . Xn .... be a series of independent fu:zy random variables, with E[IX/[122 < zxD. I f the series ~-]f~=l Var(Xk )/k2 converges, then

P Xk, EXk = 0 = 1 i=l

Acknowledgements

The author thanks W. N~ither for constructive discussions and helpful comments.

Reference

[1] Z. Artstein and R.A. Vitale, A strong law of large numbers for random compact sets, An~ Probab. 5 (1975) 879-882. [2] Ph. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets (World Scientific, Singapore, New Jersey, London, Hong Kong, 1994). [3] M. Fr~chet, Les ~i~ments al~atoires de natures quelconque darts un espaee distanci~, AnrL Inst. H. Poincar~ 10 (1948) 215-310. [4] R. Krmer, A variance of compact convex random sets, Statistics (1995) submitted. [5] R. Kruse and K.D. Meyer, Statistics with Vague Data (Reidel, Dordrecht, Boston, 1987). [6] N.N. Lyashenko, Limit theorems for sums of independent compact random subsets of euclidean space, J. Soviet. Math. 20 (1982)

2187-2196. [7] W. Nather, Linear statistical inference with random fuzzy data, Statistics (1995) submitted. [8] A. R~nyi, Probability Theory (Akadoniai Kiadr, Budapest, 1970). [9] R. Schneider, Convex Bodies, the Brunn-Minkowski Theory (Cambridge Univ. Press, Cambridge, 1993).

[10] R.A. Vitale, An alternate formulation of mean value for random geometric figures, J. Microscopy 151 (1988) 197-204. [11] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353. [12] H. Ziezold, On expected figures in the plane, in: A. Hfibler, W. Nagel, B.D. Ripley and G. Warner, eds., Geobild '89, Math. Res.,

Vol. 51 (Akademie-Vedag, Berlin, 1989) 105-110.