Fuzzy hypothesis testing with vague data using

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Jan 8, 2013 - H: h ¼ h0 against any one-sided alternative hypothesis. (Lehmann and Romano 2005). To elucidate likelihood ratio test, suppose Xi; i ¼.
Soft Comput (2013) 17:1629–1641 DOI 10.1007/s00500-012-0977-3

METHODOLOGIES AND APPLICATION

Fuzzy hypothesis testing with vague data using likelihood ratio test H. Moheb Alizadeh • A. R. Arshadi Khamseh S. M. T. Fatemi Ghomi



Published online: 8 January 2013  Springer-Verlag Berlin Heidelberg 2013

Abstract Hypothesis testing is one of the most significant facets of statistical inference, which like other situations in the real world is definitely affected by uncertain conditions. The aim of this paper is to develop hypothesis testing based on likelihood ratio test in fuzzy environment, where it is supposed that both hypotheses under study and sample data are fuzzy. The main idea is to employ Zadeh’s extension principle. In this regard, a pair of non-linear programming problems is exploited toward obtaining membership function of likelihood ratio test statistic. Afterwards, the membership function is compared with critical value of the test in order to assess acceptability of the fuzzy null hypothesis under consideration. In this step, two distinct procedures are applied. In the first procedure, a ranking method for fuzzy numbers is utilized to make an absolute decision about acceptability of fuzzy null hypothesis. From a different point of view, in the second procedure, membership degrees of fuzzy null hypothesis acceptance and rejection are first derived using resolution identity and then, a relative decision is made on fuzzy null hypothesis acceptance or rejection based on some arbitrary decision

Communicated by V. Loia. H. Moheb Alizadeh Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27695-7906, USA A. R. Arshadi Khamseh Department of Industrial Engineering, College of Engineering, Kharazmi University, 15719-14911 Tehran, Iran S. M. T. Fatemi Ghomi (&) Department of Industrial Engineering, Amirkabir University of Technology, 15916-34311 Tehran, Iran e-mail: [email protected]

rules. Flexibility of the proposed approach in testing fuzzy hypothesis with vague data is presented using some numerical examples. Keywords Hypothesis testing  Likelihood ratio test  Fuzzy random variables  Extension principle  Resolution identity

1 Introduction One of the primary purposes of statistical inference is testing hypothesis. A statistical hypothesis is a statement about population (populations) from which either one or more samples are drawn. The under test hypothesis H is called null hypothesis. A statistical procedure enabling one to accept or reject the null hypothesis H is called a test. If null hypothesis is rejected, then the alternative hypothesis K is accepted. A predetermined upper bound for the type I error that is probability of null hypothesis rejection when it is correct, is called the significance level d. If the decisions related to acceptance and rejection of null hypothesis are denoted by d0 and d1 , respectively, a non-random test assigns either d0 or d1 to each possible value x of the random variable X and hence divides the sample space into two supplement regions s0 and s1 . If x lies within the region s0 , null hypothesis is accepted; otherwise, is rejected. Therefore, the regions s0 and s1 are called acceptance and rejection (critical) regions, respectively (Lehmann and Romano 2005). In the classic approaches to hypothesis testing, all aforementioned concepts are assumed to be crisp, whereas in the real world there are many situations in which data cannot be gathered or recorded precisely. For example, suppose that h (the interested parameter) is the water level

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of a river. In order to have some ideas about h, we may measure water level in successive days. In commonplace hypothesis testing, a practitioner applies the null hypothesis ‘H: h ¼ 27 m’ against the alternative hypothesis ‘K: h  27 m’. However, it is known that the water level of a river cannot be measured in an exact way because of the fluctuation. In this case, a more realistic expression about h would be stated as ‘H: h is approximately equal to 27 m’ against ‘K: h is approximately greater than or equal to 27 m’. Hence, the methods taking into account statistical hypothesis testing in such uncertain conditions can be so worthwhile. One way for dealing with such a condition is to use the fuzzy set theory initially proposed by Zadeh (1965). In the literature, there are some efforts considering statistical hypothesis testing in fuzzy environment. Arnold (1995, 1998) proposed fuzzification of usual statistical hypothesis and considered hypothesis testing under fuzzy constraints on the type I and type II errors. Saade (1994) considered the binary hypothesis testing and discussed fuzzy likelihood function in the decision-making process applying a fuzzified version of the Bayes criterion. Grzegorzewski (2000) proposed a fuzzy test for hypothesis testing which gave the acceptability of null and alternative hypotheses. Defining the degrees of optimism and pessimism and solving optimization problems, Wu (2005) studied fuzzy hypothesis testing with vague data. Torabi and Behboodian (2007) studied likelihood ratio test for testing fuzzy hypothesis with crisp data based on whether fuzzy hypotheses are supplement to each other. In a practical context, Moheb Alizadeh et al. (2010) applied likelihood ratio test to develop multivariate variable control charts in fuzzy environment, where in control and out of control states of the process are expressed by fuzzy hypotheses. Other approaches in this area can be found in Buckley (2005), Taheri and Arefi (2009), Filzmoser and Viertl (2004), Torabi et al. (2006), Taheri and Behboodian (2001) and Kahraman et al. (2004). In statistics, likelihood ratio test has resulted in practical procedures in a great variety of problems and is still playing a major role in the development of new tests and estimates. Specifically, it poses to have the highest power among all tests with the same value of the significance level d and in many cases is uniformly most powerful test for testing the hypothesis H: h ¼ h0 against any one-sided alternative hypothesis. This paper examines fuzzy statistical hypothesis testing with fuzzy data based on this type of major statistical approach. In this regard, using a-level cuts obtained by solving a pair of non-linear programming problems resulted from Zadeh’s extension principle, the proposed approach firstly gives the membership function of likelihood ratio test statistic. In the next step, it compares the membership function given with critical value of the

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test to determine acceptability of the fuzzy null hypothesis under study. Two different procedures are proposed to accomplish this goal. While the first procedure results in an absolute decision using a ranking method of fuzzy numbers, the second one leads to a relative decision utilizing the membership degrees of fuzzy null hypothesis acceptance and rejection computed from resolution identity, and arbitrary decision rules. In this case, status of the fuzzy null hypothesis under study is divided into some intermediate modes including ‘slightly’, ‘relatively,’ and ‘completely’ accepted or rejected. Hence, the second procedure provides more flexibility for further statistical analyses and makes status of fuzzy null hypothesis clearer using obtained intermediate modes. Moreover, since likelihood ratio test is the extended version of Neyman–Pearson lemma, the proposed approach in this paper can be also exploited to evolve this lemma in fuzzy environment. This approach also has two advantages rather than the study of Torabi and Behboodian (2007). First, it is postulated in the present study that sample data drawn from the population are also fuzzy. Second, the final decision on accepting or rejecting fuzzy null hypothesis can be made either absolutely or relatively. The rest of this paper is organized as follows: In Sect. 2, some preliminary concepts are briefly introduced. The proposed approach for testing fuzzy hypothesis with fuzzy data based on likelihood ratio test is presented in Sect. 3. Section 4 gives two procedures to evaluate acceptability of fuzzy null hypothesis. Two numerical examples are discussed in Sect. 5 and finally, Sect. 6 is devoted to concluding remarks.

2 Preliminaries 2.1 Likelihood ratio test Based on definition, a class of distributions is called ‘simple’ if and only if it contains only one distribution; otherwise, it is called ‘composite’. Inspired by this definition, a statistical hypothesis is called ‘simple’ if and only if it determines only one value for unknown parameter of the population, otherwise, is called ‘composite’. Neyman– Pearson lemma is applied to build up the most efficient critical region when both null and alternative hypotheses are simple (Mood et al. 1974). Nevertheless, this lemma falls through when at least one hypothesis is composite. In such a case, likelihood ratio test is used, which has the highest power among all tests with the same value of the significance level d. Moreover, in many cases, when the unknown parameter contains only one value, this test is uniformly most powerful test for testing the hypothesis

Fuzzy hypothesis testing with vague data using likelihood ratio test

H: h ¼ h0 against any one-sided alternative hypothesis (Lehmann and Romano 2005). To elucidate likelihood ratio test, suppose Xi ; i ¼ 1; . . .; n is a random sample of size n drawn from a population whose density function in x is f ðx; hÞ. In addition, X is assumed to be the set of all possible values that h can take on. The null and alternative hypotheses under consideration are ‘H: h 2 Xh ’ and ‘K: h 2 Xk ’ respectively, where Xh and Xk are the set of all values that h can take on under each corresponding hypothesis. In this way, kðh0 ; h00 ;x1 ;. . .; xn Þ ¼ max Lðh; x1 ; . ..; xn Þ= max Lðh; x1 ; . . .; xn Þ h2Xh 0

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~ is a fuzzy number, then its a-level cut all a 2 ½0; 1. If m is a closed interval for all a 2 ½0; 1 denoted by ~ La ; ðmÞ ~ U ~ a ¼ ½ðmÞ m a : Extension principle Assume X1 ; . . .; Xn and Y are ordinary (crisp) sets and f is a point mapping from ½X1 ; . . .; Xn  to Y. The extension principle implies that f can be extended to act on fuzzy subsets of X1 ; . . .; Xn such that ~ i ; i ¼ 1; . . .; n and B ~ are fuzzy subsets of X1 ; . . .; Xn and if A Y, respectively, then (Yager 1986; Zimmermann 1996): lB~ ðyÞ ¼ sup minflA~ i ðxi Þ; i ¼ 1; . . .; n jy ¼ f ðx1 ; . . .; xn Þg x

h2X

¼ Lðh ; x1 ; . . .;xn Þ=Lðh00 ; x1 ; . . .;xn Þ

ð1Þ

is a value of the likelihood ratio test statistic K, where h0 and h00 are the maximum likelihood estimates of the parameter h when it is restricted to be a member of Xh and X, respectively. It can be easily proved that 0  k  1. When the null hypothesis H is true, that is h 2 Xh , it is expected that the numerator approaches to the denominator and consequently k ! 1. On the other hand, when the null hypothesis H is not true, the numerator would be trivial in comparison with the denominator and as a result k ! 0. Therefore, based on likelihood ratio test, the null hypothesis H is rejected if and only if k  k. If H is simple, the value of k is determined in such a way that probability of type I error is equal to the significance level d, i.e. pðK  kj h 2 Xh Þ ¼ d; otherwise, it is determined such that this probability is less than or equal to d for all h 2 Xh and equal to d for at least one value of h 2 Xh . 2.2 Fuzzy numbers ~ of X is Let X be universal set. Then a fuzzy subset A defined by its membership function as lA~ ðxÞ : x ! ½0; 1. ~ is called normal fuzzy set if there exists x such that A ~ a ¼ fx : l ~ ðxÞ  ag as the a-level lA~ ðxÞ ¼ 1. We define A A ~ for a 2 ð0; 1 and A ~ 0 as the closure of the set cut of A ~ is a convex fuzzy set if and only if fx : lA~ ðxÞ [ ag. A fx : f ðxÞ  ag is a convex set for all a, i.e. its a-level cuts ~ are convex for all a 2 ð0; 1 (Zadeh 1965). Equivalently, A is a convex fuzzy set if and only if lA~ ðkx þ ð1  kÞyÞ  min flA~ ðxÞ; lA~ ðyÞg for all k 2 ½0; 1 and x; y 2 X. In this paper, the universal set X is assumed to be real number system, that is X ¼ R. If f is a real-valued function defined on R, then f is said to be upper semi continuous if fx; f ðxÞ  ag is a closed set for each a 2 ½0; 1 (Rudin 1986). ~ is called a fuzzy number if the folThe fuzzy set m ~ is a normal and lowing three conditions are satisfied: (a) m ~ is convex fuzzy set, (b) the membership function of m upper semi continuous and (c) its a-level cut is bounded for

ð2Þ ~ is a fuzzy set with the Resolution identity If A ~a ¼ membership function lA~ ðxÞ and the a-level cut A fx : lA~ ðxÞ  ag, the following relationship is known as Resolution identity (Zadeh 1975a, b, c): lA~ ðxÞ ¼ sup a IA~ a ðxÞ

ð3Þ

where  IA~ a ðxÞ ¼

0; 1;

~a x 62 A ~ x 2 Aa

ð4Þ

2.3 Fuzzy random variables Fuzzy random variables were introduced to model and analyze ‘imprecisely valued’ measurable functions associated with the sample space of a random experiment, when the imprecision in values of these functions is formalized using fuzzy sets. They have been considered in the setting of a random experiment to model (Gil et al. 2006): • •

either a fuzzy observation of a mechanism associating a real value with each experiment outcome, or an essentially fuzzy-valued mechanism, that is, a mechanism associating a fuzzy value with each experiment outcome.

A mathematical model for the first situation was first provided by Kwakernaak (1978, 1979), which was afterward elaborated clearly by Kruse and Meyer (1987). On the other hand, Puri and Ralescu (1986) proposed another approach for the second situation to be modeled. However, Gil et al. (2006) proved that the approach proposed by Kwakernaak (1978, 1979) and Kruse and Meyer (1987) is related to that provided by Puri and Ralescu (1986), i.e. ~ is fuzzy random variable in Kwakernaak/Kruse and X Meyer’s sense, if and only if it is a fuzzy random variable in Puri and Ralescu’s sense. In this paper, to model fuzziness in observations of the random sample which is applied to decide whether fuzzy null hypothesis is accepted, the first situation is utilized

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where a fuzzy random variable is viewed as a fuzzy perception/observation of a classical real-valued random variable stated as follows: Definition 2.1 (Gil et al. 2006) Given a probability space ~ : X ! FðRÞ is said to be fuzzy ðX; A; PÞ, a mapping X random variable if and only if for all a 2 ½0; 1, the two ~ L ! R and ðXÞ ~ U ! R are realreal-valued mapping ðXÞ a a ~ L ; ðXÞ ~ U  and ~ a ¼ ½ðXÞ valued random variables, where X a a FðRÞ denotes the set of all fuzzy numbers. Inspiring the abovementioned definition, the fuzzy ran~ n can be regarded as a fuzzy per~ 1 ; . . .; X dom sample X ception of the usual random sample X1 ; . . .; Xn , in which the observations X1 ; . . .; Xn are independent and identically distributed crisp random variables. To get an overview on the development of fuzzy random variables, the interested reader is referred to Gil et al. (2006). 2.4 Fuzzy hypothesis Each statistical hypothesis defined as ‘H: h is H(h)’ is called a fuzzy hypothesis where the term ‘H: h is H(h)’ asserts that h belongs to the fuzzy set H with the membership function H(h) (Taheri and Arefi 2009). It should be noted that the commonplace statistical hypothesis ‘H: h 2 H0 ’can be viewed as a fuzzy hypothesis with the following membership function:  0; h 62 H0 HðhÞ ¼ ð5Þ 1; h 2 H0 Therefore, the predominant point in fuzzy hypothesis testing is to test the fuzzy null hypothesis ‘H: h is H(h)’ against the fuzzy alternative one ‘K: h is K(h)’ based on either a fuzzy or a crisp random sample. The definition of simple and composite hypotheses can be extended to the fuzzy environment. In this case, the fuzzy hypothesis ‘H: h is approximately equal to h0 ’ can be regarded as a crisp simple hypothesis developed into fuzzy environment. Furthermore, the fuzzy hypotheses ‘H: h is not approximately equal to h0 ’, ‘H: h is approximately less than h0 ’ and ‘H: h is approximately greater than h0 ’ can be also viewed as ‘two-sided’, ‘left one-sided’ and ‘right onesided’ crisp composite hypotheses extended to fuzzy mode respectively.

H. Moheb Alizadeh et al.

other words, the fuzzy null hypothesis ‘H: h is H(h)’ against the fuzzy alternative hypothesis ‘K: h is K(h)’ is ~ n and based ~ 1 ; . . .; X tested using the fuzzy random sample X on likelihood ratio test. As mentioned before, the main idea is to apply Zadeh’s extension principle (Yager 1986; Zimmermann 1996). In Eq. (2), n membership functions are involved. Therefore, it is hardly possible to extract lB~ ðyÞ in an explicit form. According to this equation, lB~ ðyÞ is the minimum of lA~ i ðxi Þ; i ¼ 1; . . .; n. Hence, we need lA~ i ðxi Þ  a; i ¼ 1; . . .; n and at least one lA~ i ðxi Þ; i ¼ 1; . . .; n equal to a such that y ¼ f ðx1 ; . . .; xn Þ to satisfy lB~ ðyÞ ¼ a. Since all a-level cuts form a nested structure with respect to a (Zadeh 1978), lA~ i ðxi Þ  a and lA~ i ðxi Þ ¼ a have the same domain. It is sufficient to derive the left and right shape functions of lB~ ðyÞ to extract its membership function, which is equivalent to finding the lower bound ~ L and upper bound ðBÞ ~ U of the a-level cuts of lB~ ðyÞ. ðBÞ a a ~ U are the minimum and maximum of ~ L and ðBÞ Since ðBÞ a a f ðx1 ; . . .; xn Þ respectively, they can be expressed as ~ L ¼ minff ðx1 ; . . .; xn Þ j ð~xi ÞL  xi  ð~xi ÞU ; i ¼ 1; . . .; n g ðBÞ a a a ð6Þ ~ U ¼ maxff ðx1 ; . . .; xn Þ j ð~xi ÞL  xi  ð~xi ÞU ; i ¼ 1; . . .; ng ðBÞ a a a ð7Þ Likelihood ratio test statistic defined by Eq. (1) can be viewed as a function of the sample data x1 ; . . .; xn , h0 and h00 . Therefore according to Eqs. (6) and (7), application of Zadeh’s extension principle to statistic of likelihood ratio test for deriving its membership function results in the following optimization problems:  L ~ ðKÞa ¼ min kðh0 ; h00 ;x1 ;...;xn Þjð~xi ÞLa  xi i¼1;...;n  U ~0 L U L 0 0 00 00 00 U ~ ~ ~  ð~xi Þa ; ðh Þa  h  ðh Þa ; ðh Þa  h  ðh Þa ð8Þ  U ~ ðKÞa ¼ max kðh0 ;h00 ; x1 ;.. .; xn Þj ð~xi ÞLa 

xi i¼1;...;n

0 00 ~0 L ~0 U ~00 L ~00 U  ð~xi ÞU a ; ðh Þa  h  ðh Þa ;ðh Þa  h  ðh Þa

 ð9Þ

3 Likelihood ratio test for testing fuzzy hypothesis with fuzzy data

where, kðh0 ;h00 ;x1 ; ...;xn Þ is defined by Eq. (1). As mentioned before, all a-level cuts construct a nested structure with respect to a (Zadeh 1978), i.e. if 0\ a2 \a1  1, then ½ð~xi ÞLa1 ;ð~xi ÞU xi ÞLa2 ; ð~xi ÞU a1   ½ð~ a2 ; i ¼ 1;. ..;n, 0 L 0 U 0 L 0 U 00 L ~ ~ ~ ~ ~ ~ ½ðh Þ ;ðh Þ   ½ðh Þ ;ðh Þ  and ½ðh Þ ; ðh00 ÞU   ½ðh~00 ÞL ;

In this section, it is elaborated how to develop likelihood ratio test for testing fuzzy hypothesis, where the sample data are supposed to be fuzzy random variables. In the

ðh~00 ÞU a2 . Therefore, the feasible region defined by a1 in optimization problems (8) and (9) is smaller than ~ that defined by a2 . As a result, the a-level cuts of K

a1

123

a1

a2

a2

a1

a1

a2

Fuzzy hypothesis testing with vague data using likelihood ratio test

at a1 are contained in that defined by a2 , i.e. ~ L ;ðKÞ ~ U   ½ðKÞ ~ L ;ðKÞ ~ U . Consequently, the left and ½ðKÞ a1 a1 a2 a2 right shape functions are non-decreasing and non~ increasing, respectively, which assure the convexity of K. With this property, the left shape function LðKÞ and the right shape function RðKÞ can be derived by taking ~ L and ðKÞ ~ U , respectively. In this case, the inverse of ðKÞ a a the membership function lK~ ðkÞ is established from LðKÞ and RðKÞ as follows: 8 ~ L ~ L  k  ðKÞ ðKÞ < LðKÞ; a¼0 a¼1 L ~ ~ U ð10Þ lK~ ðkÞ ¼ 1; ðKÞ  k  ðKÞ a¼1 a¼1 : U U ~ ~ RðKÞ; ðKÞ  k  ðKÞ a¼1

~ L ðKÞ a

a¼0

~ U ðKÞ a

If and cannot be solved analytically, then ~ L and ðKÞ ~ U can be gathered at numerical solutions for ðKÞ a a different a-levels to approximate the shape of LðKÞ and RðKÞ. In order to derive the critical value k of the test in fuzzy environment, we need to employ the following proposition: ~ L and ðKÞ ~ U obtained from the optiProposition 3.1 ðKÞ a a mization problems (8) and (9) are real-valued random variables. Proof It is well known that a function of random variables is, in turn, a random variable (Mood et al. 1974). Now suppose X  ¼ ½x1 ; . . .; xn ; xnþ1 ; xnþ2 T is the optimal solution vector of optimization problem (8), where for integration, h0 ¼ xnþ1 and h00 ¼ xnþ2 . Due to feasibility condition, we have ð~xi ÞLa  xi  ð~xi ÞU a ; i ¼ 1; . . .; n þ 2 (note that the decision variables h0 and h00 have been represented as xnþ1 and xnþ2 , respectively). In the optimal solution, two different situations can happen: either the optimal decision variable xi hits one of its bounds, i.e. xi ¼ ð~xi ÞLa or ð~xi ÞU a , or it falls within its respective bounds, L   i.e. ð~xi Þa \xi \ð~xi ÞU a . In the first situation, we conclude xi is a real-valued random variable because, based on Definition 2.1, ð~xi ÞLa and ð~xi ÞU a are real-valued random variables. On the other hand, if the second situation happens, one a0 -level can be found in the interval ða; 1 such that xi ¼ ð~xi ÞLa0 or ð~xi ÞU a0 . This conclusion is derived from the con~ i ; i ¼ 1; . . .; n and that, for a\a0  1 vexity property of X L we have ½ð~xi Þa0 ;ð~xi ÞU xi ÞLa ;ð~xi ÞU i.e. a0 ½ð~ a ; i¼1;...;n, L L L U U U ð~xi Þa  ð~xi Þa0  ð~xi Þ1 and ð~xi Þ1  ð~xi Þa0  ð~xi Þa . Therefore, if ð~xi ÞLa \xi  ð~xi ÞL1 , we can find a specific value of a0 2 ða; 1 such that xi ¼ ð~xi ÞLa0 . On the other hand, there exists a specific value of a0 2 ða; 1 such that xi ¼ ð~xi ÞU a0 if U U  ð~xi Þ1  xi \ð~xi Þa . In this case, it is also observed that xi is a real-valued random variable. Since, either the first or the second situation occurs for any decision variable in the

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optimal solution vector X  , it is deduced that X  is a set of ~ L : Rnþ2 ! R is a real-valued random variables. Hence, ðKÞ a function of n þ 2 random variables and consequently is, in turn, a real-valued random variable. In the same way, it is ~ U : Rnþ2 ! R is a real-valued random concluded that ðKÞ a variable. h The following proposition implies that, in fuzzy envi~ is a fuzzy ronment, the likelihood ratio test statistic K random variable. ~ n is ~ 1 ; . . .; X Proposition 3.2 If the fuzzy random sample X used for testing the fuzzy null hypothesis ‘H: h is H(h)’ against the fuzzy alternative hypothesis ‘K: h is K(h)’ based on likelihood ratio test, the obtained likelihood ratio test ~ with a-level cut of K ~ a ¼ ½ðKÞ ~ L ; ðKÞ ~ U  derived statistic K a a from optimization problems (8) and (9) is a fuzzy random variable. ~ Proof According to Proposition 3.1 and Definition 2.1, K is a fuzzy random variable. h In fuzzy environment, if one is going to evaluate a fuzzy simple null hypothesis, the critical value k should be determined such that ~  kj h is HðhÞÞ ¼ d pðK

ð11Þ

The left-hand side of Eq. (11) is interpreted as probability of rejecting fuzzy null hypothesis when it is correct. Since this probability denotes the type I error, Eq. (11) requires it to be equal to the significance level d. As a generalization from non-fuzzy mode, the fuzzy null hypothesis ‘H: h is ~  k in fuzzy environment. H(h)’ is rejected if and only if K ~  k in Eq. (11), it is required to To assess the expression K exploit a ranking method for fuzzy numbers. Here, the ranking method proposed by Buckley and Chanas (1989) is utilized which seems to be more appropriate when working with a-level cuts. Based on their proposition, the fuzzy number ~a is smaller than the fuzzy number ~b in a-level, if ~L ~ ~L ~U ð~aÞU aa ¼ ½ð~aÞLa ; ð~aÞU a \ðbÞa where ~ a  and ba ¼ ½ðbÞa ; ðbÞa . According to this proposition, the above probability is ~ U  kÞ ¼ d for a specific a-level when turned out as pððKÞ a L U ~ U is a realh 2 ½ha ; ha . Since, based on Proposition 3.1, ðKÞ a valued random variable, it can be inferred that critical value of the test in fuzzy sense is the same as that in crisp sense. It means that significance level of the test is retained at the level d in fuzzy environment.

4 Evaluating acceptability of fuzzy hypothesis ~ using After deriving the membership function of K extension principle and calculating the critical value k as

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H. Moheb Alizadeh et al.

described in the previous section, it is turn to present a procedure to be used for making a decision about whether or not the fuzzy null hypothesis under study is ultimately acceptable. In this section, two different procedures are introduced to accomplish this goal. The first procedure that is based on applying a ranking method for fuzzy numbers gives an absolute decision on status of fuzzy null hypothesis, whereas the second one which provides the membership degrees of fuzzy null hypothesis acceptance and rejection gives a relative decision using some arbitrary decision rules. 4.1 An absolute decision using fuzzy numbers ranking method ~ with critical value of the test, a In order to compare K method based on ranking fuzzy numbers is given in this ~  k, then the fuzzy section. As mentioned before, if K null hypothesis under consideration is rejected in fuzzy ~  k, there are environment. To assess the expression K a lot of methods in the literature (Abbasbandy and Asady 2006; Allahviranloo and Firozja 2010; Asady and Zendehnam 2007; Chen and Klein 1997; Ezzati et al. 2012). However, most of them require membership functions of the fuzzy numbers to be ranked. The method proposed by Chen and Klein (1997), on the other hand, is very appropriate for the present study because it is based on a-level cuts. To describe the method proposed by Chen and Klein ~ j ; j ¼ 1; . . .; m are (1997), suppose the fuzzy numbers A going to be ranked. Let h be the maximum height of lA~ j ; j ¼ 1; . . .; m. Assume h is split into n equal intervals such that ai ¼ ih=n; i ¼ 0; . . .; n. In this case, Chen and Klein (1997) proposed the following index to be used for ranking fuzzy numbers: Ij ¼

where, X is an universal set. Therefore, the aforementioned method can be suitably used for evaluating acceptability of the fuzzy null hypothesis under study. In this case, fuzzy null hypothesis is rejected if IK~  Ik ; otherwise, it is accepted, where IK~ and Ik are Chen and Klein’s indexes computed by Eq. (12) for fuzzy statistic of likelihood ratio test and critical value of the test respectively. 4.2 A relative decision using membership degrees and decision rules ~ its membership After computing the a-level cuts of K, function can be easily obtained. In its general form, sup~ is as depicted in Figs. 1 pose the membership function of K and 2, where the vertical lines specify the critical values. In each figure, two a-levels have been determined; a0 and ak levels on Fig. 1 and a00 and a0k levels on Fig. 2. It is clear ~ U  k in Fig. 1. Hence, based on the ranking that ðKÞ a0 method proposed by Buckley and Chanas (1989), it is ~  k and consequently fuzzy null concluded that K hypothesis is rejected in such an a-level. Therefore, according to the definition of Resolution Identity, ak is the greatest value for which fuzzy null hypothesis is not ~ is not less than or rejected because for any a 2 ½0; ak ; K equal to k based on the ranking method of Buckley and Chanas (1989). In the other words, for any a-level greater than ak , fuzzy null hypothesis is rejected. The value of ak is obtained using the following programming problem: ak ¼ max a s.t. ~ Uk ðKÞ

ð14Þ

a

0a1

i.hXn i Xn h Xn ~ j ÞU  c ~ j ÞU  c  ~ j ÞL  d ; ðA ½ðA ½ðA ai ai ai i¼1 i¼1 i¼1

k

ð12Þ ~ j ÞL g and d ¼ maxi;j fðA ~ j ÞU g. In this where, c ¼ mini;j fðA ai ai case, the greater value of index Ij , the more preferable the ~ j is. While this method is respective fuzzy number A authentic when n advances infinity, Chen and Klein (1997) proposed that n = 3 or 4 suffices to discriminate the differences. The critical value k of the test, which is a real number, can be viewed as a fuzzy number with membership function of lK ðkÞ : X ! f0; 1g, where K ¼ fkg and  1 if and only if xð2 XÞ ¼ k lK ðkÞ ¼ ð13Þ 0 if and only if xð2 XÞ 6¼ k

123

Membership degree

n!1

α0

αk

Likelihood ratio test statistic Fig. 1 Intersection of critical value with right-hand side of a membership function

Fuzzy hypothesis testing with vague data using likelihood ratio test

The aforementioned problem is a two-stage programming problem into which, as the first stage, the optimal solution of problem (9) should be first put. If these optimal solutions can be explicitly written as a function of a, i.e. ~ U ¼ f ðaÞ, then f ðaÞ is substituted with ðKÞ ~ U in problem ðKÞ a a (14). Otherwise, an iterative procedure is proposed here in order to compute ak as follows: 1. 2. 3. 4. 5.

Sort ascending all a-levels applied in problem (9) to ~ U as a1 ; a2 ; . . .; an . derive ðKÞ a Set i :¼ 1 and j :¼ 1. ~ U  k, set j :¼ i þ 1 and go to step 4. Otherwise, If ðKÞ ai go to step 5. ~ U  k, set i :¼ j þ 1 and go to step 3. Otherwise, If ðKÞ aj go to step 5. ~ U ; ai Þ; ðk; ak Þ and Interpolate ak using the triplet ððKÞ ai U ~ ððKÞaj ; aj Þ. In this case, ak can be calculated as follows:  .  ~ U  kÞðaj  ai ÞÞ ððKÞ ~ U  ðKÞ ~ UÞ ak ¼ aj  ððKÞ aj aj ai

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Identity, a0k is the greatest value of a 2 ½0; 1 for which fuzzy null hypothesis is rejected because for any a 2 ~ is not greater than or equal to k based on the ½0; a0k ; K ranking method proposed by Buckley and Chanas (1989). In the other words, for any a-level greater than a0k , fuzzy null hypothesis is accepted. In this way, the value of a0k is obtained using the following programming problem: a0k ¼ max a s.t. ~ Lk ðKÞ 0a1 Similar to the previous case, a two-stage programming problem is obtained here into which, as the first stage, the optimal solution of problem (8) should be put. If the obtained solutions are not explicitly a function of a, the following iterative procedure is applied to derive a0k : 1.

ð15Þ Accordingly, whenever the vertical line corresponding to the critical value crosses the right-hand side of the ~ the membership degree of membership function of K, fuzzy null hypothesis acceptance is equal to ak and the membership degree of fuzzy null hypothesis rejection (alternative hypothesis acceptance) is equal to 1. Now suppose a00 and a0k levels on Fig. 2. Since ~ L0  k, hence based on the ranking method proposed by ðKÞ a0 ~  k and Buckley and Chanas (1989), it is inferred that K consequently, fuzzy null hypothesis is accepted in a00 -level. Therefore, according to the definition of Resolution

Membership degree

k

α 0′

α k′

Likelihood ratio test statistic Fig. 2 Intersection of critical value with left hand-side of a membership function

ð16Þ

a

2. 3. 4. 5.

Sort ascending all a-levels applied in problem (8) to ~ L as a1 ; a2 ; . . .; an . derive ðKÞ a Set i :¼ 1 and j :¼ 1. ~ L  k, set j :¼ i þ 1 and go to step 4. Otherwise, If ðKÞ ai go to step 5. ~ L  k, set i :¼ j þ 1 and go to step 3. Otherwise, If ðKÞ aj go to step 5. ~ L ; ai Þ; ðk; a0 Þ and Interpolate a0k using the triplet ððKÞ k ai L 0 ~ ððKÞaj ; aj Þ. In this case, ak can be derived as follows:  .  ~ L  kÞðaj  ai ÞÞ ððKÞ ~ L  ðKÞ ~ LÞ a0k ¼ aj  ððKÞ aj aj ai ð17Þ

As a result, whenever the vertical line corresponding to the critical value intersects the left-hand side of the ~ the membership degree of membership function of K, fuzzy null hypothesis rejection is equal to a0k and the membership degree of fuzzy null hypothesis acceptance is equal to 1. It should be noted that decision making on whether fuzzy null hypothesis is accepted in fuzzy environment becomes obviously more flexible than that in the crisp mode using given membership degrees and applying some arbitrary managerial rules. The decision rules employed are defined as follows: Rule 1 when the membership degree of fuzzy null hypothesis acceptance is 1, it is said that fuzzy null hypothesis is ‘completely’ accepted if the membership degree of fuzzy null hypothesis rejection (alternative hypothesis acceptance) is less than n: It is ‘relatively’ accepted if the membership degree of fuzzy null hypothesis rejection is between n and w. Finally, it is ‘slightly’ accepted if the membership degree of fuzzy null hypothesis rejection is more than w.

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Rule 2 when the membership degree of fuzzy null hypothesis rejection is 1, it is said that fuzzy null hypothesis is ‘completely’ rejected if the membership degree of fuzzy null hypothesis acceptance is less than n0 . It is also ‘relatively’ rejected if the membership degree of fuzzy null hypothesis acceptance is between n0 and w0 . Otherwise, it is ‘slightly’ rejected. Considering the abovementioned decision rules, it is inferred that if the membership degree of fuzzy null hypothesis acceptance is 1, it will be ultimately accepted. Nevertheless, the intensity of this acceptance is specified using the membership degree of fuzzy null hypothesis rejection. In this case, the greater membership degree of fuzzy null hypothesis rejection, the less intensity the acceptance has. This inference holds when the membership degree of fuzzy null hypothesis rejection is equal to 1. In this situation, fuzzy null hypothesis will be finally rejected. However, the intensity of this rejection is determined using the membership degree of fuzzy null hypothesis acceptance, i.e. the larger membership degree of fuzzy null hypothesis acceptance, the less intensity the rejection has. The other point is that the specific values of n and w in Rule 1 and, n0 and w0 in Rule 2 are determined experimentally and intuitively by decision maker. In Rule 1, the smaller values of n and w result in a stricter decision on fuzzy null hypothesis acceptance, because, in this case, the possibility of accepting fuzzy null hypothesis with less intensity increases. Furthermore, the smaller values of n0 and w0 in Rule 2 lead to a less strict decision on fuzzy null hypothesis rejection. In the current case, the possibility of fuzzy null hypothesis rejection with less intensity increases. As the final point, it is observed that using membership degrees of fuzzy null hypothesis acceptance and rejection and decision rules help us examine status of fuzzy null hypothesis in some intermediate states. It brings about more ease for further decision making and analyses.

5 Numerical examples This section gives two numerical examples to represent how the proposed approach is performed in a practical context. ~ 1 ; . . .; X ~ n is a fuzzy random sample Example 1 Suppose X drawn from a normal population with the mean l and the unknown variance r2 . As mentioned formerly, it is fuzzy perception of a random sample from the normal distribution Nðl; r2 Þ. The accuracy of the fuzzy null hypothesis

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‘H: l is approximately equal to l0 ’ against the fuzzy alternative hypothesis ‘K: l is not approximately equal to l0 ’ is going to be evaluated using this fuzzy random sample. H and K are apparently fuzzy simple and composite hypotheses, respectively. The term ‘is approximately equal to l0 ’ in the fuzzy null hypothesis is characterized by a triangular fuzzy number as HðlÞ ¼ ðl1 ; l0 ; l2 Þ. If a tri~ is represented as ða1 ; a2 ; a3 Þ, the angular fuzzy number A lower and upper bounds of its a-level cut are computed ~ L ¼ a1 þ aða2  a1 Þ and ðAÞ ~ U ¼ a3  aða3  a2 Þ as ðAÞ a a respectively. Since variance of the normal population is unknown, fuzzy likelihood ratio statistic is attained after a few computations as follows: ~¼ K

Xn

~ 2 ~ i  XÞ ðX i¼1

. Xn

~i  l ~0 Þ2 ðX i¼1

n=2

ð18Þ

~0 is fuzzy mean of the population with the where l membership function HðlÞ. A proof of the above statistic in crisp sense is presented in Appendix A. When random sample data, sample mean and population mean are substituted by their corresponding fuzzy quantities, the above statistic is derived. In this example, K can be regarded as a function of random sample X1 ; . . .; Xn ,  and population mean l0 . Hence, based sample mean X on problems (8) and (9), the following optimization problems should be solved in order to derive its a-level cuts in fuzzy environment: ~ L ¼ minff ðX;  l0 ; Xi Þ j Xi ; X;  l0 2 AK ; i ¼ 1; . . .; ng ðKÞ a ð19Þ ~ U ¼ maxff ðX;  l0 ; Xi Þ j Xi ; X;  l0 2 AK ; i ¼ 1; . . .; ng ðKÞ a ð20Þ P

n  l0 ; Xi Þ is defined as  2 where, f ðX; i¼1 ðXi  XÞ = n=2 Pn 2 and AK indicates bounds for the i¼1 ðXi  l0 Þ

~ L  X ~ U and ~ i Þ L  Xi  ð X ~ i ÞU ; ðXÞ   ðXÞ variables as ðX a a a a l0 ÞU . In the other words, we have the ð~ l0 ÞLa  l0  ð~ a following two fractional non-linear programming problems, which can be effectively solved via the constrained variable metric method and generalized reduced gradient method (Rao 2009): hXn .Xn in=2 2 ~ L ¼ min  2 ðKÞ ðX  XÞ ðX  l Þ i i 0 a i¼1 i¼1 s.t. ~ i ÞLa  Xi  ðX ~ i ÞU ðX a ~ L  X ~ U   ðXÞ ðXÞ a a ð~ l0 ÞLa  l0  ð~ l0 Þ U a

ð21Þ

Fuzzy hypothesis testing with vague data using likelihood ratio test

~ U ¼ max ðKÞ a

hXn

 2 ðXi  XÞ i¼1

.X n

ðXi  l0 Þ2 i¼1

in=2

s.t. ~ i ÞU ~ i ÞL  X i  ðX ðX a

a

~ ~  LX  U   ðXÞ ðXÞ a a ð~ l0 ÞLa  l0  ð~ l0 ÞU a

ð22Þ

On the other hand, the critical value k should be ~ U  kÞ ¼ d. In this regard, we obtained such that pððKÞ a consider the following lemma: Lemma 1 If X1 ; . . .; Xn is a crisp random sample of a normal population with the mean l and the unknown variance r2 , we have: .Xn Xn  2 K2=n ¼ ðX  XÞ ðXi  lÞ2 i i¼1 i¼1 ð23Þ   1 ¼ 1 þ T 2 ðn  1Þ pffiffiffi   lÞ=S follows t distribution with ðn  1Þ where T ¼ nðX degree of freedom. This lemma is proved in Appendix B. In fuzzy envi~ U ¼ ½1 þ ððTÞ ~ L Þ2 =ðn  1Þn=2 for all ronment, we have ðKÞ a a L 2 ~ a Þ is a crisp random variable followa 2 ½0; 1where ððTÞ ing F distribution with F1;n1 ðdÞ critical value. Conse~ U, quently, critical value of the crisp random variable ðKÞ a which denotes in fact the critical value k, is equal to ½1 þ F1;n1 ðdÞ=ðn  1Þn=2 . ~ and the critical After deriving the membership function of K value k, it is turn to compare them to evaluate whether the fuzzy null hypothesis under study is acceptable or not. To do this, the first procedure proposed in Sect. 4.1 results in an absolute decision, whereas the second one given in Sect. 4.2 leads to a relative decision. In the latter procedure, based on whether the critical value k intersects the right- or left-hand side of the membership function lK~ , the two-stage programming problems (14) and (16) should be optimized to obtain ak and a0k , ~ L are extracted ~ U and ðKÞ respectively. In these problems, ðKÞ a a from problems (22) and (21), respectively. Considering programming problems (21) and (22), since these quantities cannot be explicitly written as functions of a, the iterative procedures presented should be exploited to obtain ak and a0k . ð33266; 33978; 34889Þ ð32093; 32617; 33255Þ ð32585; 33052; 33787Þ ð31720; 32611; 33497Þ ð32806; 33418; 33908Þ ð31977; 32455; 33034Þ ð33065; 33463; 34131Þ ð31943; 32466; 33212Þ

ð30743; 31624; 32460Þ ð32169; 33070; 33968Þ ð32415; 33127; 34072Þ ð32900; 33543; 34335Þ ð32687; 33224; 33908Þ ð30327; 30881; 31445Þ ð32185; 32597; 33186Þ ð31187; 31565; 32237Þ

ð33423; 34036; 34771Þ ð33208; 34053; 34876Þ ð31639; 32584; 33542Þ ð30945; 31838; 32739Þ ð31511; 32290; 33064Þ ð31934; 32800; 33445Þ ð33060; 33844; 34449Þ ð33464; 34157; 34974Þ

Now the example given by Wu (2005) is numerically considered with a little modification. In this example, the

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marketing department of a tire and rubber company is going to claim that the average life of a recently developed tire is equal to the prevalent average tire life known to be 32,000 miles. To try out this claim, 24 new tires are picked out and tested on 6 cars. Due to the unexpected situations, the tire lives are approximated by triangular fuzzy numbers as represented above. These fuzzy data are exploited to evaluate the fuzzy null hypothesis ‘H: the average tire life is approximately equal to 32000’ against the fuzzy alternative hypothesis ‘K: the average tire life is not approximately equal to 32000’. The term ‘is approximately equal to 32000’ is stated by a triangular fuzzy number as (30,000, 32,000, 34,000). In addition, according to the given fuzzy measures, the fuzzy sample mean is ~ ¼ ð32;218:83; 32;887:21; 33;634:38Þ. In calculated as X the present paper, the non-linear programming solver LINGO (1999) is used to solve the non-linear programming problems (21) and (22). Table 1 presents the obtained solutions of these problems at different values of a. Moreover, Fig. 3 depicts the membership function ~ of K. Critical value of the test in this example is computed as ð1 þ F1;241 ð0:05Þ=ð24  1ÞÞ24=2 ¼ 0:513. According to the first procedure for making a decision on acceptability of fuzzy null hypothesis, the membership function illustrated in Fig. 3 should be compared with the critical value 0.513 using Eq. (12). In this case, we obtain c ¼ 0; d ¼ 1; IK~ ¼ 0:465 and Ik ¼ 0:513. Therefore, it concludes that IK~  Ik and consequently, the fuzzy null hypothesis under consideration is rejected. From a different perspective, according to Fig. 3, the critical value 0.513 is placed in the right-hand side of the membership function lK~ . Hence, based on what was mentioned as the second procedure to make a decision about acceptability of fuzzy null hypothesis, it can be derived that the membership degree of fuzzy null hypothesis rejection is 1. In order to obtain the membership degree of its acceptance, the first iterative procedure given in Sect. 4.2 is utilized. In this case, we compute ak ¼ 0:919. Afterwards, the decision rules described before are applied in order to make the final decision on the status of fuzzy null hypothesis. Using Rule 2 with n0 ¼ 0:5 and w0 ¼ 0:9, it is inferred that the fuzzy null hypothesis under study is ‘slightly’ rejected. As observed, the intensity of fuzzy null hypothesis rejection is determined using membership degree of fuzzy null hypothesis acceptance. The values of n0 and w0 are determined experimentally by decision maker and almost from a managerial point of view. Wu (2005) also rejected fuzzy null hypothesis based on his proposed approach. Although using different approaches, it is observed that the conclusion in both studies is analogous, i.e. the fuzzy null hypothesis under study is rejected.

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Table 1 The a-level cuts of the fuzzy likelihood ratio test statistic in Example 1 a-levels

Quantity

~ L ðKÞ a ~ U ðKÞ a

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0

0

0

0

0

0

0

0

0

0

0.00014

1

1

1

1

1

1

1

0.997

0.926

0.636

0.00014

and the parameter c as fuzzy, the abovementioned statistic is attained in fuzzy environment. In this example, the statistic of likelihood ratio test can  and c. Hence, based on be viewed as a function of X problems (8) and (9), the following optimization problems ~ should be considered to derive a-level cuts of K:

1.0

0.8

0.6

0.4

0.2

~ L ¼ minff ðc; XÞ  j c; X  2 AK g ðKÞ a

ð27Þ

~ U ¼ maxff ðc; XÞ  j c; X  2 AK g ðKÞ a

ð28Þ

 is defined as cX  and AK specifies bounds for where, f ðc; XÞ L ~ L  X ~  U.   ðXÞ decision variables as ð~cÞ  c  ð~cÞU and ðXÞ

0.0 0.0

0.2

0.4

0.6

0.8

1.0

~ with critical value in Example 1 Fig. 3 Membership function of K

~ 1 ; . . .; X ~ n is a fuzzy random sample Example 2 Suppose X taken from an exponentially distributed population with the parameter c. It means that it is a fuzzy perception of a random sample from the exponential distribution expðcÞ. The accuracy of the fuzzy null hypothesis ‘H: c is approximately less than c0 ’ against the fuzzy alternative hypothesis ‘K: c is approximately greater than c0 ’ is going to be evaluated using this fuzzy random sample. The fuzzy hypotheses H and K are obviously an extension of the crisp composite hypotheses H: c  c0 and K: c [ c0 into fuzzy environment, respectively. The terms ‘approximately less than c0 ’ and ‘approximately greater than c0 ’ are stated by the membership functions H(c) and K(c) as follows:

HðcÞ ¼

8 < 1;

c1 c : c1 c0

;

c  c0 c0  c  c1 otherwise

ð24Þ

;

c  c0 c2  c  c0 otherwise

ð25Þ

0;

KðcÞ ¼

8 < 1;

cc2 : c0 c2

0;

where, c1 and c2 are constants. After a few computations, the fuzzy likelihood ratio statistic is computed as follows: ~ ~ ¼ ~c X  K

ð26Þ

where, ~c is fuzzy parameter of the population with the membership function H(c). Mood et al. (1974) give a proof for this statistic in crisp mode. Regarding the sample data

123

a

a

a

a

The aforementioned optimization problems are rewritten as follows: ~ L ¼ min cX  ðKÞ a s.t.

ð29Þ

ð~cÞLa  c  ð~cÞU a L ~ ~  X  U   ðXÞ ðXÞ a

~ U ðKÞ a

a

 ¼ max cX s.t.

ð30Þ

ð~cÞLa  c  ð~cÞU a L ~ ~    ðXÞ  X  ðXÞU a

a

Because of simple structure of objective functions in abovementioned problems, their optimal solutions ~ L and ðKÞ ~ ~ L ¼ ð~cÞL ðXÞ ~ U ¼ ð~cÞU ðXÞ  U if are derived as ðKÞ a a a a a a L L ~  ð~cÞa  0 and ðXÞa  0.

~  k c is HðcÞÞ ¼ d In the next step, the probability pðK should be evaluated to obtain the critical value k. Based on what was mentioned in Sect. 3, this probability is turned out as follows: ~ U  kÞ ~ U  kÞ ¼ pðð~cÞU ðXÞ pððKÞ a a a ¼

pðð~cÞU a

n X

! U ~ ðXi Þa  nkÞ

i¼1

¼ pðQ  k0 Þ ¼ d

ð31Þ

Pn ~ U where, ð~cÞLa  c  ð~cÞU cÞ U a and Q ¼ ð~ a i¼1 ðXi Þa . Since ~ i ÞU ~ i ; i ¼ 1; . . .; n are fuzzy random variables, ðX X a follows exponential distribution with the parameter c. Accordingly,

Fuzzy hypothesis testing with vague data using likelihood ratio test

Q follows gamma distribution with the parameters (n, 1). In this case, k0 is the value satisfying the following equation: Zk0

qn1 eq dq ¼ d CðnÞ

ð32Þ

0

~ and the After obtaining the membership function of K 0 critical value k , they should be compared with each other to assess whether the fuzzy null hypothesis under study is acceptable or not. To do this, if Eq. (12) is used as the first procedure, an absolute decision is resulted. Otherwise, a relative decision is made exploiting the second one. In the current procedure, the two-stage programming problems (14) and (16) should be optimized to obtain ak and a0k , respectively, based on whether the critical value k0 intersects the right- or the left-hand side of the ~ U and membership function lK~ . In these problems, ðKÞ a ~ L are derived from problems (30) and (29), ðKÞ a ~ L  0, respectively. As stated before, If ð~cÞL  0 and ðXÞ a

~ L are explicitly written as ~ U and ðKÞ then ðKÞ a a ~  L , respectively. In this case, we have ð~cÞL ðXÞ a

a ~ U ð~cÞU ð XÞ a a

and

a

ak ¼ max a s.t. ~ U  k0  ð~cÞU ðXÞ a

a

0a1

ð33Þ

and a0k ¼ max a s.t. ~  L  k0 ð~cÞL ðXÞ a

a

0a1

ð34Þ

~ 1 ; . . .; X ~ 6 is a fuzzy random Now numerically, suppose X sample drawn from an exponentially distributed population with the parameter c. This fuzzy random sample is going to be applied to evaluate accuracy of the fuzzy null hypothesis ‘H: c is approximately less than 5.5’ against the fuzzy alternative hypothesis ‘K: c is approximately greater than 5.5’. The terms ‘approximately less than 5.5’ and

1639

‘approximately greater than 5.5’ are stated by the membership functions H(c) and K(c), respectively, as follows: 8 c  5:5 < 1; HðcÞ ¼ 6c ; 5:5  c  6 ð35Þ 0:5 : 0; otherwise 8 c  5:5 < 1; KðcÞ ¼ c  4:7; 4:7  c  5:5 ð36Þ : 0; otherwise ~1 ¼ On the other hand, the sample data are gathered as X ~ 2 ¼ ð4; 4:5; 5Þ; X ~ 3 ¼ ð4; 5:2; 6:5Þ; X ~ 4 ¼ ð3; 3:6; 4Þ; ð5; 6; 7Þ; X ~ ~ X5 ¼ ð5; 5:7; 6Þ; X6 ¼ ð6; 6:5; 7Þ: Consequently, we have ~ ¼ ð4:5; 5:25; 5:92Þ. X ~ are obtained optimizing the programa-level cuts of K ming problems (29) and (30). Now the critical value k0 can be numerically obtained by evaluating the following integral: Zk0

qn1 eq dq ¼ 0:05; CðnÞ

ð37Þ

0

therefore, k0 ¼ 30:826. ~ which are given in After computing a-level cuts of K Table 2, its membership function is easily constructed. This membership function with the critical value k0 is portrayed in Fig. 4. As the first procedure, the membership function illustrated in Fig. 4 should be compared with the critical value 30.826 applying Eq. (12). In this case, the values of c and d are obtained as 0 and 35.52, respectively. Hence, we have IK~ ¼ 0:475 and Ik0 ¼ 0:868. Consequently, fuzzy null hypothesis is rejected based on this method. On the other hand, since the vertical line corresponding to the critical value crosses the right-hand side of the membership function, according to the second procedure, the fuzzy null hypothesis is accepted with the membership degree ak ¼ 0:696 and is rejected with the membership degree 1. The value of ak is computed using programming problem (33). Therefore, using Rule 2 with the same values for n0 and w0 as in the previous example, i.e. n0 ¼ 0:5 and w0 ¼ 0:9, the fuzzy null hypothesis in this example is ‘relatively’ rejected.

Table 2 The a-level cuts of the fuzzy likelihood ratio test statistic in Example 2 Quantity

~ L ðKÞ a ~ U ðKÞ a

a-levels 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0

0

0

0

0

0

0

0

0

0

0.

35.52

34.83

34.14

33.46

32.78

32.11

31.45

30.80

30.15

29.51

28.86

123

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However, likelihood ratio test can be generalized into situations in which observations of the taken sample are represented as random fuzzy variables instead of fuzzy random variables. A random fuzzy variable is defined as a fuzzy element taking random variable values. Liu (2002) characterizes the concept of random fuzzy variable.

1.0

0.8

0.6

0.4

0.2

Appendix A 0.0 25

30

35

40

~ with critical value in Example 2 Fig. 4 Membership function of K

6 Conclusion In traditional statistical inference, it is supposed that all parameters of the mathematical model and possible data are well defined. Very often, such supposition seems to be too inflexible for the real-life problems, especially while dealing with linguistic or imprecise data. To make this inflexibility free, fuzzy methods appear so beneficial to be incorporated into statistical inference. This paper was an endeavor to develop statistical hypothesis testing as a significant class of statistical inference in fuzzy environment based on likelihood ratio test. The problem was considered in its most general form, i.e. both sample data and statistical hypotheses were regarded as fuzzy. To hit this goal, a pair of non-linear programming problems was introduced based on Zadeh’s extension principle for deriving the membership function of likelihood ratio test statistic. In order to determine the status of acceptability of fuzzy null hypothesis, the abovementioned membership function was compared with critical value of the test. In the current step, two distinct procedures were introduced. The first procedure, which was based on a ranking method of fuzzy numbers, made an absolute decision, i.e. the fuzzy null hypothesis under study is absolutely accepted or rejected using this procedure. Exploiting the membership degrees of fuzzy null hypothesis acceptance and rejection computed using resolution identity, and some decision rules, on the other hand, the second procedure was able to make a relative decision, which contributes more flexibility for further inferences. Since likelihood ratio test has led to reasonable procedures in a great variety of problems, and is still playing a dominant role in the development of new tests and estimates, the approach presented in this paper provides a useful tool for benefiting advantages of likelihood ratio test in fuzzy environment.

123

This appendix gives the likelihood ratio statistic in crisp environment when one is going to evaluate the null hypothesis ‘H: l ¼ l0 ’ against the alternative hypothesis ‘K: l 6¼ l0 ’. This statistic relates to the Example 1. Suppose X1 ; . . .; Xn is a crisp random sample taken from a normally distributed population with the unknown variance r2 . In this case, likelihood ratio function is as follows: n Y f ðxi ; l; r2 Þ Lðl; r2 Þ ¼ i¼1 ! n X pffiffiffiffiffi n 2 2 ðxi  lÞ =2r ¼ ðr 2nÞ : exp  ð38Þ i¼1

Assume h ¼ ðl; r2 Þ. In this case, the above function should be maximized to obtain the likelihood ratio statistic K when h 2 Xh and h 2 X giving Lðx1 ; . . .; xn ; h0 Þ and Lðx1 ; . . .; xn ; h00 Þ, respectively. Since there is just one value for l when h 2 Xh , its likelihood ratio estimation is l0 .  when h 2 X. On the other This estimation is also equal to X hand, likelihood ratio estimation of r2 is equal to Pn ^Þ2 =n where l ^ is likelihood ratio estimation i¼1 ðxi  l of l. Therefore, likelihood ratio estimation of r2 is equal to Pn ðXi  l0 Þ2 =n when h 2 Xh and is equal to Pi¼1 n  2 i¼1 ðXi  XÞ =n when h 2 X. Hence, we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n Xn 2 Lðh0 ; x1 ; . . .; xn Þ ¼ 2 ðX  l Þ : expðn=2Þ i 0 i¼1 ð39Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi n Xn  2 Lðh00 ; x1 ; . . .; xn Þ ¼ 2 ðXi  XÞ : expðn=2Þ i¼1 ð40Þ Consequently, based on the Eq. (1), the likelihood ratio test statistic is as follows: Xn .X n n=2 2  2 K¼ ðX  XÞ ðX  l Þ ð41Þ i i 0 i¼1 i¼1 Regarding sample data, sample mean and population mean as fuzzy, Eq. (18) is obtained. h

Fuzzy hypothesis testing with vague data using likelihood ratio test

Appendix B This Appendix presents a proof of Lemma 1 introduced in P  2 ¼ ðn  1Þ S2 , the Example 1. Suppose A ¼ ni¼1 ðXi  XÞ 2 where S is the sample variance. Therefore, 2

  l0 Þ 2 Þ Kn ¼ A=ðA þ nðX   l0 Þ 2 Þ ¼ 1=ð1 þ nA1 ðX   l0 Þ 2 Þ ¼ 1=ð1 þ ðn=n  1ÞS2 ðX pffiffiffi   lÞ=S, we have Since T ¼ nðX

ð42Þ

K2=n ¼ ð1 þ T 2 =ðn  1ÞÞ1 On the other hand, ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . Pn  2 ðn  1Þ, we have i¼1 ðXi  XÞ .Xn  2 ðX  XÞ ðXi  lÞ2 i i¼1 i¼1    1 ¼ 1 þ T 2 ðn  1Þ

K2=n ¼

ð43Þ since



Xn

ð44Þ h

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