A comparison between two fuzzy estimators for the mean

Fuzzy Sets and Systems 48 (1992) 341-350 North-Holland

341

A comparison between two fuzzy estimators for the mean Stefan P. Niculescu National Water Research Institute, RRB, P.O. Box 5050, Burlington, Ontario, Canada L 7R 4A6

Reinhard Viertl lnstitut fiir Statistik und Wahrscheinlichkeitstheorie, Technische Universitiit Wien, Wiedner Hauptstrasze 8-10, A-1040 Wien, Austria Received July 1990 Revised November 1990 Abstract: In classical component reliability models with instant replacements and fuzzy data the equivalence between the associated fuzzy sample mean estimator and the fuzzy renewal estimator for the mean is proved via the propagation of fuzziness. Keywords: Fuzzy estimator; renewal theory.

1. Introduction

Let T1, T: . . . . be a sequence of strictly positive i.i.d, random variables (with respect to a given probability space (I2, ~, P)) describing the lifetimes of the items in a classical component reliability model with instant replacements. A natural problem is to evaluate the parameters of the model by using various statistical procedures. Let us assume # = f T, d P • ~+ = (0, oo). This assumption is natural for many practical cases. Generally tt is unknown and its estimation is based on the Strong Law of Large Numbers. For any t • E + let us denote by/V(t) the number of replacements in the system in the time interval (0, t], i.e. fi/(t) = ~ l(o,,l(S/) i=1

where S i = T I + ' " + T , , i • N = { 1 , 2 . . . . }, and for every set A we denote by Iz(') its indicator

function, namely

IA(X) ----

{10 f ° r x • A , for x ~ A.

Under the assumptions on the model the following results are valid: (a) Strong Law of Large Numbers (Kolmogorov [3]): a.s.

= lim S./n.

(1)

(b) Elementary Renewal Theorem (Doob [2]) a.s.

tt = lim t/lfI(t).

(2)

Based on these two results we can consider the following two asymptotic unbiased estimators for the mean g" it, = n-1 ~ W/ i-I

and 57/(0=t

~I(0,,1 ~ W/ , ti=l

\j=l

where for each i • ~, W~ are data for Ti and t is large enough to ensure the existence of all quantities involved. In practice the data can be considered as fuzzy numbers in ~ = ( - ~ , ~), the fuzziness being the result of the imprecision (non-random) of measurements. Starting from Zadeh's extension principle in fuzzy sets theory (see for example Dubois and Prade [1, p. 36] we construct the fuzzy estimators associated with/~n and h~/(t) and we compare their performances via the propagation of the fuzziness. Such a study is natural because imprecision at the level of data is propagated by statistical estimators and produces imprecision at the level of final results in the estimation procedure.

0165-0114/92/$05.00 (~) 1992--Elsevier Science Publishers B.V. All rights reserved

S.P. Niculescu, R. Viertl / Comparison between fuzzy estimators

342

2. The fuzzy sample mean and the renewal estimator for the m e a n

From the beginning let us precise some useful notions and notation. Definition 1. Let (~f, <) be a non-empty partially ordered set. A fuzzy element x in ~f corresponds by definition to a membership function qg~: ~f---~[0, 1], i.e. a function which satisfies the following assumptions: (i) there exists a • ~f such that q~x(a) = 1; (ii) Cpx(V)>I min{q~(u), q~(w)} whatever u, v, w • ~ such that u <~ v < w. For every a • ( 0 , 1 ] the a c u t of x is by definition the set

G.(x) = (u • Let

Moreover if x • ~c(~) and ~e(suppx) < ~ then C.(x) = [CL(x), C2(x)] for every a • (0, 1] and there exist

CLo(x)=

lim CL(x) = inf supp x -"~0

and

C~(x) =

lim .'.ao

C~(x)= sup suppx

which are finite too. Remark 3. Let x, y • 0~k, Xk),

X = (X 1 . . . . .

Y = (Yl . . . .

, Y*).

We say that x<-y iff xi<~yi for all i • (1 . . . . . k}. Thus (Rk, ~<) is partially ordered. Let 0 q : A ~ _ ~ and Z ~ • 3 ; ( A i ) , i • { 1 . . . . ,k}. Following Kruse and Meyer [4],

suppx = (u • ~: q~x(U) > 0 ) be the support of x and denote by ,~(~) the set of all fuzzy elements in ~ . We identify all elements of ~ ( ~ ) having the same membership function.

and its membership function is given by

Remark 1. Let (~f, <) be a given non-empty partially ordered set and x • ~ ( ~ ) .

whatever z = (zl . . . . . z,) • x / k = l Ai. Moreover C , ( Z ) = XL1 C,(Zi) for all a • (0, 11.

Cpz(Z) = min Cpz,(Zi) l~i~k

{C.(x), a • (0, 11) is called o:-cut representation of x. C,(x) ~_ C,(x) ~_ Ca(x ) ~_ suppx for a, fl • (0, 1] with a > ft. Moreover the a-cut representation of x is unique and completely characterizes x. To be more specific, this connection is given by

Cpx(U) = sup alc~(x)(U) whatever u • ~. .e(O,1]

Let us denote by ( the Lebesgue measure on and ~([~) = {x • ~ ( R ) : ~ is continuous). Throughout the paper ~ is considered as being endowed with the classical order '~<' and the usual topology on it. Remark 2. We have R ~ 3 : ( ~ ) .

If x • o ~ ( R ) and ('(supp x ) < oo then C,(x) is a finite interval and there exist CL(x)----infC,(x) and C2(x)= sup C,(x) and are finite whatever a e(0, 1].

1. Let Z i • ~c(~) be such that ~¢(suppZi)
Proposition

(Z1 . . . . . Zk). If f:Rk-->R is a continuous function then there exists a fuzzy element in f(ff~k), denoted by f ( Z ) , such that q~r(z)(X) = sup{q0z(Z): z • ~ and f (z ) = x} whatever x • f ( ~ k ) , where sup 0-- 0. The a-cut representation of f ( Z ) is given by C , ( f ( Z ) ) = f(c,(z)),

a • (0, 1].

ProoL The continuity assumptions are essential

and imply Co,(f(Z))~_f(Co,(Z)) for all a ; • (0, 1]. The other inclusions are trivial. The proof is concluded by Remark 1. Remark 4. Under the assumptions in Proposition 1, for every a e (0, 1] from the continuity of f it follows

C.(f(z))=[

min f ( z ) ,

kzeC~(Z)

max f ( z ) ] .

z~C~(Z)

d

S.P. Niculescu, R. Viertl / Comparison between fuzzy estimators

343

Remark 5. Let Z ~ • , ~ ( ~ ) ,

and

~e(sup p

q~k(CH(~-,k) + kx) = q~z,(CH(Zl) + X),

i•{1,...,k}. If Zi ) < O0 for every i • {1 . . . . . k} then there exist the fuzzy elements Z~ + • • • + Z~ and k-~(Zl + . • • + Z~) and they belong to o~(~) too.

Remark 6. Let A ~_ ~ , A an interval, and let Z • o~(~). T h e n sup{qgz(Z): z • ~ and IA(Z) = 1} = sup Cpz(Z) zqA

(5)

whatever x E ~ + and tr • (0, 1].

Proposition 2. Let (Y,)~=I ~ o~({0, 1}). Then there exists a f u z z y element in N t3 {0, ~}, denoted by X = E ? = 1 Y/, whose membership function is given by q0z(n) = sup{inf q0y,(y~):

and sup{q0z(Z): z • ~ and IA(Z) = 0} = sup Cpz(Z).

(y~)~=~ E {0, 1} ~ and ~. yi = n

z et~kA

i=1

If A 71 C~(Z) 4:0 then

whatever n E N U {0, ~}.

sup qgz(Z) = 1. z~A

In both cases there exists a fuzzy element in {0, 1} d e n o t e d by 1A(Z) such that

Proof. For every i E ~ there exists a~ • {0, 1} such hat qgy,(a~)= 1. For a = E~=~ ai we obtain qoz(a) = 1 and (i) is verified. Now let us consider the case l ~ a < o o . If 0 ~ < n < a is such that t r = q ~ z ( n ) < l then for every e > 0 there exists y = (Yi),=I E {0, 1} ~ such that inf. q~y,(yi) > (r - e. Let

q~(z)(1) = sup q~z(Z)

l l ( y ) = {i: y~ = 1 and qor,(yi) = 1},

and

I2(Y) = {i: y~ = 1 and (Pr,(Y~) < 1},

If A 71C~(Z) = 0 then

?U a

= 1

zqA

=

I3(y) = {i: Yi = 0 and qor,(y/) = 1},

SU

and

Definition 2. Let Y, Z E ~ ( ~ ) . W e say that qPz has the same shape as q0v iff there exists 8 E such that Cpy(U) = qgz(U + 6) w h a t e v e r u • ~ . We d e n o t e this situation by Z II Y. In that case the quantity D ( Z I Y) = ~ is called the delay of Z with respect to Y. F r o m Proposition 1 we deduce: 1. Let Zi • ~ ( ~ ) be such that ~¢(suppZ,)2. Let ,Y~= Zt+'''+Zk and m'k = k-l,~k. If Zi ll Z~, i • { 2 , . . . , k}, then m'k II Z~ and k =

k-~ ~'~ O(Z~

]Zl).

i=2

Moreover, k = kC

o(z, I

(Z,) +

(3)

i=2

Cpz,(cL(Zk) -- kx) = qgz,(CL(Z,) - x),

Observe that Card Ix(y) + Card I4(y) = a and Cardlm(y)~
Corollary

O(m'k I Zl)

14(y) = {i: yi = 0 and qor,(y/) < 1}.

(4)

i

i

and ~c

Ezi=n+l. i-1 As e was arbitrarily taken it follows qoz(n)~< q0z(n + 1) and consequently q0z is non-decreasing on {0 . . . . . a}. In case n > a, Card 12(y) > 0 and taking j E 12(y) and choosing zi = Yi for i 4=j and zj = 0 it follows inf qgy,(yi) ~ inf qgr,(zi) i

i

344

S.P. Niculescu, R. Viertl / Comparison between fuzzy estimators

and

If 2 ~< Card 5f(t) < oo then

oe

qJN(t)(n)

~zi=n-1. i=1

sup q~,(v)

for n = O,

1)~'t

As e was arbitrarily taken it follows
for n e {1 . . . . . =

1

for n e ~(t) U {sup LP(t)},

q~s,(t)

for n ~ ~(t),

inf sup Cps,(U)

for n : o~.

i

Let t e l l + and let ( X , ) ~ = ~ ( ~ ) be a sequence of fuzzy data corresponding to (T/)?=~. Let us assume s u p p X , = ff~+, X, II Sa for all n/> 1 and ~e(supp X~) < o~. In what follows we will use the following notation:

S,=X~+...+X,,

sup ~(t)},

u<-t

If Card LP(t) = Card ~ then { sup q~s,(V) f o r n = O , qgN(t)(n) =

n>~l,

q~s°+,(t)

for n e £~(t),

1

forn=oo.

Proof. The first part of the theorem is an immediate consequence of Proposition 2. According to Remark 6 and Proposition 1 we deduce

g'(t) = {n: t e C,(S~)}, ~ ( t ) = {n: t > Cp(S~)}, ~(t) = {n: t < C~(S~)}. The sets Le(t), $(t) and ~ ( t ) are disjoint and their union is ~. Under the assumptions in this section, $(t) is always finite. If ~ ( t ) = I~ then ~(t) is infinite. If 5¢(t) is infinite then ~ ( t ) = ~.

q~z,o,,¢s,)(1) = sup q~s,(U) = sup qgs,(U) ue(O,t]

and q~Z~o,,,(s,)(O)= sup Cps,(V) = sup q%(v) v~(O,t]

Theorem

1.

N(t) = '~ l(0,tl(S~) e ~ ( ~ / U {0, oo}). n=l

If ~ ( t ) = fJ then

q3N(t)(n ) =

I

1 qgs,(t)

=

sup{inf qvz
for n ~ {0} U ~'(t), for n ~ ~(t),

If Sg(t) = {1} then

(Yi)i~=l E

= min{sup qOs,(U), sup q0s+,(v)} = sup sup min{q0s.(U), q0s.+,(v)}.

for n = O,

13~'t

¢pN(t)(n)=

1

for n ~ { 1 } U ~(t),

qgs.(t)

for n ~ ~Y~(t),

inf sup rps,(U)

for n = oo.

i

u<~t

{0, 1} = and ~] Yi = n l) i=1

U<~I

sup Cps,(V)

v>t

for all i e[~. Coupling Proposition 1 and Corollary 1 we deduce that ~Pl~o,,j(s,)(1) is non-increasing and qh~o,,¢s,)(O) is non-decreasing as functions of i. From Proposition 2 it follows qgN(t)(n )

[inf sup qJs,(U) for n = oo. i u<~t

u~t

I):>t

If 2 ~< Card 5¢(t) < o~ and nl, n2 e ~(t), n~ < n2, then from Corollary 1 it follows C~(Sn,)<~ C~(Sn2) whatever t r e ( 0 , 1] and consequently q~s.,(t) <~ q~s.gt). If n~, n= e ~ ( t ) and n 1 < n2 then from the same result we deduce C~(S,,)<~ C~(Sn2) whatever ere(0, 1] and consequently Cps.,(t) >t cp&~(t).

345

S.P. Niculescu, R. Viertl / Comparison between fuzzy estimators

First assume ~(t) 4= 0 and 2 ~< Card ~ ( t ) < ~. From the continuity of q~x,'S it follows: (a) if 0 < n < sup ~ ( t ) then sup ¢ ~ ( u ) = 1,

Moreover, if there exists e e l + such that supp Xi ~ (e, 2) for all i ~ t~, then qgM¢o(O) = CpM(,)(2) = 0 for sufficiently large t. We use the convention t/O = ~ and t/oo = O.

u~t

Remark 7. For every i c N let us denote by W, the exact value approximated by the fuzzy value X~. Using the notation in the first section it follows

sup qgs.+,(v ) = qgs.+,(t) and consequently qgN(,)(n) = q0S°+,(t);

qgM(O(t/l~4(t)) = 1 and

(b) if n = sup 3?(0 or n ¢ ~(t) then

where m. = n-~S~, n • ~J, and t is large enough. M(t) is called the fuzzy renewal estimator for the mean, m. is the fuzzy sample mean and both are fuzzy estimators for ~.

sup Cps.(U) = sup Cps.+,(v) = 1 u<~l

q~mo(/~)= 1

ly>t

and consequently qON(t)(n ) = 1; (c) if n ~ ~ ( t ) then sup Cpsn(U) = Cps.,(t), U~t

3. Comparison between the fuzzy sample mean and the fuzzy renewal estimator for the mean

sup qgs.+,(v) = 1 I)~'f

and consequently (pu(0(n) = qgs.(t). In case ~ ( t ) = ~, (a) and (c) are the same and (b) is replaced by (b') if n = sup 3?(0 then (n + 1) ~ ~ ( t ) , sup q~s°(U) = sup q~s.+,(t) = 1 and consequently q0uCt)(n)--- 1. In all cases, q0N~,)(0) = inf q%(. ,~s,)(0) = (Pl(,,,l~s,)(0) = sup Cps,(V) p>/

The problem of comparison of mn and M(t) reduces to the comparison of the images of their membership functions. Under the assumptions on the model both /zn and t/lq(t) converge to bt as n, t---~2, except for an event of probability zero. Let ~o ~ £2 be such that P ( w ) > 0 and let W~= T,.((o), i ~ . Denote by X~ the fuzzy elements in ~ associated with W~, i ~ ~. For reasons of convenience let us assume (X~)~=I c ffc(~) be such that s u p p X i c ( e , ~), X~ II X1 for all i/> 1 and g(supp x~) < 2 for some e ~ ~ +. As a consequence of the Strong Law of Large Numbers there exists 6 ~ ~ such that

and (pN(,)(~) = inf qg~,,.,j(s,)(1)

6=limn -l~6i=/l-Wl

i

n ~

= inf sup (ps,(U). i

u~t

In case 5 f ( t ) = 0 and 37(t)={1} the proof follows by combining (b), (c) and (b'). In case Card 37(0 = Card t~ the result follows from (a).

i=2

where 6 i = W ~ - W 1 for all ii>2. Coupling Corollary 1 and Remark 3 for all n / > 2 we deduce

c b ( s . ) = nC

(XO + i=2

exists denoted M(t) = t/N(t), in Corollary

2. There

a fuzzy

element,

t / ( ~ tO {0, 2}) = {t/n: n ~ [~} to {0, ~},

and = nC

(X,) +

6, i=2

such that qgM(t)(t/n ) = CPN(t)(n) for all n c ~ tO {0, 2}.

and both of them are in N+. Under the assumptions in this section 2 ~ Card ~ ( t ) < oc for

346

S.P. Niculescu, R. Viertl / Comparison between fuzzy estimators

t large enough. We use the following notation:

Observe that

nl(t) = sup{n: C~(S,) < t},

1

n t

n2(t) = sup{n: cH(s.) < t} = sup~LP(t),

+1

n3(t) = inf{n: cL(s,) > t} = inf ~(t),

t

n(n

n4(t) = inf{n: CLo(S,) > t}. Observe that

t

1 n~) 6i

n

i=2

.+1

n+l

1

,+1

+ 1) + ~

1 "~)

i~2 ~i-na(t---)i=2

and from n > nl(t) we obtain

0 < nl(t) ~< n2(t) ~< n3(t) ~< n4(t) < 0%

t

--~ cH-[ --

lim nl(t) = oo

n

supp qJM(t)(t/') C {nl(t) + 1. . . . .

6i<00"

n i=2

Now using the continuity of the tpx,'S and the previous inequalities it follows

and n4(t)},

lim Al(t)

= O.

where the support is considered in the classical manner. It appears natural to compare qgu(,) with q%'.,,r A good characterization of the relative position of the corresponding image points is given by

t-..}oo

A(t) =

From n2(t) < n < n3(t) we deduce

max

nl(t)
Theorem

Icpu(t)(t/n) - cpm.,,,,(t/n)l.

2. Under

the assumptions

in this

Similarily, A2(t) =

max

n2(t)
1 --

CIf(x,) + - ~ 6, <~ t n i=2

section, lim

<.

n

(PX,

(t

~i

"

cIH(X1 ) + 1 (~i" n i=2

1

Proof. Obviously

CL(mn4(t)) = c L ( x 1 )

A(t) = max{Al(t), A2(t ), A3(t ), A4(t)}

and 1

CH(m"'(')) = Cln(Xl) + ~ max

ICpMe)(t/n) - qgm.,,,,(t/n)l,

max

IqgM(t)(t/n)- Cpm.,o,(t/n)l,

max

ICpu(o(t/n) -- q~m.,>(t/n)l,

.4(0

"1- ~ / = 2 Z

where n i(t)
n4(t) i=2

As a consequence of Corollary 1 we obtain

A ( t ) = O.

Al(t ) =

6i

(~i

n4(t)

i=2Z~i"

Now observe that A2(t ) = A3(t ) =

n2(t)
A4(t) = IqJM(,)(t/n2(t)) -- CPm.,o)(t/n2(t))[. Using Theorem 1 and Corollary 1 it follows that

=

Cr((m.,(') )

=

n i__~z 6i - -n4(t) i=2 5i

--

( cH(xI)

"{--1i__~2 ~i ) n

afro q9.... (t) - q0x,( t 1 "~ ) 6i ) max -,(0<-<-2(0 n~t) i=2 ( t 1 "~ l ) = max qgx, 6i -,(0<-<-~(0 n + 1 n + 1 i=2

=

1

n4(t) ,=2 6i) •

0

as t---~ o0. Then from the continuity of the qOx,'S we deduce lim A2(t) = O.

S.P. Niculescu,R. Viertl / Comparisonbetweenfuzzy estimators Similarly,

A3(t ) =

cL(x3)

max

"30). . . . . (t)

=

max

cL(xs) =

Cps.(t) - tpx,

(t

= cL(x14)

~X 1

-- --

n3(t)<~n<~n4(t)

n

)

i=2

n4(t)

i=2

=

347

2.3,

cL(x4) = 1.6,

cL(x,1) = 1.4,

CL(X,o) = 0.5,

6i

c1L(x6) = cL(x24) cL(xs)

(~i

=

~-- C L ( x 1 7 ) =

CL(x9) = cL(x25)

=

1.8, 1, 2.4,

CL(x,2) = CL(Xz2) = 0.7, which, coupled with the continuity of qgx'S, implies lim A3(/) = 0.

CL(x,3) = C~(X,9) = 1.2,

cL(x,5) = cL(X2o) = CL(x2s)= 1.9, CL(x,8) = 1.7,

cL(x21) = 1.3,

CL(x23) = 0.9,

cL(x26)

cL(x29) = 2.6,

CL(X3o) = 1.1.

t~

Further, A4(/)~ 1 -

qgm"4"'(n2(t{+ 1) t

t

Using the continuity of the l i m , ~ A2(t) = 0 it follows that

q~x,'S and

lim z~4(t) = 0,

=

0.4,

For t = 11 the membership function of N(t) (dots) is represented in Figure l(a). The comparison between the membership functions associated with M(t) and mn,0) is performed in Figure l(b). The similar features for t = 35 are given in Figures l(c) and l(d). For t = 35 we can observe the tendency of the image points of qOM~,) to superpose on the graph of qg,,,,,~.

which completes the proof.

Remark 8. Theorem 2 shows that for large values of t and corresponding values for n the fuzzy sample mean and the fuzzy renewal estimator for the mean are equivalent via the propagation of fuzziness.

Example 2. A sample of 28 fuzzy data is considered, with the same asymmetric triangular shape membership functions and characterized by CoL(X1) = 0.3,

CL(xI) = CH(xI) = 0.4,

C"o(X,) = 0.6,

CL(x2) = cL(Xz2) = 0.7,

cL(x3) = 1.3,

4. Examples EL(x4) = EL(x9)

The following examples illustrate the results in the previous sections. Let us assume that the assumptions in Section 3 are fulfilled. In order to avoid very complicated figures we will describe the fuzzy numbers using c~-cuts.

C~(xs) = C~(X28) = 1.9, CL(XT) = 2.3,

CL(X,o) = 0.9, cL(x,3) =

Example 1. A sample of 30 fuzzy data is considered, with the same asymmetric trapezoid membership functions and characterized by cL(x1) = EL(x16) = 0.1,

C~(X,) = 0.4,

C~(X,) = 0.3,

cH(x,) = 0.5,

CL(X2) = CL(X7) = CL(x27) = 0.8,

~- E L ( x 2 3 ) = 1.6,

CIL(X6)= 0.8,

cL(x8) = CL(x12) = 1.4,

CL(x,o = 2,

cL(x2,) = 1.8,

cL(x14)

= 2.4,

CL(xI5) = 0.5,

CL(/16) = c1L(x24) = 0.6,

CL(x17) = 2.2,

CL(x18) = CL(x26) = 1,

CIf(xz5)= 1.5. For t = 30 the membership function of N(t)

cL(x19)

= 1.1,

C~(X2o)=2.1,

(dots) is represented in Figure 2(a). The comparison between the membership functions

S.P, Niculescu, R. Viertl / Comparison between fuzzy estimators

348 a

o

--

0

i0

~ON(t)

. .~

0

1

2

, t=ll

,

t=ll

n 4 (t)

d

C

B

--

0

20

. %- .

30

~ON(t)

,

-

. t m.a.

,

.

1

, t=34

. )

2

~(t)

, t=34

~m

'

n4(t)

Fig, 1. Membership functions of Example 1. a

O

6

.....

i

. 7t

~ e . °•

• e.*.

- r • )

15

25 ~N(t)

'

1

2

~M(t)

t=30

'

~°m

, t=30

n4(t)

d 1

-

-

-

o

•

P.

•

•

.

.

.

.

.

.

20

4.

30

~N(t)

' t=36

=

.

)

0

,

.:

.

.

.

.

~

1

~oM(t)

Fig. 2. Membership functions of Example 2.

_e

.

.

.

2

, ~om

n4(t)

, t=36

.

)

S.P. Niculescu, R. Viertl

/

associated with M(t) (dots) and m,,(o is performed in Figure 2(b). The similar features for t = 36 are given in Figures 2(c) and 2(d).

ct(x~) = c~1(x,3) = cy(x~,) = cy(x~) = c1L(x46) = 0 . 7 , CL(x2)

Example 3. A sample of 50 fuzzy data is considered, with the same symmetric triangular shape membership functions and characterized by

=

CL(xI9)

=

CIL(X25)

--- CIL(X3I)

= e L ( x 3 6 ) = cIL(X48) = 0.9, cIL(X9)

c L ( x , ) = O. 1,

= CIL(XIs)

=

fIE(X24)

--- CIL(X35)

= c~(x37)

=

c~,(x45) = o.5,

CL(X,o) = c L ( x 1 5 ) = e L ( X 4 0 ) = c L ( x 4 4 )

c1L(x1) = c1H(x1) -~- c L ( x I 7 ) = ElL(X28)

= CL(Xso) = 1,

~---C L ( x 2 9 ) = c L ( x 4 7 ) ~---0.2,

c~(x,) =

349

Comparison between fuzzy estimators

cL(x,2) = 1.4,

0.3,

C L ( x 1 6 ) = C L ( x 2 6 ) = CL(X3o) =

c L ( x 3 ) = ElL(X14) ~- cIL(X34) = c L ( x 4 2 )

cL(x23) = 1.2,

cL(x~)=

0.6,

1.3.

= cL(x43) = 0.4, For t = 21 the membership function of N(t) (dots) is represented in Figure 3(a). The comparison between the membership functions associated with M(t) (dots) and mn,(t) is performed in Figure 3(b). The similar features for t = 28 are given in Figures 3(c) and 3(d). We

cIL(X4) = c L ( x 2 0 ) = CL(X22) = c L ( x 3 9 )

= cL(x49) = 0.3, c L ( X s ) = C L ( X s ) = c L ( x 4 1 ) = 0.8, c L ( x 6 ) ~-

cL( x ' 1) = cIL(X27) = c L ( x 3 3 ) = l . 1,

B

l

I

30 ~ON(t)

0

, t=21

~M(t)

'

~0m

, t=21

n4(t)

Q Q o o Q o Q

i _ i ,.-o

•

-~ 7 . . . . . . . . . . . . .

• )

1

4O

~/q(t)

'

t=28

~n(t)

Fig. 3. Membership functions of Example 3.

~0m

, t=28

n4(t)

350

S.P. Niculescu, R. Viertl / Comparison between fuzzy estimators

can observe an almost perfect superposition of the image points of q~M(t) on the graph of tp,,.,,~.

Remark 9. The previous examples underline the effect of the shape of the membership functions of the fuzzy data on the speed of approaching between the image points of q0M~,)and the graph of q0,~n,~,~. It seems that the asymmetry plays some role in braking this phenomenon. The relationship between /~n and ~/(t) as t, n-~ oo is also illustrated. An important element to be pointed out and which made possible the derivation of a Theorem 2 type result was the same nature of these estimators.

Acknowledgements The research was financially supported by the Fonds zur Ftrderung der Wissenschaftlichen Forschung, Vienna, under projects No. 6350 and No. 7717-Phy. The authors are indebted to the referees for

suggestions which led to improvement in the presentation of the paper.

References [1] D. Dubois and H. Prade, Fuzzy Sets and Systems. Theory and Applications. (Academic Press, New York-London, 1980). [2] J.L. Doob, Renewal theory from the point of view of the theory of probability, Trans. Amer. Math. Soc. 63 (1948) 422-439. [3] A. Kolmogorov, Uber die Summen durch den Zufail bestimmer unabh~ingiger Grtszen, Math. Ann. 99 (1928) 303-319. [4] R. Kruse and K.D. Meyer, Statistics with Vague Data (Reidel, Dordrecht-Boston, 1987). [5] G. Mihoc and S.P. Niculescu Renewal Processes (Editura Academiei, Bucharest, 1982) (in Romanian). [6] R. Viertl, Modelling of fuzzy measurements in reliability estimation, in: V. Colombari, Ed., Reliability Data Collection and Use in Risk and Availability Assessment (Springer-Verlag, Berlin, 1989) 206-211. [7] R. Viertl, Statistical inference for fuzzy data in environmetrics, Environmentrics 1 (1990) 37-42.