Independent fuzzy random variables and their application

FUI Y sets and systems ELSEVIER

Fuzzy Sets and Systems 82 (1996) 335-350

Independent fuzzy random variables and their application Chao-Ming

Hwang*,

Jing-Shing Yao

Department of Applied Mathematics, Chinese Culture University, Hwakang, Yangminshan, Taipei, Taiwan,ROC Received November 1994; revised June 1995

Abstract In this paper, we define the probability measure P()~I (wl) ..... )~,(w.)) of fuzzy sets )fl (wl) ..... )~.(w.) and induce some properties of it. We also compare our probability measure with that of Zadeh (1968). Moreover, we have different results about the independence which derive from these two probability measures. Under our probability measure, we also define MLE to estimate unknown parameter. We also provide two examples to demonstrate how to use MLE to estimate unknown parameter,

Keywords: Random variable; Fuzzy random variable; Independence of fuzzy random variable; Maximum likelihood estimate

1. Introduction K w a k e r n a a k [2], Pari and Ralescu [3], and Wang and Zhang [4] provided the concept of a fuzzy r.v. In this paper, we apply the concept of the probability measure of the r.v. X with p.d.f, f ( x ) and the probability measure of r.v.'s X 1 , . . . , X , with joint p.d.f, g ( x l , . . . , x,) to extend to the probability measure of fuzzy r.v.'s )~1 . . . . . )~, (as defined in Definition 2.2) and have some results of the union operation v and bounded sum operation @ [7] of the measure. Moreover, we discuss the concept of the independence of fuzzy r.v.'s X~ .... , X,. Suppose that there are n independent r.v.'s X1 .... , )~,, we can induce two cases of independent fuzzy sets, one defined by Definition 2.4 and the other derived by Zadeh [6], respectively, and find that the probability measure of the first case is greater than or equal to the probability measure of the second case. In Section 3, if the p.d.f, of r.v. X is f ( x l 0), where 0 is an unknown parameter, we introduce a method to find the estimation of 0 by random sample of size n with one vague data associated with p.d.f, f ( x I O) and provide two examples, one for the normal, r.v. X ~ N(0, 1) and the other r.v. X ~ b(1, 0), to illustrate the method of the estimation of 0. It is the intention of our study to apply the results to the data communication system, the production line system, and so on. For example, we draw n data every 3 seconds to select one from the production line system

*Corresponding author 0165-0114/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved SSDI 0165-01 14(95)00234-0

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336

and the ith section of the selecting process is interrupted for a m o m e n t ; hence the ith data is vague. Therefore, we need to deal with r a n d o m sample of size n with one vague data and have to m a k e some statistical decisions a b o u t them.

2. Probability of fuzzy random variables and some properties Let X : (2 ~ R, be a r.v. with p.d.f, f(x), where (f2, B ' P ) is a probability space and B* be a Borel field of R. F o r A e B*,

p(X A) = fAf(x)dx = f

(2.1)

~ Ca(x)f(x)dx ,

where CA is a characteristic function of A. Let Xj: f2 ~ R, j = 1, 2, ..., n, be r.v.'s and g ( x l , . . . , x , ) be the joint p.d.f, of r.v.'s X~ .... , X,. If CAj is a characteristic function of Aj, j = 1, 2 , . . . , n, then

CA1 (X1)

A

CA2(X2) /k "'" /k CA.(Xn) ~. CA1 (X1)'CA:(X2)"" CA.(Xn) "~XI, x z . . . . .

x , e R.

Hence,

f~-~ "'" f~-oc CA1(X1) A "'" A CA"(xn)g(xI''"'xn)dxI ""dXn

= f~oo'" fToo C&(Xl) "" CA"(xn)g(xl'""xn)dXl "'dxn" F o r Aj e B*, j = i, 2 . . . . . n, P ( X l e A1 . . . . .

A.) = fa 1x --. xAn f g(x, . . . . .

x,)dxl ...dx,

C a , ( X l ) ^ "" ^ C A . ( X . ) O ( X l , . . . , x . ) d x a

CAI (X 1 ) " " CAn(X,)O(XI,..., Xn) d X l

... dx., or

... d x , .

(2.2)

(2.2*)

If r.v.'s X1 . . . . . X , are independent then

P(X1 G A 1 , . . . , X n ~ An) = f i P ( X i e A~)

(2.3)

j=l

for any A i e B*, j = 1, 2 .... , n. We try to extend the results of (2.1), (2.2), (2.2*) and (2.3) to fuzzy p h e n o m e n o n by extension principle. In this paper, it suffices to consider the continuous case only, since the discrete case is treated entirely similarly with integrals replaced by s u m m a t i o n signs.

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337

Let X:f2 ~ R be a r.v. with p.d.f, f(x), where (t2, B'P) is a probability space and B* be a Borel field of R. We, as well as, Pari and Ralescu [3], Wang and Zhang [4], introduced the concept of a fuzzy r.v. as a function .~:t2 -.+ F(R) and F(R) denotes all piecewise continuous functions (fuzzy subsets of R) )7(w):R ~ [0, 1] (subject to certain measurable conditions) or all discrete functions (fuzzy subsets of R) 3~(w):R --* [0, 1] (subject to certain measurable conditions). For a fuzzy r.v. J( and w e f2, let :~(w) be a fuzzy set with the membership function .X(w)(x). By extension principle, (2.1) and (2.2), we have the following definitions.

Definition 2.1. For a fuzzy r.v. J~ and w e f2, ig(J~(w)) is called the probability measure of fuzzy set defined as /~(,g(w)) = f_~

,~(w)(x)f(x)dx,

X(w), (2.4)

(this is the extension of(2.1)) where X(w)(x) is a membership function of fuzzy set )7(w). By the definition of a fuzzy r.v. X, the membership function X(w)(x) is a discrete function or continuous function, Hence, Y((w)(x) is a measurable function and )~ (w)(X) is a classical r.v. The expectation of )~ (w)(X) is S ~ ,1~(w)(x)f(x)dx. Therefore, by (2.4), we have EX(w)(X) =/~0~(w)).

Definition 2.2. For any wjef2 and fuzzy r.v.'s ,g~, j = 1,2z...,n, P(£1(WI) . . . . . £n(Wn)) is called the probability measure of the intersection of fuzzy sets -gl (wl) ..... X.(w.) (the same as/3 (,~1 (wl) A "" A -g.(W.))), defined as

f;

P(£~(w,) ....,£.(w.)) . . . .

£,(w~)(x,) ^ £2(w~)(x~) CO

--c~

^ .., ^ £ , ( w , ) ( x , ) ' g ( x l , . . . ,

x,)dXl ... dx,

(2.5)

(this is the extension of(2.2)) and by the sense in Zadeh [6],/~ (I]~.= 1 .~(wj)) is called the probability measure of the product of fuzzy sets -~1 (w:) ..... :~,(w,), which is defined as .... , x , ) d x l

...dx,

(2.6)

(this is the extension of (2.2*)) where ,~i(wj)(xj) is a membership function of fuzzy set ,gj(wj) for j = 1, 2 .... , n. Obviously, in fuzzy theory, we have £ 1 (w,)(x~) ^ £2(w2)(x2) ^ ... ^ 2.(w.)(x.) >>.£1 (w~)(Xl). 22(w2)(x~)... £ . ( w . ) ( x . ) .

Thus,

J-

( ~ Xl(wl)(xOA "" ^X,(w.)(x,)g(x., .... x,)dx:...dx. ~ f~o "" i~ Xl(Wl)(X,) "" -~,(w,)(x.)9(xl,...,x.)dx: ... dx,.

Hence, by (2.5) and (2.6), we have

,ff(.Xl (w: ), ... , ,Y.(w.)) >~F ( fi ,~,(w,) ). j=l

(2.7)

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338

Hence, the two probability measures are equal in the classical sense (see (2.2) and (2.2*)), but /~(Xl(w~), ..., )?,(w,))>~/~ (jO~)~j(wj)) in vague sense (from (2.7)). (Only under the condition which there would be at most one fuzzy set and the rest the crisp sets among fuzzy sets X1 (w,), ..., 3~. (w,), P()~I (wl), ..., X.(w.)) = P(I]~=I -~i(wj))) • Note: Obviously, 0 ~
CA(x)=

1, x~A, O, otherwise.

Hence, characteristic function Ca (x) is a special case of membership function and crisp set A is a special case of fuzzy set and/5(A) = P(A).

2.1. Some properties of the probability measure on fuzzy sets X1 (wl),...,)~,(w,) In the following, we consider some properties of the probability measure on fuzzy sets )~1 (wl) . . . . . )~, (w,) defined by (2.5). For w, w' e f2 and )~(w) ^ )~(w') = 0, by (2.4), we have P(.~(w) v ..Y(w'))= f ~

()((w)v

X(w'))(x)f(x)dx.

d - - ~3

If )~(w) ^ )f(w') = 0, then, for each x, we have either Let U and V be defined as follows: u = {xl~(w)(x)

>>.~(w')(x)},

v = {xlg(w)(x)

< .¢(w')(x)}.

X(w)(x)

or

X(w')(x)

or both equal to 0.

It is clear that (i) if x e U then )~ (w')(x) = 0. (ii) if x e V then X(w)(x) = 0 and R = U ~ V, Uc~ V = 0. Hence, we have

f~-o~ff~(w)(x)f (x)dx = fv X (w)(x)f (x)dx and

f~-oo~(w')(x)f(x)dx= fv iY(w')(x)f(x)dx. Thus,

P(X(w) v X(w')) = P(:~(w)) + P(:~(w')).

(2.8)

Let )~j: f2 ~ F(R), j = 1, 2, ..., n, be fuzzy r.v.'s and )~1 (w l) ^ )~1 (w'l) = 0. This implies that there would be 0 in either Xl(wl)(xl) or Xl(w'l)(xl) or both for each xl.

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339

By (2.5), we have

P((271 (wl) v g l (w'l)), g~ (w~) ..... ~?.(w.)) = f ~ m ""f~_oo ( X I ( W 1 ) V 21(W'1))(XI)A g2(W2)(X2) A "'" A g n ( W n ) ( X n ) g ( X I ' ' " ' x n ) d x I

""dxn"

Similarly, we get P ( ( g l (wl) v gl(w'l)), g2(w2) . . . . . g.(w.)) = P ( 2 1 ( w l ) , g2(w2), •.., -~.(w,)) + P(:~I (Wl), ' Xz(w2),.. ~ ., )~.(w.)).

(2.9)

Proposition 2.1. L e t X j : f2 ~ F(R), j = 1, 2 . . . . . n, be f u z z y r.v.'s and Xi(wi) ^ X i ( w i ) = 0; then t

P(gl(W1)

.... , X i - l(Wi - 1), (X.i (wi) v Xi(wi)), ..., Xn(wn))

= P(g,(wl)

.... , gi-~(w,

+ P (~ X l~( w l ) ,

.

1), g ~ ( w , ) . . . . , ~ . ( w . ) )

. . , X , - x ( w , - 1 ) , X~i ( w , ) , '

...,

(2.10)

X,(w,)).

We consider some operations on fuzzy sets. By Zimmermann [7], we have the following definitions.

Definition 2.3. (1) The algebraic sum C = A + / ~ is defined as C = {(x,/~,i + ~(x))lx ~ R}, where ~a + ~(x) = ~,~(x) + #~(x) - ~,~(x). ~ ( x ) . (2) The bounded sum C = A @ / ~ is defined as C = { ( x , # a . ~ ( x ) ) [ x e R } , where # a e ~ ( x ) = min {1,/~,7(x) + #~(x)}. (3) The bounded difference C = A Q/~ is defined as C = { ( x , # 7 ~ e ~ ( x ) ) l x e R } , where #,~et~(x) = min {0, #a(x) + #/~(x) - 1}. (4) The algebraic product C = A./~ is defined as C = { (x, # a ~(x))] x ~ R}, where/~,i. O(x) =/~,i(x)'#~(x). If ~71 (w i ) ^ 371 (W'l) = 0, there would be 0 in either 3~ i (w i )(x 1) or X1 (w 1) (x 1) o r both for each x i and it is from Definition 2.3 that ()~1(Wl) @ Xl(W'l))(xl) = min {1, 3~l(Wl)(Xl) + -~l (W'l)(Xl)}, = )~1 (Wl)(Xl) + )71 (W'l)(Xl). Thus, we have p- ( ( X l-( W l )

(~ X~l ( W l ) ,) , X 2 ( w 2 ) , . . . ,

= ffo~ "" f ~

Xn(wn))

(gl(wi)q~,~l(w'l))(xl)

^ g ~ ( w 2 ) ( x 2 ) /x "" A X . ( w . ) ( x , ) 9 ( x l

..... x.)dxl

... d x .

= P ( X l (wl),)~2(w2), ..., X,(w,)) + P ( X 1 (wl), X2(w2) . . . . . 3~.(w,)).

Similarly, we get.

Proposition 2.2. L e t X ~ " f2 --+ F ( R ), j = 1, 2 . . . . . n, be f u z z y r.v.'s and X, i ( w i ) ^ X i ( w i ) = 0; then ~

P~( X~i ( w l ) ,

.. . , g i - i ( w i - l ) , ( g i ( w l ) ~

: P(gl(W1),

..., X i - l ( W i - 1 ) , X i ( w i )

X~~ ( w ~ ') ) . . . . .

r

g.(w.))

. . . . . .~,(W,)) + P(Xx(wl)~~

. . . . . )(/-a(w/-a), Xi(wi),- ' ..., X,(w.)).

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For )~1 (w~1)) ^ £1(w~ 2)) = 0 and X~2 ( "w 2(~h, ^ ) ~ (w~2)) = 0, it implies that

, (1)-i, + £ , (w~~)) x--1 (wl(1)~, (~ £ 1 (w~2)) = x"1 ~wl and

-- (i)'~1 @ X2 -- ~w2 , (2)~ X2(w2 , = .~2(W(1)) =~_x~2(W(22)). Thus, we have

P((£I (w~') ~ (£I (w~2))), (£~(w', I>) (D (£~(w(~2))).... , £ . ( w . ) ) = f~O ...

f__oO(.z~I(W1 /

~ , (1)~

o~ ^ "'" ^ X . ( w . ) ( x . ) O ( X l ,

..., x,)dxl

... d x . .

Let Uj and V j , j = 1, 2, be defined as follows: Uj = { x j l S j ( w ~ l ) ) ( x j )

~ £j(w~2))(xj)},

Vj = ( x j l X j ( w ~ I ' ) ( X j )

<

Xj(W~2')(Xj))

and R = U j u Vj, U j n Vj = 0, j = I, 2. We have the following results: (i) If (xl, x2)e U1 x U2, then -Y1 (w~Z))(xl) -- O, X~2 ( w (2) 2 ) (X 2 ) = O. (ii) If (xl, x2)e U1 x 1/2, then )~1 (w~e))(xl) = O, X 2 ( w ~ l ) ) ( x z ) = O. ~ (i) .~ (2)~ (iii) If(xx, x2)e Vx x U2, then XI(W 1 )(X1) = 0, 2(W2 , (X2) = O. ~ (1) (iv) If (X1, X2) ~ V 1 x V2, then X1 (W1 )(Xl) ~-- O, xY2(W(21))(.X'2) ~- O. It is clear from (i) that we have

f ?co "" f aoc¢(£1 (W~I))(Xl) A X2(W(21))(-~2) A ~r3(W3)(X3)A ... A £ n ( W n ) ( X n ) ) g ( X 1 , " ' , x n ) d x l

... d.~n

=

1 xUzxRx ... xR A "'" A ,'Yn(Wn)(Xn))g(Xl, . . . , x n ) d X l ... dxl.

Similarly, we have

(X~w2 , • 2

25 (w~))),

Y:~(w~), ..., £.(w.))

2

= L 2

i=1j=1

" ,W(1)~ -(2)-~ (i)) Proposition 2.3. L e t X~:f2--* F ( R ) , j = 1,2 . . . . . n, be f u z z y r.v.'s and X i ( i , A X i ( w i , = O, X j ( w j ! ^ )~j(w~2)) = O, i < j, then

~ P- -( X l ( W l ) , . . . , ( X i ( w 2 =

(1) i ) ( ~ X i-( w i(2) )), .... (£j(w~l,) ~ X j -.w(2),~ ( j i,,...,2n(Wn))

2

~ pO~l(Wl),... , X~i ( w (P) i ) . . . . . X~j ( w j (q)) . . . . . gn(Wn))" p=l q=l

By mathematical induction, we have the following proposition.

(2.11)

341

C-M. Hwang, J.-S. Yao / Fuzzy Sets and @stems 82 (1996) 335-350

Proposition 2.4. Let Xj:f2 --. F(R), j = 1, 2 .... , n, be fuzzy r.v.'s and Xi(wl 1)) ^ Xi(wl 2)) = O, i = 1,2 . . . . . k, 2 <.k <.n. Then

- - (w~') + 2~ (~2~)), (2~ (~<~'>)+ 2~ ("w(~ P((X, 2 ,, . ...,(~?~(w~') . . . • g~(w~))),

2 2 i~=1i2=1

, ~(w.))

2 (2.12) it~=l

2.2. Some properties o f the independence of fuzzy r.v. 's

In the following, we consider some properties of the independence of fuzzy r.v.'s . ~ .... , X, with the probability measure/~(Xt (wl),..., X,(w~)) defined in Definition 2.2. By extension principle and (2.3), we have the following definition. Definition 2.4. Let "('i: f2 ~ F (R), j = 1, 2 ..... n, be fuzzy r.v.'s. (1) For any fixed wj ~ f2, j = 1, 2 , . . . , n, if

P(g,(w~),...,

g, ( w , ) ) = I~I

P(g~(wj)),

(2.13)

j=l

then we say that fuzzy sets .gl(W~) .... , .~,(w,) are independent. (2) For any wj e f2, j = 1, 2 .... , n, if

P(g~ (w,),..., ~,(w,)) = I-I P(gj(wj)),

(2.14)

j=l

then we say that fuzzy r.v.'s , ~ ..... )~, are independent. Employing Zadeh's idea, we have another way to define independence of a fuzzy sets X1 (wl), ..., X, (w,). The fuzzy sets X1 (wl) .... , )~,(w,) are independent if and only if P

j=l

Sj(wj)

=

j=l

P(Xj(wj)) =

j=l

Xj(wj)(xi)f(xj)dxj

if #(xl,..., x,) = I-I f(xj)j=l

To define independence of fuzzy r.v.'s X1 .... , .~,, we say that fuzzy r.v.'s . ~ .... , X, are independent if and only if P

(fi) Xj(wj) j=l

=

fi

P(gj(wj) )

j=l

( = fi f oo X j ( w j ) ( x j ) f ( x j ) d x i ifg(xl . . . . . j=l oo

x,) =

jO1f ( x i ) ) "=

for any wj e f2. The following theorem shows some properties of n independent fuzzy r.v.'s .~1 .... , .~, which is defined in Definition 2.4.

Theorem 2.1. Let X ~: f2 ~ F (R), j = 1, 2 . . . . . n, be fuzzy r.v.' s and X j(wj)(xj) be the membership function of fuzzy set Xj(wj) for j = 1, 2 , . . . , n. The following statements are equivalent. (1) Fuzzy r.v.'s X I . . . . . ,Y, are independent. (2) O(Xl .... , x,) = I-I~= 1 f (xj) and for each w~ ~ f2, j = 1, 2 . . . . . n,

gl(wl)(Xl) ^ g2(w~)(x2) ^ ... ^ g.(w.)(x.) = gl(Wl)(x~)" g2(w2)(x2) ... g'.(w.)(x.) for V(x~ . . . . . x~) e R".

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342 II

(3) g(xl ..... x,) = I J j = l f ( x j ) and there would be at most one membership function and the rest are characteristic functions amon9 X1 (Wl) (xl), ... , X , (w, (x,) for each wj ~ I2, j = 1, 2 , . . . , n. (4) For each wl ~ I2, j = 1, 2 . . . . . n, fuzzy sets .I~1(Wl) . . . . . X , ( w , ) are independent. Proof. By Definitions 2.2 and 2.4, we get that (1)¢,.(2). If X l ( w ) ( x x ) / x X 2 ( w ) ( x 2 ) ^ . . . / x .~.(w)(x.) = X1 (w)(xO" .l~2(w)(x2) ... )~.(w)(x.), there would be at most one membership function and the rest are characteristic functions among )~1 (w)(xl) .... , .~,(w)(x.). Hence (2) ~ (3). It is clear that (3) ~ (2). By (2) of Definition 2.4, it is clear that (4) ~ (1). [] Example 2.1. Suppose we toss an unbiased coin three times independently, we have O = {H, T}. If the first toss comes up H, we win about 50 dollars and if it comes up T, we win about 30 dollars. If the second toss comes up H or T, we win 10 dollars. If the third toss comes up H or T, we win 30 dollars. Moreover, we set 1~ (H)(x) = [-1 - (x - 50) z] +, where

[a] + =

{0

)~1 (T)(x) = [1 - (x - 30) z] +

ifa~>0 if a < 0 ,

and )~2 (w)(x) = CA (X), X3 (w)(x) = CB(X), where CA and Ca are characteristic functions of the classical sets A = {10}, B = {30}, w ~ f2. It is clear that the fuzzy sets ) ~ (H),)~2 (n), 1~3 (H) are independent. Similarly, for any w~, w2, wa ~ t2, the fuzzy sets )~1 (wl), )~2 (w2), X3 (wa) are independent. Hence, by (2) of Definition 2.4, fuzzy r.v.'s X~, X:, Xa are independent. Note: In Example 2.1, )~1 is a fuzzy r.v. and ~'Y2 and -~3 are classical r.v.'s because X l ( w ~ ) ( x l ) is a membership function, and )~2 (w2) (x2), X3 (wa) (x3) are characteristic functions (special case of membership functions). Similar to [6.], we define the conditional probability of fuzzy sets P ( X i , ( w q ) , . . . , . ~ i , . ( w j I Xi.+~ (wi.+,) .... , Xi.+~(wi.~)) as follows.

Definition 2.5. Let Xj: f2 ~ F(R), j = 1, 2 .... , n, be a fuzzy r.v.'s. The conditional probability of fuzzy sets P(-~i~ (wi~) . . . . . .~i.. (wi.~) l Xi.+ , (wi.+ ,) . . . . . X i . +~(wi.,~) ) defined as

P (-L~(w~,),..., -¢i~(w~)l $~.~ (wi.+,),..., ~.+~ (wi.+~)) P(g~(w~) .... , $,.~,(w,.+~))

1 ~
where N = {1, 2 .... , n} and P(Xi.~,(wi.+~), ...,-~i.+~(wi,+~)) > O. In the following discussion, we take the independence of fuzzy r.v.'s )~1, ..., )~, defined in Definition 2.4. Theorem 2.2. Let Xj : ~ ~ F(R), j --- 1, 2 .... , n, be fuzzy r.v.'s. The following statements hold: (1) I f fuzzy r.v.'s X1 . . . . . X , are independent, every, subcollection Xi . . . . . . Xi.+, of X1 . . . . . X , are also independent. (2) Fuzzy r.v.'s X1 .... , .~, are independent if and only if

F(Y,,, (w,,) ..... si~ (w,..)I g~.+ 1(wi.+ 1) ..... .~i.+,(w~. +~)) = P(Xi,(wil) .... , .~im(wi,.)),

where N = { 1, 2 ..... n}.

1 <~ ia < i2 < "" < im+h <~ n, m 6 N, h ~ N,

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343

Proof. (1) Since, fuzzy r.v.'s A'I .... , )~, are independent, it is from Theorem 2.1 that there would be at most one membership function and the rest are characteristic functions among-X1 (w)(xl ), ..:,-~, (w)(x,). Hence, there would be at most one fuzzy set and the rest are classical sets among Xil (wil), ..., Xi.+~(wi.+~). Thus, by Theorem 2.1, fuzzy r.v.'s )~i,, ...,~ 3~i. +h are independent. (2) Since fuzzy r.v.'s X1, ..., X, are independent, then from Theorem 2.1, Definition 2.5 and (1) of Theorem 2.2, we get P ( X i l (wil), ... , Xim(Wim)[ Xi,+t (Wi.+l), ... , Xi=+h(Wi,+h))

P(£~(wil),..., ~.+~(w,.÷~)) = P(g~(w~,), ~(w~.)). = P(£~.~, (w~.~) ..... g~.+~(w~.+~)) ""' Conversely, from Definition 2.5, we get

P(~,,(wi,) .... , g,..Aw,.+,)) = P(gi,(w,,),..., . ~ ( w J ) .

P(g~..,(w~.+),..., ~.+~(w~..~)).

Thus,

P(gl (w,) ..... g,(w.)) = P(g~ (wl))" P(£2(w2) ..... g,(w.)) = P ( ~ (wl))" P($'2 (w2)). P(g3(w~),..., g,(w.))

= ( I P(XAw~)) • j=l

It is clear from Definition 2.4 that fuzzy r.v.'s ) ~ , . . . , .~, are independent.

[]

Proposition 2.5. L e t X ~: f2 -~ F(R), j = 1, 2 . . . . . n be f u z z y r.v.'s. I f f u z z y r.v.'s X1 . . . . . X , are independent, then f u z z y sets

(g~,,.,(w~,,.,) @ g~,.~,(w~,.~) ® -.. ® g~, ..... (w~,..... )),

(gi,...(wi,. .) @ Si,..~,(wi,.D @ "'" @ Si, ..... (wi,..... )) are independent, where 1 <~ kj <~ n,j = 1, 2 .... , m and (i(L ~),..., i(,,,,,~)) is a permutation o f 1, 2 .... , n.

Proof. Since fuzzy r.v.'s Xx,..., )~. are independent, it is from Theorem 2.1 that there would be at most one fuzzy set and the rest are classical sets among Xa(w),..., J~.(w). Hence, there would be at most one membership function and the rest are characteristic functions among

(g~,,,(wi,.,) @ g~,.~,(wi,,,~) @ '.. ® Xi, ..... (wi,,.,,,))(xk~),

(Xil2.1)(Wi(2,1,)1~ Xi[2,2)(W[(2,2))(~ (~,.,,,(w~,.,,) @ ~,..~,(wi,..~,)

"'" @ Xi{ ....

(Wi,..... ))(Xk2)"

@ ".. @ -~i, ..... (wi~..... ))(xk.).

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Hence, by Theorem 2.1,

(~i,,.,,(wi,,.,,) ¢ £~,,.~,(wi,.2,) ¢ "'" ¢ 2~, ..... (w~,,.,,,)), (2,,,.,,(w,,,.,) ¢ £',,,.,,(w,,2 ,,) ¢ ... ¢ 2,,2.,~,(w,,~,2,)), (2,,.,,,(w,,.,,,) ¢ $,,.~,(w,,.~,) ¢ ... • 2,,.,.,(w,, ..... )) are independent.

Proposition 2.6.

[] L e t X j : I 2 ~ F ( R ) , j = 1, 2, ..., n, be f u z z y r.v.'s. I f f u z z y r.v.'s -~1 . . . . . .~, are independent,

then

(1) F u z z y sets

($i,,,,,(w,,, ,,)" 2~,, ~,(w~,, ~,) ... 2~,,,,,(w~,,,,,)),

(2~,..,,(w~,..,,)" 2i,. 2,(wi,. ~,) ." 2~,.,.,(wi,..,.,)) are independent, where 1 <~ kj <~ n, j = 1, 2 .... , m and (i(1,1), ..., i(m,km)) is a permutation o f 1, 2 . . . . . n. (2) F u z z y sets

(2~,, ,,(w~,, ,,) + 2~,, ~,(wi,, ~,) + -.. + 2~,,.,,,(w~,.,,,)),

(2~,.,,(w~,..,,) + 2~,..2,(w,,..~,) + ... + 2i, ..... (w~,..... )) are independent, where 1 <<.kj <<.n , j = 1, 2 .... , m and (i(1.~), ..., i(~,km)) is a permutation o f 1, 2 . . . . . n. (3) F u z z y sets

(2i,,.,,(w/,,.,,) O ~i,,.~,(wi,,.~,) G "'" O 2i, ..... (wi,,.,,,)), ($ i(,,l) (Wi(~.l)) 0 $i(2,2)(Wi(2.2)) 0 "'" 0 $ i(:~,,2,(Wi(2,k2)))'

(2'(..,l(Wi(nl.l}) ~ 2[(m,2,(Wi(m.2)) ~ "'" ~ S'[(..... (Wi(m,k.))) are independent, where 1 <~ k i <~ n, j = 1, 2 . . . . . m and (i(1.1), ..., i(,,k,)) is a permutation of 1, 2 . . . . . n.

The proof of Proposition 2.6 is the same as that of Proposition 2.5.

3. Maximum Hkelihood of random sample with one vague data Let Xi be a fuzzy r.v. and X j, j = 1, 2 , . . . , n, j ~ i, be classical r.v.'s. With the above results, we introduce the concept of maximum likelihood estimate.

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C.-M. Hwang, J.-S. Yao / Fuzzy Sets and Systems 82 (1996) 3 3 5 - 3 5 0

Definition 3,1. Let X be a r.v. with p.d.f, f(xlO), where 0 is an unknown parameter and 0 e O c_ R r. (1) For any fixed i, let )~: t2 ~ F(R) be a fuzzy random variable associated with p.d.f, f(x]O). Hence, P(Xi(w)) = S~_o~£i(w)(xl) f (xiIO)dxi. Let X1 .... , X i - 1 , Xi+ 1,..., X , be classical random variables distributed as X with p.d.f, f(xlO). If X1, ..., X~- I, Xi(w), X~+ I, ..., X , are independent, we say that X1, ..., X i - 1 , £i(w), Xi+ 1, ..., X , are random sample of size n with one vague data £~(w) associated with p.d.f, f(x[O). (2) Let X1, ..., X i - 1 , £i(w), Xi+l .... , X, be random sample of size n with one vague data )~(w) associated with p.d.f, f(x[O). L(Olx1,...,Xi_l,Si(w),xi+l,...,xn) is called the likelihood function of X1 = x l , ..., X i - 1 = Xi- l, £i(w), Xi+ l = xi+ l . . . . . X , = x,, defined as follows. (i) If f(x[O) is discrete then L ( O l x l , . . . , X~-l, £,(w), xi+l, ..., x,) -- ~ £ i ( w ) ( x i ) f ( x i l O ) xi

fi

f(x~lO).

j = l,j # i

The definition of the likelihood function y~, £~(w)(x~)f(x~lO) 1-17= 1,j # l f(xslO), which is the probability of X1 = x l .... , X ~_ ~ = x~_ 1, X ~(w), X~+ 1 = x~÷ 1. . . . . £ , = x, is the same as that of the likelihood function of discrete r.v.'s X1 .... , X. in classical statistics. (ii) If f ( x l O ) is continuous then g(O[xx,...,xi-l,£~(w),xi+l,...,x~)=

£i(w)(xdf(xilO)dxi

f(xslO). i = 1,j ¢ i

(3) If L (01 x l .... , x i - 1, £ i (w), xi + 1..... x,) = max0 ~ o L (0 ]xl ..... xi 1, £ i (w), xi + 1 . . . . . x,), the estimate is called a maximum likelihood estimate of 0.

Example 3.1. Let £1, X2,-.., Xn be random sample of size n with one vague data £1 (w) associated with N(0,1), where 0 is an unknown parameter and suppose the membership function £ 1 ( w ) ( x l ) = e x p [ - ½ ( X l - re(w))2], where re(w) is a known real number (only dependent on w). According to Definition 3.1, we have L(O I £ 1 (w), x2 .... , xn)

-

e x p [ - ½ ( X x - m(w))2-1exp[ -½(Xx - 0)2? dXl I~ Qo

=

j=2

~exp

j = 2 ( x s - t)2 e x p [ - l ( 0 - m ( w ) )

~1

exp [ -½ (x s - 0)2]

~

2]exp

-~(0-02

,

n

where t = Ej=2 x j ( n - 1). The likelihood equation is (d/d0) l n L ( O l £ 1 ( w ) , x2 .... , x , ) = 0. Solving the equation for 0, we get the maximum likelihood estimate

O=

2E=2 xj + m(w) 2n-- 1

The distribution of maximum likelihood estimator 2Z'j=2X,+m(w) 2n--i-

(2(n-1)O+m(w)

isN

-2n--T

4(n-l)) ' (-~n-- ~

"

Example 3.2. Consider the experiment of tossing a coin where the probability of a head on an individual toss is 0. Suppose that for each toss that comes up head, it is denoted by H, but for each toss that comes up tail, it

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is denoted by T. Hence, (2 = {H, T}. Let X : O ~ {0, 1} be a r.v. and X ( H ) = 1, X ( T ) = O. Hence, r.v. X has p.d.f, f ( x [ O) = 0~(1 - 0) 1 -~, x = O, 1, and 0 < 0 < 1. Let )~1, X2 .... , X , be r a n d o m sample with one vague data )~1 (w) associated with p.d.f, f(x[O). The probability of )~1 (w), X2 = xz . . . . . X , = x , is /~ (~3~1(W), X 2 =

x 2 ....

,

X . = x,[O) = [)~a(w)(0)(1 - 0) + )~l(w)(1)(0)] f i 0~J(1

-

O)1-xj,

j=2

where xj = 0, 1, and j = 2, 3 . . . . . n, and the likelihood function

L(Op~I(w), x2 . . . .

, x . ) = 2 1 (w)(0)0'(1 - 0)" ' + ~

( w ) ( 1 ) 0 '+ 1(1 - 0) " - 1 - '

(3.1)

M

where t = ~j= 2 xj. (1) If X1 (w)(0) = )~1 (w)(1)(# 0), then the likelihood function L(O !21 (w), x2, ..., x,) = )~1 (w)(0)0'(1 - O)"-t.

Solving the likelihood function (d/d0) L(OJX1 (w), x2 .... , x.) = 0, we get the m a x i m u m likelihood estimate t

0---n-1

n __ ~ j = 2

Xj

n-1

which is the same result as the M L E by the n - 1 stable data without vague data )~1 (w) in crisp statistics. (2) Suppose )~1 (w)(0) # )~1 (w)(1). (2.1) When t = 0, then the likelihood function L(Ol~l(W), x2 .... , Xn ) = )~I(W)(0)( 1 _ O)n + 21(W)(1)0( 1 __ o)n 1

and the likelihood equation d d---OL(OIX1 (w), xz .... , x,) = (1 - 0)"-2 n [ 2 , (w)(0) - 2 , (w)(1)] [0 - B , ] , where B1 ~

nX1(w)(O)-21(w)(1)

n [ 2 1 ( w ) ( 0 ) - - .~l(w)(1)]

(a) If X l ( w ) ( 0 ) > X l ( w ) ( 1 ) , it follows that B1 > 1 and d L(OIX1 (w), x2 .... , x,) < O. d---O Thus, we get 0 = 0. (b) If Xl (w)(0) < X1 (w)(1), then we discuss two cases as follows: (i) If )~1 (w)(0) < -gl (w)(1) ~< n)~l (w)(0), it follows that B1 < 0 and d ~ L ( O I ~ I (w)' x2, ..., xn) < O. Thus we get 0 = O. (ii) If n21(w)(O ) < 21(W)(1), it follows that 0 < B 1 < 1. Thus, we get =

Xx(w)(1) - n21(w)(O ) n L ~ I (w)(1) - ~g~(w)(0)]"

C.-M. Hwang, J.-S. Yao / Fuzzy Sets and Systems 82 (1996) 335-350

347

(2.2) When t = n - 1, then the likelihood function L(0lJ?a (w), x2 . . . . . x,) = J?l(w)(0)0"-l(1 - 0) + )?l (w) (1) 0" and the likelihood equation d L (01)~ x (w), x2, ..., x,) = 0"- 2 n [)~1 (w)(0) - )~1 (w)(1)] [Bz - 0], dO where

in - 1)21 (w)(0) B2 = n [)~1 (w)(0) - 3~1 (w)(1)]" (a) If n)~l (w)(1) < )~1 (w)(0), it follows that 0 < BE < 1. Thus, we get

0=

( n - 1)J?l(w)(0) n [-~1 (w)(0) - -~1 (w)(1)]"

(b) If )~1 (w)(0) < )~1 (w)(1), then we discuss two cases as follows: (i) If )~(w)(1) < )~l(w)(0) ~< nJ~(w)(1), it follows that B2 t> 1. Thus, we get 0 = 1. (ii) If )~1 (w)(0) < ) ~ (w)(1), it follows that B z < O. Thus, we get 0 = 1. (2.3) When 0 < t < n - 1, it is from (3.1) that the likelihood equation d

dO

L(O) = 0 ' - 1(1 - 0)"-*- 2 {n [)~, (w)(0) - )~, (w)(1)] 02

+ [ - ( n + t)J~l(w)(0) + (t + 1))~x(w)(1)] 0 ÷ Xl(w)(0)t} = 0'- 1(1 - 0)"-'- 2 n IX1 (w)(0) - )~1 (w)(1)] [0 - A] [0 - B3],

(3.2)

where A-

in + t)£1 (w)(0) - (t + 1)£1(w)(1) + 2n [)~1 (w)(0) - )~x (w)(1)]

(3.3)

B3 _ (n ÷ t)J~l(W)(0 ) --(t ÷ 1))~1 (W)(1) -- X/~ 2n [)~1 (w)(0) -- )~l(W)(1)]

(3.4)

D = [(n - t) J~l (W)(0) - (n + t))~l (w)(1)] 2 + 4 t ( n - 1 - t ) X ~ (w)(0))~l (w)(1).

(3.5)

and

It is clear that D/> 0. (a) When )~1 (w)(1) > )~l(w)(0), the following statement is true. A < 0 if and only if - (n + t)J~l (w)(0) + (t + 1)Xl(w)(1) < x/-D.

(3.6)

(a.1) If --(n + t)J~l(w)(0) + (t + 1))~l(w)(1) < 0, it implies that A < 0. (a.2) If - ( n + t))~l (w)(0) + (t + 1))~1 (w)(1) > 0, the following statement is true: - ( n + 1))~1 (w)(0) + (t + 1))~l(w)(1) < x / ~ if and only if [ - ( n + t)Xl(w)(0) + (t + 1))~l(W)(1)] 2 < D. After some operations,

C.-M. Hwang,J.-S. Yao / FuzzySets and Systems82 (1996)335-350

348

the following statement is also true: [ - ( n + t ) X l ( w ) ( 0 ) + ( t + 1))~l(W)(1)] 2 < D if and only if O < - 4 n [)~1 (w)(0) - )~i (w)(1)] J~l (w)(0)t. Hence, we get A < 0. By (a.1) and (a.2), we get A < 0. Similarly, we can get 0 < B3 < 1. Thus, we get A < 0 < Ba < 1. Finally, we get

O=B 3 --

--(n + t) Z~"l (W)(0) + (t + 1))~1(W)(1) + v / D 2n [X1 (w)(1) - J~l (w)(0)]

(b) When J~l(w)(1)
0= B3 =

(n + t)g~(w)(O) - ( t + 1)2~(w)(1)-v/-O

2n [)~1 (w)(0) - )~1 (w)(1)]

As a summary, we have the following results. (A) If )~l (w)(0) = J ~ (w)(1) ( # 0), then

0_

1 n

l j= 2

Xj.

(B) I f t = 0 a n d Xl(w)(0) > )~l (w)(1), then 0 = 0. (C) If t = 0 and )~1 (w)(0) < )~L(w)(1) ~ n)~l (w)(0), then 0 = 0. (D) If t = 0 and )~l (w)(1) > nXa (w)(0), then

n [)Tl(w)(1) - )~l(W)(0)]

<

"

(E) I f t = n - 1 and )~l(w)(0) < J~l (w)(1), then 0 = 1. (F) If t = n - 1 and )~l (w)(1) < Xt(w)(0) ~< nJ~l (w)(1), then 0 = 1. (G) If t = n - 1 and -~1 (w)(0) > n X i (w)(1), then (n - 1))~i (w)(0) 0 = n [)~l (w)(0) - l~l (w)(1)] (H) I f 0 < t < n - l ,

(.1) >

.

n

then

0 = (n + t))~i (w)(0)- (t + 1))(1 (w)(1)2n [J~l (w)(0) - Xl(w)(1)]

4. C o n c l u s i o n

(1) In the paper, we defined the probability measure P(J~I

(Wl) .... ,

)~,(w,)) as

P(XI(W1)'"" "~n(Wn))= f oo'" f ~- ~I(W1)(X1) ^ "" ^ X , ( w , ) ( x , ) ' O ( x l , ..., x . ) d x l ... dx..

which is different from the probability measure defined by Zadeh [6] as

f:

xl,W,,Xl)

..... x.)dxl...dx..

C.-M. Hwang, J.-S. Yao / Fuzzy Sets and Systems 82 (1996) 335-350

349

The advantage of our definition of probability measure is that we have (from (2.7))

_P(371(w~)..... 37n(Wn))>~P ( fi 37j(Wj)). j=l

But under our probability measure to define independence, we found that at most one of the fuzzy random variables is vague, and under Zadeh's probability measure, all the fuzzy random variables can be vague. (2) In Example 3.1 the MLE 0 of 0 by random sample of size n with one vague data 37 1(w) from N (0, 1) is the weighted average of m (w) (satisfying 371 (w)(m (w)) = 1) and two times of sum of xa's, forj = 2 to n. That is, 0 = m (w) + 2 ~ = z xi 2n- 1 which is different from n

= Yj=lXj n

which is the M LE of 0 by n size of classical random sample from N (0, 1). By the fuzzy phenomena, we see that the M L E 0, where n ~ = re(w) + 2y.j= 2 xj

2n- 1 is reasonable. (3) In Example 3.2, if 371 (w)(0) = 371 (w)(1), we have

__x~

0 = ~. j=2 n

1

that is different from n

X" = 2 j = l x j n

which is the M L E of 0 by classical random sample size n from b(1, 0). Furthermore, the former is the same as the M L E by the n - 1 stable data without vague data 371 (w) in crisp statistics. Hence, if we consider the MLE of 0 by random sample size n with one vague data 371(w) from b(1, 0) and 371(w)(0) = 371(w)(1), then the fuzzy sample is useless. If x2 = x3 . . . . . x, = 0 and 371(w)(1) < Xl(w)(0) (or 3 7 1 ( W ) ( 0 ) < 37,(w)(1) ~< n,~l(w)(0)), then we get 0 = 0 . This means that P(X = 0 ) = 1 a.s. When x2 = x3 . . . . . xn = 1 and 371 (w)(0) < 371 (w)(1) (or J7 1(w)(1) < 371 (w)(0) ~< n371 (w)(1)), we get 0 = 1. This means that P(X = 1) = 1 a.s. For other conditions, we can explain in the same way and all results are reasonable.

Acknowledgements The authors would like to thank the referees for valuable comments and suggestions.

References [1] S. Khalili,Independentfuzzyevents,J. Math.Anal.Appl.67 (1979) 412-420. [2] H. Kwakernaak,Fuzzy random variables:definitionand theorem, Inform.Sci. 15 (1978) 1-29.

350 [3] [4] [5] [6] [7]

C-M. Hwang, J.-S. Yao / Fuzzy Sets and Systems 82 (1996) 335-350

M.L. Pail and D.A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986) 409-422. G. Wang and Y. Zbang, The theory of fuzzy stochastic processes, Fuzzy Sets and Systems 51 (1992) 161-178. L.A. Zadeh, Probability measures of fuzzy events, J. Math. Anal. Appl. 23 (1968) 421-427. Y. Zhang, Fuzzy random variables, J. Harbin Archit. Civil En O. lnst. 3 (1989) 12-21. H.J. Zimmermann, Fuzzy Set Theory and its Application (Kluwer Academic Publishers, Boston, 2nd edn.).