An Uniformly Minimum Variance Unbiased Point Estimator Using Fuzzy Observations

This paper proposes a new method for uniformly minimum variance unbiased fuzzy point estimation. For this purpose we make use of a uniformly minimum variance unbiased estimator and we develop this new method for a fuzzy random sample ~X1,...,~Xn  is induced by  X1,,...,Xn  onthe same probability space.


Introduction
Statistical analysis in traditional form is based on crispness of data, random variables (RV's), point estimations, hypotheses, parameters, and so on.But there are many other situations in which the above mentioned concepts are imprecise.The point estimation approaches are frequently used in statistical inference.On the other hand, the theory of fuzzy sets is a well known tool for formulation and analysis of imprecise and subjective concepts.Therefore, the uniformly minimum variance unbiased fuzzy estimator (UMVUFE) with fuzzy data can be important.The problem of point estimation for an unknown parameter, using fuzzy data, is developed in different approaches.Kruse (1984) and Kruse and Meyer (1987) explained some methods for point and interval estimation for examples of fuzzy random variables (FRV's).Buckley (1983) studied the problem of estimation, with fuzzy data by a fuzzy decision making approach.Okuda (1987) considered fuzzy observations to estimate moments and parameters and he discusses maximum likelihood estimators and the loss of information due to fuzziness.Viertl (1996) studied nonparametric methods in estimation using fuzzy data.Lopez-Diaz and Gil (1998) derived some statistical inference methods, and studied their applications specially in statistical decision theory with fuzzy losses and fuzzy utilities.Cai et al. (1991) proposed a method for estimating parameters of membership functions through defining a likelihood function and studied its applications in fuzzy software reliability modelling.Cai (1993) discussed parameter estimation methods for normal membership functions.Lubiano et al. (1999) and Sadeghpour and Gien (2002) studied a Rao-Blackwell type theorem for FRV's.Lopez-Diaz and Gil (1998) defined a fuzzy unbiased estimator of the sample mean in random sampling with replacement from a finite population.Garcia Austrian Journal of Statistics, Vol. 36 (2007), No. 4, 307-317 et al. (2001) illustrated estimating the expected value of a FRV in the stratified random sampling.Some methods of statistical inference with fuzzy data are reviewed by Viertl (2002aViertl ( , 2002b)).
There are some researches regarding the Bayesian point estimation methods combined with ideas from fuzzy set theory.Hryniewicz (2002) proposed the notion of fuzzy Bayes point estimation for fuzzy data.Gertner and Zhu (1997), based on two extensions of likelihood function, generalized Bayesian estimates for use when sample information and prior distribution are fuzzy.They applied their method to forest survey.Uemura (1991Uemura ( , 1993) ) formulated the fuzzy Bayes decision rule to facilitate determination of the loss function of a Bayes decision rule in a fuzzy environment.Wu (2003) studied fuzzy estimators of fuzzy parameters based on FRV's.Finally, Hong-Zhong et al. (2006) proposed a new method to determine the membership function of the parameter estimate of a multi-parameter distribution.This method can be used to determine the membership function of Bayesian estimates of a multi-parameter distribution.
In this paper we organize the matter in the following way.In Section 2 we describe some basic concepts of canonical fuzzy numbers and FRV's.Also, we apply the ranking fuzzy numbers based on signed distance (Yao and Wu, 2000) between fuzzy canonical numbers.In Section 3 we summarize the research results report in the literature on uniformly minimum variance unbiased estimators (UMVUE's).Section 4 is devoted to describe UMVUE's using fuzzy data.Finally, some numerical examples are presented in Section 5 in order to illustrate our proposed method.

Preliminaries
Let (Ω, F, P ) be a probability space.A RV X is a measurable function from (Ω, F, P ) to (X , B, P X ), where P X is the probability measure induced by X and is called the distribution of X, i.e., If P X is dominated by a σ−finite measure υ, then by the Radon-Nikodym theorem (see Billingsley, 1995) we have where f (x|θ) is the Radon-Nikodym derivative of P X with respect to υ and is called probability density function (PDF) of X with respect to υ.In a statistical context the measure υ is usually a "counting measure" or a "Lebesgue measure", hence P X (A) is calculated by x∈A f (x|θ) or A f (x|θ)dx, respectively.Let S X = {x ∈ X |f (x|θ) > 0} be the "support" or "sample space" of X, then a fuzzy subset x of S X is defined by its membership function µ x : S X → [0, 1].We denote by xα = {x : µ x(x) ≥ α} the α-cut set of x and x0 is the closure of the set {x : µ x(x) > 0}.x is called normal fuzzy set if there exist x ∈ S X such that µ x(x) = 1 and is called convex fuzzy set if The fuzzy set x is called a fuzzy number if x is a normal convex fuzzy set and its α-cut set is bounded for all α = 0. x is called a closed fuzzy number if x is a fuzzy number and its membership function µ x is upper semicontinous.x is called a bounded fuzzy number if x is a fuzzy number and its membership function µ x has compact support.
If x is a closed and bounded fuzzy number with x L α = inf{x : x ∈ xα } and x U α = sup{x : x ∈ xα } and its membership function is strictly increasing on [x L α , x L 1 ] and strictly decreasing on [x U 1 , x U α ], then x is called canonical fuzzy number.Given a real number x ∈ S X , we can induce a fuzzy number x with membership function µ x(r) such that µ x(x) = 1 and µ x(r) < 1 for r = x.Let X be a RV with support S X and F(S X ) be a set of all fuzzy real numbers induced by the real numbers S X .If Definition 2.2 Let X be a RV with cumulative distribution function (CDF) F (x) and let X be a FRV induced by X.The fuzzy function F (x) is called fuzzy cumulative distribution function of the FRV X, whenever its membership function equals where ] contains all of the CDF of each x L β and x U β for β ≥ α.To assert the fuzzy expectation of a FRV X, we first define E L α ( X) and E U α ( X) as ] will contain all of the expectations of each RV X L β and X U β for β ≥ α.
Definition 2.3 Let X be a FRV induced by X.Then the fuzzy expectation of X is a fuzzy number Ẽ( X) with membership function Several ranking methods have been proposed so far by Cheng (1998), Modarres and Sadi-Nezhad (2001), and Nojavan and Ghazanfari (2006).Here we use another ranking system for canonical fuzzy numbers which is very realistic and is defined by Yao and Wu (2000) as follows.d(ã, b) denotes the distance between ã and b.

The UMVUE with Crisp Data
In this section, we describe concepts of UMVUE's with crisp data.Let X 1 , . . ., X n be a random sample of size n, where the X i 's have PDF f (x|θ) with unknown parameter θ, θ ∈ Θ, and x 1 , . . ., x n are realizations of X 1 , . . ., X n , respectively.Recall that an estimator T (X 1 , . . ., X n ) of θ is unbiased iff E[T (X 1 , . . ., X n )] = θ for any θ ∈ Θ.If an unbiased estimator of θ exists, then θ is called an estimable parameter.
The derivation of such an UMVUE is relatively simple if there exist a complete sufficient statistic for θ ∈ Θ.
Theorem 3.1 (Lehmann-Scheffé) Suppose that there exists a complete sufficient statistic T (X 1 , . . ., X n ) for θ ∈ Θ.If θ is estimable, then there exists an unique unbiased estimator of g(θ) of the form h(T ) with a Borel function h.Furthermore, h(T ) is the unique UMVUE of θ.
There are two typically ways to derive a UMVUE when a complete sufficient statistic is available.The first one is to solve for h when the distribution of T is available.The second method to derive an UMVUE when there exists a complete sufficient statistic (used here), is to condition on T , i.e., if U (X 1 , . . ., X n ) is an unbiased estimator of θ, then E[(U (X 1 , . . ., X n )|T )] is the UMVUE of θ.To apply this method, we do not need the distribution of T but need to work out the conditional expectation E[U (X 1 , . . ., X n |T )].
From the uniqueness of the UMVUE, it does not matter which U (X 1 , . . ., X n ) is used.We should choose U (X 1 , . . ., X n ) to make the calculation of E[U (X 1 , . . ., X n )|T ] as easy as possible.For a review in more details, see Shao (2003).

An UMVUE with Fuzzy Data
Now we introduce concepts of an UMVUFE.Definition 4.1 Let X and Ỹ be two FRV's induced by X and Y .We say that X and Definition 4.2 We say that X and Ỹ are identically distributed iff X L α , Y L α are identically distributed, and Let X = ( X1 , . . ., Xn ) be a fuzzy random sample induced by X = (X 1 , . . ., X n ) on the sample probability space (Ω, F, P ), and with membership functions µ Xi (y).
In traditional statistics, parameter estimates are functions of the sample.Formally, based on the sample X for a RV X with sample space S X , PDF f (x|θ) and parameter space Θ, estimators are functions δ(X) defined on the sample space S X 1 × • • • × S Xn .For a realized sample x = (x 1 , . . ., x n ) an estimation θ of the parameter θ is obtained by For a fuzzy random sample X induced by X with membership functions µ Xi (y) the function δ(x), where x = (x 1 , . . ., xn ), becomes a fuzzy element θ ∈ Θ, whose membership function µ δ(˜x) is derived by applying the extension principle (Klir and Yuan, 1995), i.e.
For a fuzzy random sample X, the fuzzy number δ( X) is called a fuzzy statistic, provided that it is not a function of any unknown parameter.Let Xi and X i have CDF F and F , respectively.A fuzzy statistic δ( X) is said to be a (point) estimator of θ, if δ : where the interval Definition 4.3 The fuzzy estimator δ( X) is unbiased for θ iff for all α ∈ [0, 1] and there exists a θ 0 ∈ θα such that E(δ 0 ) = θ 0 .Let D be a nonempty set of all fuzzy unbiased estimators δ( X) for θ.
Theorem 4.1 Let X = (X 1 , . . ., X n ) be a random sample and X = ( X1 , . . ., Xn ) be a fuzzy random sample induced by X.If Ũ ( X) is a fuzzy unbiased estimator for θ and T ( X) a fuzzy sufficient statistics for θ, then the UMVUFE δ * ( X) has membership function in which Proof: We first prove that δ * is a canonical fuzzy number.Define S α = {(x 1 , . . ., x n ) : We know that xi 's are canonical fuzzy numbers, k is continuous, S α is connected, closed, and bounded implying that the range of k is a closed and bounded interval of real numbers.Define δ * )].Now we know that Ũ ( X) is an unbiased fuzzy estimator.By definition 4.3 there exists a θ 0 ∈ θα such that E[δ * α (X)] = E[U (X)] = θ 0 .As a result, δ * ( X) is a fuzzy unbiased estimator.

Numerical Examples
Now we illustrate the proposed approach for some distributions.
Example 5.1 Let X be a RV from a N (θ, 1) population, i.e.
Consider x1 , . . ., xn with triangular membership functions (x i − a, x i , x i + b) given by for each 1 ≤ i ≤ n and a, b ≥ 0. We can interpret the canonical fuzzy numbers xi as the values of "near to x i ".We have , and we note that in crisp form for this example the UMVUE is δ * (X) = X and also µ δ * (x) (x) = 1.We derive and let xi be some fuzzy observations with membership functions µ xi (y) = exp(−(y − x i ) 2 ) , x i − 0.5 ≤ y ≤ x i + 0.5 0 , otherwise for each 1 ≤ i ≤ n.We can interpret the canonical fuzzy numbers xi as the values of "near to x i ".We have such that We note that in crisp form for this example the UMVUE is δ * (X) = X (1) − 1/n and also and xi are fuzzy observations with triangular membership functions (x i − a, x i , x i + b).
We have Example 5.5 Let X be a RV with PDF Let x1 and x2 be two fuzzy canonical numbers with triangular membership functions (x i − a , x i , x i + b ), then we have Furthermore, the parameter estimate at any α-cut level can be calculated with respect to membership function µ δ * (x) (y) = sup 0≤α≤1 αI [E L α ,E U α ] (y) .

Conclusion
From the above discussion, it is clear that under Yao-Wu signed distance, the UMVUF estimator propose a canonical fuzzy number.The UMVUF estimator has the smallest fuzzy variance of the other fuzzy estimator for fuzzy parameter θ.