A Simple but Efficient Approach for Testing Fuzzy Hypotheses
 Abbas Parchami^{1},
 S. Mahmoud Taheri^{2}Email author,
 Bahram Sadeghpour Gildeh^{3} and
 Mashaallah Mashinchi^{1}
DOI: 10.1186/s4046701500428
© Parchami et al. 2015
Received: 27 August 2015
Accepted: 9 December 2015
Published: 20 January 2016
Abstract
In this paper, a new method is proposed for testing fuzzy hypotheses based on the following two generalized pvalues: (1) the generalized pvalue of null fuzzy hypothesis against alternative fuzzy hypothesis and (2) the generalized pvalue of alternative fuzzy hypothesis against null fuzzy hypothesis. In the proposed method, each generalized pvalue is formulated on the basis of Zadeh’s probability measure of fuzzy events. The introduced pvalue method has several advantages over the common pvalue methods for testing fuzzy hypotheses. A few illustrative examples and also an agricultural example, based on a realworld data set, are given to clarify the proposed method.
Keywords
Fuzzy statistics Fuzzy hypothesis Testing hypothesis pvalueIntroduction
Also, Geyer and Meeden [9] investigated the concepts of fuzzy pvalue and fuzzy confidence interval when both the hypotheses and data are crisp.
Unlike these studies, another efficient and simple pvaluebased method for testing fuzzy hypotheses is presented in this paper. The proposed method is on the basis of the probability measure of fuzzy event introduced by Zadeh [22]. Also, it must be mentioned that all results of this study coincide with the results of testing classical hypotheses, when the hypotheses reduce to two crisp sets on the parameter space.
This paper is organized as follows. Some preliminaries, motivations and basic definitions about testing fuzzy hypotheses are reviewed in Section “Fuzzy Hypotheses: Motivation and Basic Definitions”. In Section “Testing Fuzzy Hypotheses Based on a New pValueBased Approach”, we present a new pvaluebased approach for testing fuzzy hypotheses. Some illustrative examples are given in Section “Illustrative Examples”. An agricultural applied example is presented in Section “Application to Agricultural Studies”. Also, a conclusion is given in the final section.
Fuzzy Hypotheses: Motivation and Basic Definitions
Here, we are going to briefly review some basic concepts which are needed or developed through this paper.
Testing Statistical Hypotheses

(i) H _{0} :θ=θ _{0} versus H _{1} :θ=θ _{1} (θ _{0}>θ _{1})

(ii) H _{0} :θ=θ _{0} versus H _{1} :θ=θ _{1} (θ _{0}<θ _{1})

(iii) H _{0} :θ≥θ _{0} versus H _{1} :θ<θ _{0}

(iv) H _{0} :θ≤θ _{0} versus H _{1} :θ>θ _{0}

(v) H _{0} :θ=θ _{0} versus H _{1} :θ≠θ _{0}
where t _{ l }, t _{ r }, or t _{1} and t _{2} are certain quantiles of the distribution of T, so that α _{ ϕ }=α. In case (c), we may obtain t _{1} and t _{2} by the equal tails method, so that P _{ θ }(T≤t _{1})=P _{ θ }(T≥t _{2})=α/2. The hypothesis H _{0} is rejected if the value of t=t(x) falls into the rejection region. In usual tests, the critical regions of testing hypotheses (i) and (iii) are of form (1.a), the critical region of testing hypotheses (ii) and (iv) are of form (1.b) and the critical region of testing hypothesis (v) is of form (1.c). For more details, see [8, 11, 15].
Fuzzy Hypotheses: Motivation
Fuzzy Hypothesis and Its Boundary
Here, we review some basic concepts about fuzzy hypotheses from Taheri and Behboodian [18] and Parchami et al. [15], which are used in Section “Testing Fuzzy Hypotheses Based on a New pValueBased Approach”.
Definition 1.
Any hypothesis of form “\(\tilde {H}:\,\theta \) is H(θ)” is called a fuzzy hypothesis, where “ θ is H(θ)” implies that θ is in a fuzzy set of Θ, the parameter space, with membership function H(θ).
Note that the ordinary hypothesis “ H: θ=θ _{0}” is a fuzzy hypothesis with the membership function H(θ)=1 at θ=θ _{0}, and zero otherwise, i.e. the indicator function of the crisp set {θ _{0}}.
Definition 2.
(See also [1, 2]) (a) Fuzzy hypothesis \(\tilde {H}:\,\theta \) is H(θ) is called a fuzzy onesided hypothesis, if there exists θ _{1}∈Θ so that: (i) H(θ)=1 for θ≤θ _{1} (θ≥θ _{1}) and (ii) H is an decreasing (increasing) function of θ for θ>θ _{1} (θ<θ _{1}).
(b) Fuzzy hypothesis \(\tilde {H}:\,\theta \) is H(θ) is called a fuzzy twosided hypothesis, if there exists an interval [θ _{1},θ _{2}]⊂Θ so that: (i) H(θ)=1 for θ∈[θ _{1},θ _{2}] and (ii) H is an increasing function of θ for θ≤θ _{1} and is a decreasing function for θ≥θ _{2}.
Definition 3.

(i) \(H_{b}(\theta)=\bigg \{ \begin {array}{lcc} H(\theta) &\ \ \ \text {for}&\ \ \ \theta \leq \theta _{1}, \\ \ \ 0 &\ \ \ \text {for}&\ \ \ \theta >\theta _{1} \end {array} \bigg \}\), if \(\tilde {H}\) is onesided and H is increasing,

(ii) \(H_{b}(\theta)=\bigg \{ \begin {array}{lcc} H(\theta) &\ \ \ \text {for}&\ \ \ \theta \geq \theta _{1}, \\ \ \ 0 &\ \ \ \text {for}&\ \ \ \theta <\theta _{1} \end {array} \bigg \}\), if \(\tilde {H}\) is onesided and H is decreasing,

(iii) H _{ b }(θ)=H(θ), if \(\tilde {H}\) is twosided.
Example 1.
Probability Measure Under a Fuzzy Hypothesis
Definition 4.
where \(H^{*}(\theta)=\frac {H(\theta)}{\int _{\theta } H(\theta)d\theta }\) is the normalized membership function of H(θ). Replace integration by summation in discrete cases.
Remark 1.
(Torabi and Behboodian [20]) (a) The normalized membership function is not necessarily a membership function, i.e. it may be greater than 1 for some values of θ.(b) Note that \(f(x;\tilde {H})\) in Definition 4 is a p.d.f., since \(f(x;\tilde {H})\) is nonnegative and \(\int _{x}\,f(x;\tilde {H})dx=1\). (c) If H is the crisp hypothesis H:θ=θ _{0}, then \(f(x;\tilde H)=f(x;\theta _{0})\).
Example 2.
The major advantage of Definition 4 is that the weighted p.d.f. can integrate all possible p.d.f.s with different weights. The value of H ^{∗}(θ) can be understood as the weight of f(x;θ), and the weighted p.d.f. can let different possible f(x;θ)s play different roles in this integration (e.g. see Fig. 2 in Example 2).
Testing Fuzzy Hypotheses Based on a New pValueBased Approach
The pValue Approach

(i) \(\ \ \bigg \{ \begin {array}{l} \tilde {H}_{0}:\theta \ \mathrm {is \ approximately} \ \theta _{0}, \\ \tilde {H}_{1}:\theta \ \mathrm {is \ approximately} \ \theta _{1}, \end {array} \ \ \mathrm {where \ Def}(H_{0})>\text {Def}(H_{1}),\)

(ii) \(\ \ \bigg \{ \begin {array}{l} \tilde {H}_{0}:\theta \ \mathrm {is \ approximately} \ \theta _{0}, \\ \tilde {H}_{1}:\theta \ \mathrm {is \ approximately} \ \theta _{1}, \end {array} \ \ \mathrm {where \ Def}(H_{0})<\text {Def}(H_{1}),\)

(iii) \(\ \ \bigg \{ \begin {array}{l} \tilde {H}_{0}:\theta \ \mathrm {is \ approximately \ bigger \ than} \ \theta _{0}, \\ \tilde {H}_{1}:\theta \ \mathrm {is \ approximately \ smaller \ than} \ \theta _{0}, \end {array}\)

(iv) \(\ \ \bigg \{ \begin {array}{l} \tilde {H}_{0}:\theta \ \mathrm {is \ approximately \ smaller \ than} \ \theta _{0}, \\ \tilde {H}_{1}:\theta \ \mathrm {is \ approximately \ bigger \ than} \ \theta _{0}, \end {array}\)

(v)\( \ \ \bigg \{ \begin {array}{l} \tilde {H}_{0}:\theta \ \mathrm {is \ near \ to} \ \theta _{0}, \\ \tilde {H}_{1}:\theta \ \mathrm {is \ away \ from} \ \theta _{0}, \end {array}\)
where θ _{0} and θ _{1} are two known numbers and Def(.) is a defuzzifier function. It is obvious that critical regions of testing fuzzy hypotheses are similar to the critical regions of testing precise hypotheses which are formulated by Relations (1). In other words, the critical regions of testing fuzzy hypotheses (i) and (iii) are of form (1.a), the critical regions of testing fuzzy hypotheses (ii) and (iv) are of form (1.b) and the critical region of testing fuzzy hypothesis (v) is of form (1.c). It must be noted that the critical regions of testing fuzzy hypotheses (i) and (ii) must be determined after defuzzification of the hypotheses, and it depends on the defuzzifier function.
Considering Definitions 1 and 3, one can assert that “\(\tilde {H}_{b}:\,\theta \ \text {is} \ H_{b}(\theta)\)” is a fuzzy hypothesis, and therefore we can generalize the classical pvalue for testing fuzzy hypothesis \(\tilde {H}_{0}\) against \(\tilde {H}_{1}\) as follows (see Subsection “Probability Measure Under a Fuzzy Hypothesis”, for more details about the probability measure under a fuzzy hypothesis).
Definition 5.
where \(H^{*}_{0\,b}(\theta)=\frac {H_{0\,b}(\theta)}{\int _{\theta } H_{0\,b}(\theta)d\theta }\) is the normalized membership function of the boundary in the fuzzy null hypothesis, t is the observed value of test statistic (T) and \(m_{H_{0\,b}}\) is the median of the weighted distribution of T(X) under the boundary of the fuzzy null hypothesis \(\tilde {H}_{0\,b}\). Replacement of integration by summation is needed in discrete case.
Remark 2.
In contrast with previous pvalue methods in fuzzy environments (reviewed in Section “Introduction”), the proposed pvalue in this study is a real number on unit interval which is formulated according to the probability measure under fuzzy hypothesis.
Remark 3.
where θ _{0} is the boundary of the null hypothesis and \(m_{\theta _{0}}\) is the median of T under θ _{0}, see [8] and page 381 of [11].
Decision Rule
In testing fuzzy hypothesis \( \tilde {H}_{0} \) against \( \tilde {H}_{1} \), suppose that p _{01} is the pvalue in testing \( \tilde {H}_{0} \) against \( \tilde {H}_{1} \) and p _{10} is the pvalue in testing \( \tilde {H}_{1} \) versus \( \tilde {H}_{0} \). Now we are going to extend the proposed method of Emadi and Arghami [5] for testing fuzzy hypotheses. It must be mentioned that the proposed decision rule omit several weaknesses of the classical pvaluebased tests which are point out in the next subsection.
in which I(.) is the indicator function.
Advantages of the Proposed Method
 1.
The proposed decision rule is a function of both fuzzy hypotheses \( \tilde {H}_{0} \) and \( \tilde {H}_{1} \), while the current fuzzy pvalue methods are based on null fuzzy hypothesis (e.g. see Remark 7 in [14]).
 2.
The proposed decision rule is symmetric with respect to the hypotheses. In other words, the acceptance (rejection) \( \tilde {H}_{0} \) versus \( \tilde {H}_{1} \) in this method is equivalent to the rejection (acceptance) \( \tilde {H}_{1} \) versus \( \tilde {H}_{0} \), while the previous pvaluebased methods in fuzzy environments do not have such reasonable property (see [16] for investigation on misleading statistical evidence by the current pvalue methods). Also, there exist such a symmetry in the introduced confidence factor, since \( \text {CF}(\tilde {H}_{0})=1\text {CF}(\tilde {H}_{1}) \).
 3.
When \( \tilde {H}_{1} \longrightarrow \tilde {H}_{0} \), then p _{10}→p _{01} and therefore, \( \text {CF} \longrightarrow \frac {1}{2} \) which indicates a similar supporting data from both hypotheses. It must be mentioned that not only the usual pvalue methods for testing crisp hypotheses do not have this property but also the fuzzy pvaluebased methods for fuzzy environments do not have such property, e.g. see [8, 17].
 4.
The previous fuzzy pvaluebased methods may not lead the user to a clear decision in twosided tests, when the fuzziness of the data or/and the fuzziness of the null hypothesis is/are high (for more details, see [7, 8, 14, 15]). This problem is solved in the proposed approach by omitting nodecision area in Formula (4).
Illustrative Examples
Example 3.
Therefore, p _{01}<p _{10}, and so we accept \(\tilde {H}_{1}\) against \(\tilde {H}_{0}\) with confidence factor \(\text {CF}=\frac {0.572}{0.156 + 0.572}=0.786\). Note that on the basis of the classical pvalue method, one accepts \(\tilde {H}_{0}\) against \(\tilde {H}_{1}\) at any significance level α<p _{01}=0.156. Although in this example, the result of the proposed method is in conflict with the result of the classical significance tests (e.g. at level 0.05), but we assert that the proposed method is much better according to the comparison of two grey surfaces in Fig. 3.
Example 4.
The results of five different tests in Example 4
Test  H _{0}(μ)  H _{1}(μ)  p _{01}  p _{10}  Accepted  CF 

number  hypothesis  
1  \(\left \{ \begin {array}{lr} \mu 3 \ \ & \ \ 3<\mu \leq 4 \\ 5\mu \ \ & \ \ 4<\mu \leq 5 \\ 0 \ \ &\ \ o.w. \end {array} \right.\)  \(\left \{ \begin {array}{lr} \mu \ \ & \ \ 0<\mu \leq 1 \\ 2\mu \ \ & \ \ 1<\mu \leq 2 \\ 0 \ \ &\ \ o.w. \end {array} \right.\)  0.019  0.244  \(\tilde {H}_{1}\)  0.927 
2  \(\left \{ \begin {array}{lr} \mu 3 \ \ & \ \ 3<\mu \leq 4 \\ 5\mu \ \ & \ \ 4<\mu \leq 5 \\ 0 \ \ &\ \ o.w. \end {array} \right.\)  \(\left \{ \begin {array}{lr} \mu +5 \ \ & \ \ 5<\mu \leq 4 \\ 3\mu \ \ & \ \ 4<\mu \leq 3 \\ 0 \ \ &\ \ o.w. \end {array} \right.\)  0.019  ≃0  \(\tilde {H}_{0}\)  0.99 
3  \(\left \{ \begin {array}{lr} \mu 3 \ \ & \ \ 3<\mu \leq 4 \\ 5\mu \ \ & \ \ 4<\mu \leq 5 \\ 0 \ \ &\ \ o.w. \end {array} \right.\)  \(\left \{ \begin {array}{lr} \frac {\mu +1}{2} \ \ & \ \ 1<\mu \leq 1 \\ \frac {3\mu }{2} \ \ & \ \ 1<\mu \leq 3 \\ 0 \ \ &\ \ o.w. \end {array} \right.\)  0.019  0.283  \(\tilde {H}_{1}\)  0.937 
4  \(\left \{ \begin {array}{lr} \frac {\mu 2}{2} \ \ & \ \ 2<\mu \leq 4 \\ \frac {6\mu }{2} \ \ & \ \ 4<\mu \leq 6 \\ 0 \ \ &\ \ o.w. \end {array} \right.\)  \(\left \{ \begin {array}{lr} \frac {\mu +1}{2} \ \ & \ \ 1<\mu \leq 1 \\ \frac {3\mu }{2} \ \ & \ \ 1<\mu \leq 3 \\ 0 \ \ &\ \ o.w. \end {array} \right.\)  0.041  0.283  \(\tilde {H}_{1}\)  0.873 
5  \(\left \{ \begin {array}{lr} \frac {\mu +4}{8} \ \ & \ \ 4<\mu \leq 4 \\ 5\mu \ \ & \ \ 4<\mu \leq 5 \\ 0 \ \ &\ \ o.w. \end {array} \right.\)  \(\left \{ \begin {array}{lr} \mu \ \ & \ \ 1<\mu \leq 1 \\ \frac {5\mu }{4} \ \ & \ \ 1<\mu \leq 3 \\ 0 \ \ &\ \ o.w. \end {array} \right.\)  0.527  0.452  \(\tilde {H}_{0}\)  0.538 
To compare the results of several testing fuzzy hypotheses based on the proposed approach, five different tests are considered in this example by changing the fuzzy hypotheses (see Fig. 4). The membership functions of fuzzy hypotheses and the results of tests are presented in Table 1. Comparing the result of test 1 with the result of test 2 shows the sensitivity of the proposed approach to the location of the alternative fuzzy hypothesis (compare the first graph with the second one in Fig. 4). Also, comparing the result of test 1 with the result of test 3 shows the sensitivity of the proposed approach to the fuzziness of \( \tilde {H}_{1} \). Instead of the current fuzzy pvalue methods, the results of tests 1–5 show that the proposed decision rule in this paper is a function of both fuzzy hypotheses \( \tilde {H}_{0} \) and \( \tilde {H}_{1} \) (see the first advantage from Subsection “Advantages of the Proposed Method”).
Regarding to the used defuzzifier function \(\text {Def}(H)=\frac {\int _{\theta } \theta H(\theta) d\theta }{\int _{\theta } H(\theta) d\theta }\) in this example, the form of critical region is (1.b) for calculating p _{01} in test 5 (since Def(H _{0})<Def(H _{1})), while in tests 1–4, the form of critical region is of form (1.a).
Example 5.
Similarly, \( p_{10}=P_{\tilde {H}_{1\,b}} (\bar {X}\leq 1327) = \int _{\mu }H^{*}_{1\,b}(\mu) \ P_{\mu }(\bar {X}\leq 1327) \ d\mu = 2.95\times 10^{7} \). Therefore, \(\tilde {H}_{0}\) is strongly accepted against \(\tilde {H}_{1}\) with confidence factor \(\text {CF}=\frac {0.285}{0.285+2.95\times 10^{7}}=0.999\).
Example 6.
Therefore, p _{01}<p _{10}, and \(\tilde {H}_{1}\) is accepted against \(\tilde {H}_{0}\) with confidence factor \(\text {CF}=\frac {0.895}{0.059+0.895}=0.938\).
Application to Agricultural Studies
This applied example was conducted on an agriculturally polluted soil with CdNO_{3} salt in a laboratory at Tehran University, Iran [10]. Suppose that we are going to have an investigation on the amount of cadmium (Cd) absorption in a plant from a polluted soil with CdNO_{3} salt. The unknown parameter is the amount of Cd absorption in a plant (in terms of mg.kg ^{−1} dry matter) from soil which we denoted it with μ. The optimum range of Cd absorbed in a plant has been proposed by Pais and Benton [12] as [0.05, 0.2], and also its maximum has been specified with 3 mg.kg ^{−1} dry matter. The experimenter wants to investigate on the following question: Whether the mean of Cd uptake coincides with the proposed suitable amounts by Pais and Benton or not?
Now, the experimenter can test fuzzy hypotheses \(\tilde {H}_{0}:\,\mu \) is H _{0}(μ), against \(\tilde {H}_{1}:\,\mu \) is H _{1}(μ) without facing any contradiction in the result, where the membership functions of \(\tilde {H}_{0}(\mu)\) and \(\tilde {H}_{1}(\mu)=1\tilde {H}_{0}(\mu)\) are shown in Fig. 7. This is an advantage of considering fuzzy hypotheses rather than crisp hypotheses in some practical problems.
In order to test fuzzy hypothesis \(\tilde {H}_{0}\) against \(\tilde {H}_{1}\), the experimenter has recorded the following data for the amount of Cd absorption via below radish parts from a random sample of size n=25 pots: 1.33, 1.42, 1.56, 1.85, 1.89, 2.96, 2.97, 1.60, 2.02, 1.78, 1.98, 1.86, 1.63, 1.56, 2.12, 1.35, 1.52, 1.32, 2.06, 1.38, 1.09, 2.50, 1.61, 1.46 and 2.18; see [13].
The sample mean is \(\bar {x} = 1.80\) mg.kg ^{−1} dry matter and the sample standard deviation is s=0.479 mg.kg ^{−1} dry matter. In this study, we assume that X _{ i }∼N(μ,s ^{2}), for i=1,…,25, in which the unknown variance parameter can be estimated by a maximum likelihood estimator \( s^{2}= \frac {1}{n} \sum _{i=1}^{n} (x_{i} \bar {x})^{2} \). Note that the normal distribution assumption for random variable X _{ i } comes from the essence of random variable X _{ i }, which is rooted from nature. For instance, one can accept that the weight of seeds picked from a particular plant type, the absorption amount of heavy metals through the roots of a plant in a special greenhouse experiment, or the growth rate of plants in a specific time period, are all normal random variables with suitable means and variances.
Therefore, p _{01}<p _{10}, and so \(\tilde {H}_{1}\) is accepted against \(\tilde {H}_{0}\) with confidence factor \(\text {CF}=\frac {0.328}{0.185+0.328}=0.639\). In other words, considering the confidence factor 0.639, one can assert that the mean absorption Cd in the lower radish parts does not coincides with the proposed amounts by Pais and Benton [12] and so it is not suitable.
Note that if the experimenter decides to solve this problem by classical pvalue method, first he/she must formulate the problem by one of the following hypotheses:
Test 1: H _{0}:μ≥0.2 against H _{1}:μ<0.2 and
Test 2: H _{0}:μ≥3 against H _{1}:μ<3,
while the results of tests 1 and 2 are in conflict with each other, for more details, see Tables 3–5 of [13]. It must be noted that the presented contradiction in the result of tests 1 and 2 comes from the difference between the null hypotheses in the two; in other words, it comes from very vague proposed information by Pais and Benton [12].
Conclusions
In this paper, a new pvaluebased approach was presented for testing statistical hypotheses when the hypotheses are fuzzy rather than crisp. In contrast with the commonly pvaluebased approach for testing fuzzy hypotheses, the decision rule in this approach is based on two pvalues: (1) the pvalue of testing fuzzy null hypothesis against fuzzy alternative hypothesis and (2) the pvalue of testing fuzzy alternative hypothesis against fuzzy null hypothesis. On the basis of this idea, therefore, the introduced method has several advantages over the common methods. The main advantage is that the proposed method is based on both the null and alternative hypotheses. Several numerical examples and also an agricultural example were provided to illustrate the performance of the method. The study of testing fuzzy hypotheses in the framework of uncertainty theory is a potential topic for future work. Also, the study of testing fuzzy hypotheses using the introduced pvalue and based on the paradigm of evidential statistics [16] is another potential topic for more research.
Declarations
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Authors’ Affiliations
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