Reversing the order of integration in iterated

Journal of Statistical Planning and Inference 74 (1998) 11–29

Reversing the order of integration in iterated expectations of fuzzy random variables, and statistical applications Miguel Lopez-Daz, Mara Angeles Gil ∗ Departamento de Estadstica, I.O. y D.M., Facultad de Ciencias, Universidad de Oviedo, C/Calvo Sotelo, s/n, 33071 Oviedo, Spain Received 5 August 1996; accepted 20 February 1998

Abstract In this paper conditions are given to compute iterated expectations of fuzzy random variables irrespectively of the order of integration. To this purpose, some studies about the measurability and the integrable boundedness of the fuzzy expected value of a fuzzy random variable with respect to a regular conditional probability are rst developed. The conclusions obtained are applied later to statistical problems involving fuzzy random variables, like those concerning some hierarchical models and mixture distributions (more precisely, some techniques to obtain c 1998 Elsevier fuzzy unbiased estimators and the Bayesian analysis of statistical problems).  Science B.V. All rights reserved. AMS classication: primary 28B20; 62H12; 62H15; 62H99; secondary 03E72; 26B99; 60D05 Keywords: Estimation; Fuzzy expected value; Fuzzy losses; Fuzzy random variable; Fuzzy utilities; Hierarchical models; Hypothesis testing; Mixture distributions; Regular conditional probability; Statistical decision problem

1. Introduction Fuzzy random variables (as intended by Puri and Ralescu, 1986) have been introduced as an extension of real- or vectorial-valued random variables based on measurable set-valued functions (usually referred to as random sets) which also extend. Fuzzy random variables formalize an existing imprecise quantication process associated with a random experiment and assessing a fuzzy value to each experimental outcome. One of the most common and useful characteristics of a random variable is its expected value. The concept of expectation plays an important role as a central notion in Probability Theory, Statistics and Bayesian Decision Theory. ∗

Corresponding author. Tel.: 349 8510 3356; fax.: +349 8510 3356; e-mail: [email protected].

c 1998 Elsevier Science B.V. All rights reserved. 0378-3758/98/$ – see front matter  PII: S 0 3 7 8 - 3 7 5 8 ( 9 8 ) 0 0 1 0 0 - 1

12

M. Lopez-Daz, M.A. Gil / Journal of Statistical Planning and Inference 74 (1998) 11–29

The fuzzy expected value of a fuzzy random variable has been presented (Puri and Ralescu, 1986) as an extension of the expectation of a real- or vectorial-valued random variable based on the Aumann integral of measurable set-valued functions (Aumann, 1965). In practical calculations, the expectations of random variables dened from product spaces have often to be evaluated by computing iterated expectations. The rst target of this paper is to justify a similar process in the fuzzy case, as well as to state hypotheses under which the order of integration in iterated fuzzy expectations does not matter. The second aim is to apply the results of the rst study to some statistical problems involving fuzzy random variables, which in fact have motivated the rst objective. In this way, the paper is organized as follows: Section 2 contains preliminary denitions and results which will be used later; in Section 3, we describe the main probabilistic aspects of the paper, namely, the analysis of the measurability and integrable boundedness of fuzzy expected values of fuzzy random variables with respect to regular conditional probabilities, and a study concerning the order of integration in iterated expectations of fuzzy random variables is stated; in Section 4, we present some statistical applications of the conclusions in Section 3; in Section 4.1 we extend the double expectation theorem for hierarchical models of fuzzy random variables, and apply it to state a mechanism to construct a fuzzy unbiased estimator from another fuzzy unbiased estimator and to dene a certain fuzzy unbiased estimator of the sample mean in random sampling with replacement from a nite population; in Section 4.2 conditions are established for the equivalence of the extensive and normal forms of the Bayesian analysis of a decision problem with fuzzy-valued utilities or losses, and implications of this equivalence for the Bayesian analysis of estimation and hypothesis testing problems are examined and illustrated with some examples. Finally, some remarks and an appendix containing the proofs of the main results in Sections 3 and 4 are included.

2. Preliminaries Let (Rn ; |·|) be the Euclidean space with the associated norm, and let Kc (Rn ) be the class of all nonempty compact convex subsets of Rn : Let (Kc (Rn ); dH ) be the Hausdor metric space, that is, if A; B ∈ Kc (Rn ); dH (A; B) = max{(A; B); (B; A)} with (A; B) = supa∈A inf b∈B |a − b|: Let Fc (Rn ) denote the class of the fuzzy subsets of Rn ; V : Rn → [0; 1]; such that: (i) V is upper semicontinuous (i.e., the -level sets of V; V = {x ∈ Rn : V(x)¿}; are closed for all 0¡61); (ii) V is normal (i.e., V1 = ∅); (iii) V is convex (i.e., V is a convex subset of Rn for all 0¡61); and (iv) the closed convex hull of the support of V; V0 = Cl [co {x ∈ Rn : V(x)¿0} ] (which under the assumption (iii) equals the closure of the support of V), is compact. Let (; a; P) be a probability space, and let F :  → P(Rn ) be a set-valued function such that F(!) is closed for all ! ∈ : Dierent a-measurability denitions have

M. Lopez-Daz, M.A. Gil / Journal of Statistical Planning and Inference 74 (1998) 11–29

13

been proposed for set-valued functions (see, for instance, Himmelberg, 1975; Hiai and Umegaki, 1977). Throughout this paper we will make use of the weakest one, which is that considered by Puri and Ralescu (1986) to formalize fuzzy random variables. In accordance with this denition, Gr (F) = {(!; x) ∈  × Rn : x ∈ F(!)} ∈ a ⊗ BRn ; where BRn is the Borel -eld of Rn . As shown by Debreu (1966), Castaing (1969) and Himmelberg (1975), all measurability denitions are equivalent whenever the probability space (; a; P) is complete. If F :  → P(Rn ) is a measurable set-valued function, F is said to be integrably bounded if there exists an L1 (P) function h :  → R such that dH (F(!); {0}) = sup |x|6h(!) x∈F(!)

for all ! ∈ : If we denote by S(F) = {f :  → Rn ; f(!) ∈ F(!) a.s. [P]; f ∈ L1 (P)}; then if F is integrably bounded the set S(F) is nonempty and bounded. The concept of fuzzy random variable we now recall has been introduced by Puri and Ralescu (1986) to formalize an existing fuzzy-valued quantication process associated with a random experiment. Denition 2.1. Given a measurable space (; a), a mapping X :  → Fc (Rn ) is said to be a fuzzy random variable associated with (; a) if the set-valued function X :  → Kc (Rn ) such that X (!) = (X (!)) for all ! ∈  is a-measurable for all  ∈ [0; 1]. To guarantee the existence of the expected value of a fuzzy random variable, the following condition will be assumed: Denition 2.2. Given a probability space (; a; P), a fuzzy random variable associated with (; a) is said to be an integrably bounded fuzzy random variable with respect to (; a; P) if the set-valued functions X are integrably bounded for all  ∈ [0; 1]: Obviously, whenever X0 is an integrably bounded measurable set-valued function, then X is also integrably bounded for all  ∈ [0; 1]. On the other hand, if (; a; P) is a complete probability space, the measurability of X for all  ∈ (0; 1] ensures the measurability of X0 . From now on, we will denote by (; a; P) the set of all integrably bounded fuzzy random variables with respect to (; a; P): As we have commented above, the concept of fuzzy expected value of an integrably bounded fuzzy random variable is based on Aumann’s integral of a set-valued function. If F :  → Kc (Rn ) is an integrably bounded measurable set-valued function with respect to a probability space (; a;P), then the Aumann integral of F with respect to P is  given by the set  F dP = {  f dP : f ∈ S(F)}: On the basis of this notion, Puri and Ralescu (1986) have introduced as a value representing the ‘central tendency’ of a fuzzy random variable the following one:

14

M. Lopez-Daz, M.A. Gil / Journal of Statistical Planning and Inference 74 (1998) 11–29

Denition 2.3. If X is an integrably bounded fuzzy random variable with respect to the probability space (; a; P), the fuzzy expected value of X is the unique fuzzy set    such that (E(X=P)) of Fc (Rn ), E(X=P),  =  X dP for all  ∈ (0; 1]:  E(X=P) has been proven (Puri and Ralescu, 1986) to belong to Fc (Rn ). On the other hand, if X is a simple fuzzy random variable, that is, X can be expressed as follows: X (·) =

m ] 

ICi (·)  Vi ;

i=1 n  m where {Ci }m i=1 ⊂ a is a partition of ; and {Vi }i = 1 ⊂ Fc (R ); then the convexity of n the elements of Fc (R ) implies the equality of the Aumann and the Debreu integrals of the functions X (see Byrne, 1978) and, consequently, the fuzzy expected value of X can be equivalently dened by

 E(X=P) =

m ] 

P(Ci )  Vi :

i=1

 (-alternatively denoted by ⊕- and  being the fuzzy sum and product, respectively, based on Zadeh’s extension principle, (Zadeh, 1975).) The following supporting result refers to the measurability and integrable boundedness, and the fuzzy expected value, of the fuzzy sum of fuzzy random variables (see, for instance, Stojakovic, 1992). Lemma 2.1. Let (; a; P) be a complete probability space and let X; Y ∈ (; a; P):  ⊕ Y=P) = E(X=P)   Then; X ⊕ Y ∈  (; a; P); and E(X ⊕ E(Y=P); where X ⊕ Y is dened by (X ⊕ Y )(!) = X (!) ⊕ Y (!) for all ! ∈ : The following result will be used also later, and is similar to one stated by Puri and Ralescu (1986). Lemma 2.2. Let (; a; P) be a probability space; let Xm ; X ∈ (; a; P); m¿1; such that a.s. [P] : limm→∞ dH ((Xm (!)) ; (X (!)) ) = 0 for all  ∈ (0; 1]: If there exists h ∈ L1 (P) with supx∈(Xm (!))0 |x|6h(!); for all m¿1; and ! ∈ ; then limm→∞ dH ((E  (Xm =P)) ; (E(X=P))  ) = 0 for all  ∈ (0; 1]:

3. Main results In this section, conditions are given to guarantee the measurability and integrable boundedness of the fuzzy expected value of the projections of an integrable bounded fuzzy random variable with respect to a regular conditional probability. Then, we state

M. Lopez-Daz, M.A. Gil / Journal of Statistical Planning and Inference 74 (1998) 11–29

15

hypotheses under which the order of integration in iterated expectations of fuzzy random variables does not matter. Results in this section are presented in accordance with the order they can be obtained, as can be shown in the appendix. From now on, we will assume that (j ; aj ; Pj ); j = 1; 2; are two complete probability spaces induced by two measurable functions Yj ; j = 1; 2; dened from a given probability space (; G; Q), and P will denote the probability measure induced by (Y1 ; Y2 ) on the measurable space (1 × 2 ; a1 ⊗ a2 ): We will also suppose that (Y1 ; Y2 ) is such that there exists a regular conditional probability (cf. Breiman, 1968) for Y2 given Y1 = !1 , P!1 , for each !1 ∈ 1 , that is, (1) P!1 is a probability measure on (2 ; a2 ) for each xed !1 ∈ 1 ; and (2) for each xed A2 ∈ a2 ; the mapping g : 1 → [0; 1] such that g(!1 ) = P!1 (A2 ) is a random variable associated with (1 ; a1 ) satisfying that for each A1 ∈ a1  P(Y1 ∈ A1 ; Y2 ∈ A2 ) =

P!1 (A2 ) dP1 A1

and there exists also a regular conditional probability for Y1 given Y2 = !2 ; P!2 , for each !2 ∈ 2 . In consequence, if B ∈ a1 ⊗ a2 , then (cf. Ash, 1975)  P(B) =

 P!1 (B!1 ) dP1 =

1

P!2 (B!2 ) dP2 : 2

1 Let 0 (P) denote the set of real-valued functions f : 1 × 2 → R; f ∈ L1 (P), such that for all (!1 ; !2 ) ∈ 1 × 2 ; we have that f!j ∈ L1 (P!j ) and the mapping gj : j → R, such that gj (!j ) = 3−j f!j dP!j for all !j ∈ j , satises that gj ∈ L1 (Pj ); j = 1; 2: Denition 3.1. The set 0 (1 × 2 ; a1 ⊗ a2 ; P) is the set of all fuzzy random variables 1 X ∈ (1 × 2 ; a1 ⊗ a2 ; P) such that there exists f ∈ 0 (P) with sup

|x|6f(!1 ; !2 )

for all (!1 ; !2 ) ∈ 1 × 2 :

x∈(X (!1 ; !2 ))0

Of course, any simple fuzzy random variable belongs to 0 (1 × 2 ; a1 ⊗ a2 ; P): The problem analyzed in the rst part of this section can be now formulated as follows: given X ∈ 0 (1 × 2 ; a1 ⊗ a2 ; P); we can dene a mapping assigning to  !1 =P!1 ), where X!1 is the !1 -projection of each !1 ∈ 1 the fuzzy expected value E(X X ; we are now going to establish hypotheses to ensure the measurability and integrable boundedness of the preceding mapping with respect to (1 ; a1 ; P1 ). To this purpose, a result examining the measurability and integrable boundedness of the projection X!1 is rst stated.

16

M. Lopez-Daz, M.A. Gil / Journal of Statistical Planning and Inference 74 (1998) 11–29

Lemma 3.1. Let X ∈ 0 (1 ×2 ; a1 ⊗a2 ; P). For each !1 ∈ 1 we dene the mapping X!1 : 2 → Fc (Rn ) given by X!1 (!2 ) = X (!1 ; !2 ) for all !2 ∈ 2 : Then; the mapping X!1 ∈ (2 ; a2 ; P!1 ) for each !1 ∈ 1 . The main result in this part is now stated for simple fuzzy random variables. Proposition 3.2. Let X ∈ 0 (1 × 2 ; a1 ⊗ a2 ; P) be a simple fuzzy random variable.  !1 =P!1 ) for each If the mapping v1 : 1 → Fc (Rn ) is dened so that v1 (!1 ) =E(X 1 !1 ∈ 1 ; then v ∈ (1 ; a1 ; P1 ): The preceding result can be generalized by using the following one, which can be immediately deduced from Lopez–Daz and Gil (1997). Lemma 3.3. Let (; a; P) be a probability space and let X ∈ (; a; P). Then; there exists a sequence of simple fuzzy random variables {Xm }m ⊂ (; a; P) and a mapping h ∈ L1 (P) with supx∈(Xm (!))0 |x|6h(!) for all ! ∈ ; and m¿1; such that a.s. [P] lim dH ((Xm (!)) ; (X (!)) ) = 0 for all  ∈ (0; 1];

m→∞

where h is a linear function of h0 such that supx∈X0 (!) |x|6h0 (!) for all ! ∈ : The key question in the rst part of this section is now stated as follows: Theorem 3.4. Let X ∈ 0 (1 × 2 ; a1 ⊗ a2 ; P). If we dene the mapping v1 : 1 →  !1 =P!1 ) for each !1 ∈ 1 ; then v1 ∈ (1 ; a1 ; P1 ): Fc (Rn ) such that v1 (!1 ) = E(X We are now are going to establish the main results in relation to the order of integration in iterated expectations of fuzzy random variables. A valuable result supporting this aim is that concerning the fuzzy expected value of a fuzzy random variable which is obtained by considering the fuzzy product of a classical random variable and an element of Fc (Rn ). Lemma 3.5. Let (; a; P) be a probability space; let X :  → R+ be an L1 (P)-random variable; and let V ∈ Fc (Rn ): The function Y ∈ (; a; P) such that Y (!) = X (!)  V  for all ! ∈ ; satises that E(Y=P) = E(X=P)  V: In the following result we state the main probabilistic conclusion in this paper for simple fuzzy random variables. Proposition 3.6. Let X ∈ 0 (1 × 2 ; a1 ⊗ a2 ; P) be a simple fuzzy random variable.  !1 =P!1 ) for Let vj : j → Fc (Rn ); j = 1; 2; be the functions such that v1 (!1 ) =E(X  !2 =P!2 ) for each !2 ∈ 2 : Then; E(v  1 =P1 ) = E(v  2 =P2 ) = each !1 ∈ 1 ; and v2 (!2 ) =E(X  E(X=P):

M. Lopez-Daz, M.A. Gil / Journal of Statistical Planning and Inference 74 (1998) 11–29

17

The last result can be generalized by using Lemma 3.3 as follows: Theorem 3.7. Let X ∈ 0 (1 × 2 ; a1 ⊗ a2 ; P): If we dene the mappings v j : j →  !1 =P!1 ) for each !1 ∈ 1 ; and v2 (!2 ) = Fc (Rn ); j = 1; 2; such that v1 (!1 ) =E(X  !2 =P!2 ) for each !2 ∈ 2 ; then; E(X (i) vi ∈ (i ; ai ; Pi ); i = 1; 2;  1 =P1 ) =E(v  2 =P2 ) = E(X=P)  or; equivalently; (ii) E(v     !2 =P!2 )=P2 ) =E(X=P):  (iii) E(E(X!1 =P!1 )=P1 ) =E(E(X

4. Some statistical applications of the iterated expectation of fuzzy random variables The results obtained in Section 3 can be applied to extend some interesting conclusions concerning statistical problems involving real-valued random variables, to statistical problems involving fuzzy random variables. Among these applications we pay special attention to the hierarchical models and mixture distributions regarding fuzzy random variables, which will be widely applicable in statistics. We now describe some of these applications, and illustrate them with a few examples. 4.1. The double expectation theorem of fuzzy random variables and its application to fuzzy unbiased estimation In statistics we often nd processes which may be modeled by a nite sequence of random variables placed in a hierarchy. In particular, the computation of a fuzzy random variable can be signicantly simplied sometimes by using the extension of the so-called double expectation theorem, which is presented in the following result. Theorem 4.1. Let (; A; P) be a complete probability space and let X :  → Fc (Rn ); Y :  → Fc (Rn ) be two integrably bounded fuzzy random variables. Let X and Y be the -elds in Fc (Rn ) induced from A by X and Y; respectively; and let PX and PY be the probability measures induced from P by X and Y; respectively. Consider the product probability space (Fc (Rn ) × Fc (Rn ); X ⊗ Y ; PX ⊗ PY ); and let X ∗ : Fc (Rn ) × Fc (Rn ) → Fc (Rn ) be the integrably bounded fuzzy random variable such that X ∗ (˜x; y) ˜ = x˜ ; for all x˜ ; y˜ ∈ Fc (Rn ): Assume that when Y = y˜ the conditional probability distribution induced by X is given by a regular conditional probability dis  ∗ =Py˜) = y) ˜ =E(X tribution on (Fc (Rn ); X ) denoted by Py˜: Then; if we identify E(X=Y y˜  E(X=Y   E(X=Y  = y)=P ˜ Y ); we obtain that and E( )=PY ) = E(   E(X=Y  E(X=P) = E( )=PY ): On the basis of the preceding theorem we can dene a technique for obtaining a new fuzzy unbiased estimator from a given unbiased estimator. This technique

18

M. Lopez-Daz, M.A. Gil / Journal of Statistical Planning and Inference 74 (1998) 11–29

generalizes one of the basic ideas contained in Rao–Blackwell Theorem (see, for instance, Dudewicz and Mishra, 1988). Thus, Theorem 4.2. Let X and Y be two fuzzy random variables satisfying the conditions in  ) =V (that is; X is a fuzzy unbiased Theorem 4:1; and let V ∈ Fc (Rn ) such that E(X estimator – in the fuzzy expected value sense – of V). Let ’(Y ) be dened so that  ’(y) ˜ =E(X=Y = y) ˜ for all value y˜ of Y . Then; ’(Y ) is a fuzzy random variable and  it is a fuzzy unbiased estimator of V; that is; E(’(Y )) =V. In statistical problems involving fuzzy random variables some distributions arise from a hierarchical structure, and a fuzzy random variable X whose distribution depends on a random variable can be referred to as a mixture distribution of fuzzy random variables. In this way, the following example illustrates the conclusions in Theorem 4.1, and the use of a mixture distribution of fuzzy random variables (actually, one of them is a real-valued one). Example 4.1. Let X be a fuzzy random variable which in a population of N sampling units, U1 ; : : : ; UN , takes on the values x˜ 1 = X (U1 ); : : : ; x˜ N = X (UN ) with x˜ 1 ; : : : ; x˜ N ∈ Fc (R). If a sample [] of size n is chosen at random and without replacement from the overall population, and U1 ; : : : ; Un are the units in it, then the fuzzy sample mean X n which associates with [] the fuzzy value

X n [] =

n ] 1   X (Ui ) n i=1

N is a fuzzy unbiased estimator of the fuzzy population mean, X = (1=N )  ] j=1 X (Uj ):   Thus, E(X n ) = X , where E(X n ) is the fuzzy expected value of the sample mean X n

computed on the space of the Nn distinct possible random samples without replacement of size n from the given population. Assume that a sample of size n is now chosen at random and with replacement from the overall population, and let  be the eective sample size (see, for instance, Thompson, 1992), that is, the number of distinct units contained in the sample. Let X denote the sample mean for the distinct units. X is a fuzzy random variable whose distribution depends on a classical random variable , and X and  satisfy the conditions in Theorem 4.1. Then, X is a fuzzy unbiased estimator of X , since on the basis of Theorem 4.1 we can conclude that if P is the distribution of  on the space of the N +n−1

distinct possible random samples with replacement of size n from the given n population, the fuzzy expected value of X on this space is given by   ) = E(  E(X   = = k)=P ) E(X

M. Lopez-Daz, M.A. Gil / Journal of Statistical Planning and Inference 74 (1998) 11–29

19

  = = k) = E(X  ), because it is equivalent to the expected value of the sample and E(X  ) = X , mean X k in a random sample without replacement of size k. Consequently, E(X whence X is a fuzzy unbiased estimator of X in the random sampling with replacement. 4.2. The equivalence of the extensive and normal forms of the Bayesian analysis of statistical problems with fuzzy losses or utilities Mixture distributions of fuzzy random variables appear also when we consider decision-oriented statistical problems in a Bayesian context. In Gil and Lopez-Daz (1996) we have introduced a handy model to deal with single-stage decision problems when values assessed to the consequences of decisions are assumed to be fuzzy. This model has been stated so that, on one hand, the existence of fuzzy utility functions representing the preference pattern of the decision maker can be established in terms of an axiomatic development and, on the other hand, conditions for the equivalence of the normal and extensive forms of the Bayesian analysis (extending studies of the real-valued case, like those in Brown and Purves, 1973) can be given. The study about the conditions for the last equivalence is one of the motivations of the study we have carried out in Section 3. Therefore, the ‘translation’ of the conclusions in Theorem 3.7 to the statistical decision problem with fuzzy utilities or losses will serve us to give another application of this theorem. The concept of fuzzy utility function (in the fuzzy expected utility approach) can be formalized as follows: Denition 4.2. Consider a single-stage statistical decision problem with state space and action space A. A function U : × A → Fc (R) is said to be a fuzzy utility function if • Ua is a fuzzy random variable associated with the measurable space ( ; E); for all a ∈ A; and • if a prior distribution on ( ; E) is assumed; then for any two actions a1 ; a2 ∈A for which the projections Ua1 and Ua2 are integrably bounded fuzzy random variables associated with ( ; E; ); a2 is considered to be not preferred to a1 if; and only if;  a2 = ); where ¡ is an appropriate ranking of fuzzy values.  a1 = ) ¡ E(U E(U If categorical choices among actions are required, a ranking method which has been proven to t properly the denition above is that introduced by Campos and Gonzalez (1989), which can be particularized to elements in Fc (R) as follows: Denition 4.3. If A; B ∈ Fc (R); A is said to be greater than or equal to B in accordance with the -average ranking method; and will be denoted by A¿ B; if and only VL (A)¿VL (B);

20

M. Lopez-Daz, M.A. Gil / Journal of Statistical Planning and Inference 74 (1998) 11–29

where for each A ∈ Fc (R); with A = [a1 ; a2 ] for each  ∈ (0; 1]; the -average value of A is given by  1 [a2 + (1 − )a1 ] d; VL (A) = 0

where  ∈ [0; 1]: The parameter  is a subjective degree of optimism–pessimism. In a utility (loss) context,  = 1 ( = 0) reects the highest optimism (that is, for each  ∈ (0; 1], A is considered to be represented by its upper extreme, a2 ), and  = 0 ( = 1) reects the highest pessimism (that is, for each  ∈ (0; 1], A is considered to be represented by its lower extreme, a1 ). The -average value satises many useful properties. On the basis of some of them (see Campos and Gonzalez, 1989; Lopez-Daz and Gil, 1998) one can conclude that any set of axioms guaranteeing the existence of a bounded real-valued utility function on × A, which is unique up to increasing linear transformations, also ensures the existence of a class of fuzzy utility functions on × A (see Gil and Lopez-Daz, 1996). In a decision problem involving the observation of the value of X , the decision maker must choose a decision rule, which is a mapping : X → A. The class of all decision rules in the decision problem involving observations from X will be denoted by X . To compare decision rules in X , we can dene the function U : × X → Fc (R) such that U (; x) = U (; (x)). The aim of the extension of the Bayesian analysis in normal form of the statistical decision problems based on the model above presented, is to choose as an ‘optimal’ decision rule (Bayes decision rule) in X any B ∈ X which ‘maximizes’ (in the -average sense) the iterated expectation  E(U   =P )= ); E( where U is the -projection of U . The ‘value’ of the statistical decision problem in this analysis can be quantied by means of the iterated fuzzy expectation  E(U  B =P )= ). E(  To look for a ‘maximizing’ function in X is a very complex task, in practice. However, under certain conditions a Bayes decision rule in a statistical decision problem involving fuzzy utilities can be easily determined, since this special rule reduces to carry out ‘maximizations’ in A. More precisely, Theorem 4.3. Consider a statistical decision problem with state space , action space A, and a random experiment X ≡ (X; BX ; P ),  ∈ , and assume that we have dened a fuzzy utility function U to represent decision maker’s preferences. Let be the prior distribution on the measurable space ( ; E), so that ( ; E; ) is a complete probability space. Then, if U ∈ 0 ( × X; E ⊗ BX ; P) for all ∈ X (P being the

M. Lopez-Daz, M.A. Gil / Journal of Statistical Planning and Inference 74 (1998) 11–29

21

joint distribution of  and X ), the decision rule B ∈ X such that B (x) = ax for each X ∈ X, where  a = x )  ax = x ) ¿ E(U E(U

for all a ∈ A

( x being the posterior distribution given x) is a Bayes decision rule for this problem. The aim of the immediate extension of the extensive form of Bayesian analysis of the statistical decision problem above considered is to choose for each x ∈ X an action ax (Bayes posterior action given x) which ‘maximizes’ in A the (posterior)  a = x ). The ‘value’ of the statistical decision problem in this fuzzy expected utility E(U second type of Bayesian analysis can be quantied by means of the iterated fuzzy expectation  E(U  ax = x )=PX ) E( (PX being the marginal-predictive-experimental distribution). The equivalence of the normal and the extensive forms is given in the following result Theorem 4.4. Consider a statistical decision problem with state space , action space A, and a random experiment X ≡ (X; BX ; P );  ∈ ; and assume that we have dened a fuzzy utility function U to represent the decision maker’s preferences. Let be the prior distribution on the measurable space ( ; E), so that ( ; E; ) is a complete probability space. If U ∈0 ( × X; E ⊗ BX ; P) for all ∈ X , then  E(U  ax = x )=PX ):  E(U  B =P )= ) = E( E(  The preceding results indicate that operationally we will always proceed by max ax = x ) for each x. This conclusion is not limited to imizing the expected value E(U standard decision problems with fuzzy utilities, but it can be applied also to inferential problems with fuzzy-valued losses in a Bayesian context. In this way, regarding the estimation problem we have that Theorem 4.5. Let X ≡ (X; BX ; P ),  ∈ , be a random experiment and assume that  is an unknown parameter and L : × → Fc (R) is a fuzzy loss function (which will be intended as a mapping such that for all  ∈ the projection L is an integrable bounded fuzzy random variable with respect to the probability space ( ; E; x ) for each x ∈ X). Then, if LT : × X → Fc (R) such that LT (; x) = L(; T (x)) satises that LT ∈0 ( × X; E ⊗ BX ; P) for all estimators T (X ) of , then the estimator TB (X )  T (x) = x ) for all  ∈ is a Bayes estimator   = x )¿ E(L such that for each x ∈ X, E(L B for this problem. This result is now illustrated as follows (see Gil et al., 1998, for the detailed calculations):

22

M. Lopez-Daz, M.A. Gil / Journal of Statistical Planning and Inference 74 (1998) 11–29

Fig. 1. Version of the fuzzy modier APPROXIMATELY in Example 4.4.

Example 4.4. A poll is taken to estimate the proportion p of people in a large population who approve of a political decision. Consider a noninformative prior (i.e., a uniform prior distribution on [0; 1]) for p, and the following fuzzy loss function: L(p; p ) = APPROXIMATELY |p − p | for all p; p ∈ [0; 1]; where the linguistic modier APPROXIMATELY is assumed to be identied, for instance, with a narrower version than the triangular Tri(|p − p |=2; |p − p |; 3|p − p |=2), as that given by L(p; p ) = G(|p − p |; |p − p |=2) with G(; )(t) =

e−|t−| − e− 1 − e−

for all t ∈ R;

where  and  denote the ‘center’ and the ‘width’, respectively (see Fig. 1). Assume that to help in estimating p the pollster can interview 20 people and nd the proportion of them who approve of the decision (that is, a binomial sampling – experiment X – is performed). Since the beta distribution is conjugate prior for the binomial distribution B(20; p), we can immediately conclude that x is a beta distribution Be (x + 1; 21 − x), for each x ∈ X = {0; 1; : : : ; 20}. On the other hand, for each x ∈ {0; 1; : : : ; 20} and p ∈ [0; 1], we have that (when  = 0:5)  0:5   |p − p | d x (p): VL (E(Lp = x )) = [0;1]

The Bayes estimator of p in this example will then be the one associating with each x ∈ {0; 1; : : : ; 20} the median m(x) of the distribution x , which can be obtained simply

M. Lopez-Daz, M.A. Gil / Journal of Statistical Planning and Inference 74 (1998) 11–29

23

by using most statistical packages to get the inverse beta distribution function in 0:5 for each x. On the other hand, regarding the hypothesis testing problem we have that Theorem 4.6. Let X ≡ (X; BX ; P ),  ∈ , be a random experiment and assume that  is an unknown parameter, a state of nature, or a subindex. Consider the hypotheses Hi :  ∈ i ; i = 1; : : : ; k, where { 1 ; : : : ; N } is a partition of by elements of the -eld E. Let A = {a1 ; : : : ; ak }, with ai = acceptance of the hypothesis Hi (i = 1; : : : ; k), and L : × A → Fc (R) be a fuzzy loss function (which will be intended as a mapping such that for all ai the projection Lai is an integrable bounded fuzzy random variable with respect to the probability space ( ; E; x ) for each x ∈ X). Then, if L : × X → Fc (R) such that L (; x) = L(; (x)) satises that L ∈0 ( × X; E ⊗ BX ; P) for all tests (X ) of , then the test B (X ) such that for each   (x) = x ) for all i ∈ {1; : : : ; k} is a Bayes test for this problem.  ai = x ) ¿ E(L x ∈ X; E(L B This result can be illustrated by means of the following example (see Gil et al., 1998, for the detailed calculations): Example 4.5. A neurologist has to test the following hypotheses regarding his most serious patients: hypothesis H1 : ‘the patient requires exploratory brain surgery’, hypothesis H2 : ‘the patient requires a preventive nonsurgical treatment with drugs’, and hypothesis H3 : ‘the patient does not require either treatment or surgery’. Obviously, the losses of right classications are null. The losses of wrong classications are diverse: an unnecessary operation means resources are wasted and the health of the patient may be prejudiced; a preventive treatment means superuous expenses and possible side eects, if the patient does not require either preventive treatment or surgery, and may be insucient if the surgery is really required; if a patient requiring surgery does not get it on time and no preventive treatment is applied, the time lost until clear symptoms appear may be crucial. Assume that, because of the imprecise nature of the hypotheses Hi and states i (which in this example coincide for each i ∈ {1; 2; 3}), the neurologist considers the following fuzzy loss function: L(1 ; a1 ) = L(2 ; a2 ) = L(3 ; a3 ) = 0; L(1 ; a2 ) = VERY DANGEROUS; L(2 ; a1 ) = INCONVENIENT; L(3 ; a1 ) = EXCESSIVE;

L(1 ; a3 ) = EXTREMELY DANGEROUS;

L(2 ; a3 ) = DANGEROUS;

L(3 ; a2 ) = UNSUITABLE:

If the preceding fuzzy values are expressed in terms of fuzzy numbers as, for instance, L(2 ; a1 ) = (0:1; 0:6), L(3 ; a1 ) = (0:1; 0:7) ( being the well-known PI

24

M. Lopez-Daz, M.A. Gil / Journal of Statistical Planning and Inference 74 (1998) 11–29

Fig. 2. Fuzzy losses of wrong classications in Example 4.5.

curve – cf. Zadeh, 1976; Cox, 1994), ⎧ 1 − 12(t − 1)2 ⎪ ⎪ ⎨ L(2 ; a3 )(t) = 20t 2 − 24t + 7 ⎪ ⎪ ⎩ 0 ⎧ 2 5t − 8t + 3 ⎪ ⎪ ⎨ L(3 ; a2 )(t) = 1 − 3t 2 ⎪ ⎪ ⎩ 0

if t ∈ [0:75; 1]; if t ∈ [0:7; 0:75]; otherwise;

if t ∈ [0:5; 0:6]; if t ∈ [0; 0:5]; otherwise

and L(1 ; a2 ) and L(1 ; a3 ) are both obtained from L(2 ; a2 ) by applying the linguistic modiers VERY and EXTREMELY (cf. Zadeh, 1976; Cox, 1994), that is L(1 ; a2 ) = [L(2 ; a3 )]2 and L(1 ; a3 ) = [L(2 ; a3 )]3 (see Fig. 2). In practice, the neurologist should not base his decision only on the prior information, but he must revise it in view of sample information concerning the patient to be classied. In particular, assume that the neurologist can base his decision on a score X obtained from a combination of several clinical tests. Suppose that past experiences have shown that X is normally distributed with mean 120 and variance 64 for those who require surgery, with mean 110 and variance 64 for those who require treatment, and mean 100 and variance 64 for those who do not require either treatment with drugs or surgery. If we consider the hypothesis testing problem with the preceding fuzzy losses, then for each x ∈ R the posterior distributions x are given by

x (1 ) = [1 + 0:6 e−20(x−115)=128 + 0:4 e−40(x−110)=128 ]−1 ;

x (2 ) = [1 + 1:666667 e20(x−115)=128 + 1:333333 e−20(x−105)=128 ]−1 ;

x (3 ) = [1 + 2:5 e40(x−110)=128 + 1:5 e20(x−105)=128 ]−1

M. Lopez-Daz, M.A. Gil / Journal of Statistical Planning and Inference 74 (1998) 11–29

25

and, if we consider the ranking ¿0:5 , a Bayes test will be that accepting H3 if x695:9840, H2 if 95:9840¡x6109:7764, and H1 otherwise. In Gil and Lopez-Daz (1996) and Gil et al. (1998) we can see that although the estimation and testing problems reduce to the Bayesian ones with the real-valued loss function VL ◦ L, the ‘value’ of the statistical problems will be essentially fuzzy. Thus, fuzziness is temporarily ‘put a side’ to establish a crisp ranking of the fuzzy expected losses associated with estimates or test conclusions, but fuzziness is nally recovered to state the value of the problem. 5. Concluding remarks The studies in Section 4.1 can be enlarged by comparing the central -mean-squared deviation (see Lubiano et al., 1998) associated with the fuzzy unbiased estimators dened as the expectations of another fuzzy unbiased estimator, with the central mean-squared deviation of the latter. This enlargement will mean the extension of the main objective of the Rao–Blackwell Theorem. On the other hand, the extension of several well-known statistical results and applications derived from the double expectation theorem (see, for instance, Casella and Berger, 1990) can be also examined. Acknowledgements The research in this paper was supported in part by DGES Grant No. PB95-1049. Their nancial support is gratefully acknowledged. The authors are also grateful to the referee and the Associate Editor dealing with the rst version of this paper, as well as to the Executive Editor for their helpful suggestions. Appendix In this section proofs of the main results in Sections 3 and 4 are gathered. Proof of Lemma 3.1. For all  ∈ [0; 1], and for all !1 ∈ 1 , we have that the !1 projection of Gr(X ), (Gr(X ))!1 , is given by (Gr(X ))!1 = {(!1 ; !2 ; x) ∈ 2 × Rn : x ∈ (X (!1 ; !2 )) }!1 = {(!2 ; x) ∈ 2 × Rn : x ∈ (X!1 (!2 )) } = Gr((X!1 ) ): Since X ∈ (1 × 2 ; a1 ⊗ a2 ; P), then (Gr(X ))!1 = Gr((X!1 ) ) ∈ a2 ⊗ BRn : On the 1 other hand, there exists h ∈ 0 (P) such that for all (!1 ; !2 ) ∈ 1 × 2 sup x∈(X!1 (!2 ))0

|x|6h!1 (!2 );

where h!1 ∈ L1 (P!1 ) for all !1 ∈ 1 , which gives the result.

26

M. Lopez-Daz, M.A. Gil / Journal of Statistical Planning and Inference 74 (1998) 11–29

Proof of Proposition 3.2. Since X is a simple fuzzy random variable, it can be exm m  pressed as X (· ; ·)= ] i=1 ICi (· ; ·)  Vi , where {Ci }i=1 ⊂ a1 ⊗ a2 is a partition of 1 × 2 , m n i } ⊂ Fc (R ). and {V i=1 m  Then, we have that X!1 (·) = ] i=1 ICi!1 (·)  Vi for all !1 ∈ 1 : In virtue of Lemma 3.1, X!1 ∈ (2 ; a2 ; P!1 ), and it is a simple fuzzy random variable whose fuzzy expected value can be expressed as  !1 =P!1 ) = v1 (!1 ) = E(X

m ] 

i : P!1 (Ci!1 )  V

i=1

Since P!1 is a regular conditional probability, then the mapping mB : 1 → R with mB (!1 ) = P!1 (B!1 ) is a classical random variable for all B ∈ a1 ⊗ a2 (see, for instance, Ash, 1975), whence the mappings pi : 1 → R such that pi (!1 ) = P!1 (Ci!1 ) are L1 (P1 )random variables, and hence the mappings Vi 1 : 1 → Fc (Rn ) with Vi 1 (!1 ) = P!1 (Ci!1 ) i belong to (1 ; a1 ; P1 ); i = 1; : : : ; m: V In accordance with Lemma 2.1  !1 =P2 ) = v1 (!1 ) = E(X

m ] 

i ∈  (1 ; a1 ; P1 ): P2 (Ci!1 )  V

i=1

Proof of Theorem 3.4. In accordance with Lemma 3.3, there exists a sequence of sim1 ple fuzzy random variables {Xm }m ⊂ 0 (1 × 2 ; a1 ⊗ a2 ; P) and a mapping h ∈ 0 (P) such that for all (!1 ; !2 ) ∈ 1 × 2 ; m¿1, sup

|x|6h(!1 ; !2 )

x∈(Xm (!1 ; !2 ))0

and a.s. [P]: lim dH ((Xm (!1 ; !2 )) ; (X (!1 ; !2 )) ) = 0

m→∞

for all  ∈ (0; 1]:

Then, for all  ∈ (0; 1] lim dH ((Xm!1 (!2 )) ; (X!1 (!2 )) )

m→∞

a:s: [P!1 ]

=

0

a:s: [P1 ]

and supx∈(Xm! (!2 ))0 |x|6h!1 (!2 ) with h!1 ∈ L1 (P!1 ): 1 On the basis of Lemma 2.2  m!1 =P!1 )) ; (E(X  !1 =P!1 )) ) = 0 lim dH ((E(X

m→∞

for all  ∈ (0; 1] a:s: [P1 ];

that is, 1 (!1 )) ; (v1 (!1 )) ) = 0 lim dH ((vm

m→∞

for all  ∈ (0; 1] a:s: [P1 ];

 m!1 =P!1 ), so that for all x ∈ Rn where vm (!1 ) = E(X 1 lim d(x; (vm (!1 )) ) = d(x; (v1 (!1 )) )

m→∞

for all  ∈ (0; 1] a:s: [P1 ]

(d being dened in terms of the Euclidean norm in R).

M. Lopez-Daz, M.A. Gil / Journal of Statistical Planning and Inference 74 (1998) 11–29

27

Since (1 ; a1 ; P1 ) is a complete probability space and vm ∈(1 ; a1 ; P1 ), for all 1 (!1 )) ) are a1 -measurable, and hence for all x ∈ Rn x ∈ Rn the mappings !1 → d(x; (vm 1 the mapping !1 → d(x; (v (!1 )) ) is also a1 -measurable (cf. Hiai and Umegaki, 1977), whence v1 is a fuzzy random variable. Furthermore, v1 is integrably bounded, since supx∈(X! (!2 ))0 |x|6h!1 (!2 ) for all 1 (!1 ; !2 ) ∈ 1 × 2 and, consequently,  sup |x|6 h!1 (!2 ) dP!1 = g(!1 )  x∈

2

2

(X!1 )0 dP!1

with g ∈ L1 (P1 ), whence v1 ∈  (1 ; a1 ; P1 ): Proof of Proposition 3.6. Since X is a simple fuzzy random variable, then for all !1 ∈ 1 we have that  !1 =P!1 ) = v1 (!1 ) = E(X

m ] 

i ; P!1 (Ci!1 )  V

i=1

whence, because of the linearity of the expected value, and on the basis of Lemma 4.1,  1 =P1 ) = E(v

m ] 

 !1 (Ci!1 )  V i =P1 ) = E(P

i=1

m ] 

i : E(P!1 (Ci!1 )=P1 )  V

i=1

In a similar way, we have that  2 =P2 ) = E(v

m ] 

i E(P!2 (Ci!2 )=P2 )  V

i=1

and since X is a simple fuzzy random variable  E(X=P) =

m ] 

i : P(Ci )  V

i=1

Furthermore, Ci ∈ a1 ⊗ a2 , so that P(Ci ) = E(P!1 (Ci!1 )=P1 ) = E(P!2 (Ci!2 )=P2 ); which proves the result. Proof of Theorem 3.7. Indeed, let {Xm }m ⊂ 0 (1 × 2 ; a1 ⊗ a2 ; P1 ⊗ P2 ) be a sequence of simple fuzzy random variables satisfying the conditions indicated in Lemma 3.3. 1 (!1 ) = If we consider the mappings vmj : j → Fc (Rn ); j = 1; 2; m¿1, with vm j 2   E(Xm!1 =P!1 ) and vm (!2 ) = E(Xm!2 =P!2 ), then (see Theorem 3.4) vm ∈(j ; aj ; Pj ) and v j ∈(j ; aj ; Pj ); j = 1; 2: On the other hand, and by following the proof of Theorem 3.4 we have that 1 lim dH ((vm (!1 )) ; (v1 (!1 )) ) = 0

m→∞

for all  ∈ (0; 1] a:s: [P1 ]

28

M. Lopez-Daz, M.A. Gil / Journal of Statistical Planning and Inference 74 (1998) 11–29

and

 sup

|x|6

1 (! )) x∈(vm 1 0

h!1 (!2 ) dP!1 = g(!1 ) ∈ L1 (P1 ); 2

whence for all  ∈ (0; 1] 1  m  1 =P1 )) ) = 0: lim dH ((E(v =P1 )) ; (E(v

m→∞

In a similar way, we have for all  ∈ (0; 1] that 2  m  2 =P2 )) ) = 0 =P2 )) ; (E(v lim dH ((E(v

m→∞

and, analogously,  m =P)) ; (E(X=P))  lim dH ((E(X  ) = 0:

m→∞

Since Xm are simple fuzzy random variables, we have that for all  ∈ (0; 1] 1 2  m  m  m =P)) ; (E(v =P1 )) = (E(v =P2 )) = (E(X

whence  1 =P1 ) = E(v  2 =P2 ) = E(X=P)  E(v ; which proves the result. Proof of Theorem 4.1. Indeed, in accordance with Theorem 3.7, we have that  E(X  x˜∗ =Px˜)=PX ) = E(  E(X  y∗˜ =Py˜ )=PY ): E( On the other hand,  E(X  x˜∗ =Px˜ )=PX ) = E(X=P  E( X)  ∗ =Py˜ ), then  ∗ =Y = y) and if E(X ˜ denotes E(X y˜ y˜    ∗  E(X=Y  E(X=P ˜ Y ) = E( )=P) X ) = E(E(Xy˜ =Y = y)=P which proves the result. References Ash, R., 1975. Real Analysis and Probability. Academic Press, New York. Aumann, R.J., 1965. Integrals of set-valued functions. J. Math. Anal. Appl. 12, 1–12. Breiman, L., 1968. Probability. Addison-Wesley, Reading, MA. Brown, L.D., Purves, R., 1973. Measurable selection of extrema. Ann. Statist. 1, 902–912. Byrne, C., 1978. Remarks on the set-valued integrals of Debreu and Aumann. J. Math. Anal. Appl. 78, 243–246. Campos, L.M., Gonzalez, A., 1989. A subjective approach for ranking fuzzy numbers. Fuzzy Sets and Systems 29, 145–153.

M. Lopez-Daz, M.A. Gil / Journal of Statistical Planning and Inference 74 (1998) 11–29

29

Casella, G., Berger, R.L., 1990. Statistical Inference. Wadsworth & Brooks=Cole, Pacic Grove. Castaing, C., 1969. Le theoreme de Dunford-Pettis generalise. C. R. Acad. Sci. Paris, Ser. A 268, 327–329. Cox, E., 1994. The Fuzzy Systems Handbook. Academic Press, Cambridge. Debreu, G., 1966. Integration of correspondences. Proc. 5th Berkeley Symp. on Math. Statist. and Probability, vol. II, Part I, pp. 351–372. Dudewicz, E.J., Mishra, S.N., 1988. Modern Mathematical Statistics. Wiley, New York. Gil, M.A., Lopez-Daz, M., 1996. Fundamentals and Bayesian Analyses of decision problems with fuzzyvalued utilities. Int. J. Approx. Reason. 15, 203–224. Gil, M.A., Lopez-Daz, M., Rodrguez-Mu˜niz, L.J., 1998. An improvement of a comparison of experiments in statistical decision problems with fuzzy utities. IEEE Trans. Systems. Man, Cybernet., in press. Hiai, F., Umegaki, H., 1977. Integrals, conditional expectations, and martingales of multivalued functions. J. Multivariate Anal. 7, 149–182. Himmelberg, C.J., 1975. Measurable relations. Fund. Math. 87, 53–72. Lopez-Daz, M., Gil M.A., 1997. Constructive denitions of fuzzy random variables. Statist. Probab. Lett. 36, 135–143. Lopez-Daz, M., Gil, M.A., 1998. The -average value of the expected value of a fuzzy random variables. Fuzzy Sets and Systems, in press. Lubiano, M.A., Gil, M.A., Lopez-Daz, M., Lopez, M.T., 1998. The -mean squared dispersion associated with a fuzzy random variable. Fuzzy Sets and Systems, in press. Puri, M.L., Ralescu, D.A., 1986. Fuzzy random variables. J. Math. Anal. Appl. 114, 409 – 422. Stojakovic, M., 1992. Fuzzy conditional expectation. Fuzzy Sets and Systems 52, 53– 60. Thompson, S.K., 1992. Sampling. Wiley, New York. Zadeh, L.A., 1975. The concept of a linguistic variable and its application to approximate reasoning. Inform. Sci. Part 1 8, 199 –249; Part 2 8, 301–353; Part 3 9, 43–80. Zadeh, L.A., 1976. A fuzzy-algorithmic approach to the denition of complex or imprecise concepts Int. J. Man-Mach. Stud. 8, 249–291.