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Non-parametric Statistical Tests for Fuzzy Observations: Fuzzy Test Statistic Approach
Int. J. Fuzzy Log. Intell. Syst. 2017;17(3):145-153
Published online September 30, 2017
© 2017 Korean Institute of Intelligent Systems.

S. Mahmoud Taheri1, and G. Hesamian2

1Faculty of Engineering Science, College of Engineering, University of Tehran, Tehran, Iran, 2Department of Statistics, Payame Noor University of Tehran, Tehran, Iran
Correspondence to: S. Mahmoud Taheri (sm_taheri@ut.ac.ir)
Received August 25, 2017; Revised September 20, 2017; Accepted September 20, 2017.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

A general approach to the problem of testing statistical non-parametric tests is proposed, for the case when the available data are fuzzy and the level of significance is given as a fuzzy number. To do this, the usual concepts of test statistic and critical value are extended to the fuzzy test statistic and fuzzy critical value, by using the α-cuts approach. The method of decision making (to accept or reject the hypothesis of interest) is based on a suitable ranking method. A numerical example is prepaired to clarify the proposed approach.

Keywords : Location problem, scale problem, Fuzzy critical value, Fuzzy significance level, Preference degree
1. Introduction

Non-parametric approaches, including non-parametric tests, provide inferential procedures to statistics based on some weak assumptions regarding the nature of the underlying population distributions. A particular class of non-parametric tests is composed of two-sample tests. Such tests are commonly based on crisp (exact/nonfuzzy) observations. But, in real world, there are many situations in which the available data are imprecise (vague/fuzzy) rather than precise (crisp). For instance, in the source water studies, the water level of a river cannot be measured in an exact way because of the fluctuation. In this case, the level of water may be reported as imprecise quantities such as: “about 180 (cm)”, “approximately 145 (cm)”, etc. As another example, in survival analysis, we may not determine an exact value for the lifetime of a certain virus. A virus may active complectly over a certain period but losing in effect for some time, and finally go dead complectly at a certain time. In such case, we may report the lifetimes as imprecise quantities such as: “approximately 40 (h)”, “approximately 55 (h)”, and the like. To perform suitable statistical methods for dealing with imprecise observations, first we need to model such data, and then, extend the usual approach to imprecise environment. Fuzzy set theory seems to have suitable tools for modeling these data and providing appropriate statistical methods based on such data.

After introducing fuzzy set theory, there have been a lot of attempts for developing fuzzy statistical methods. But, as the authors know, there have been a few works on non-parametric approach in fuzzy environment. Concerning the purposes of this article, let us briefly review some of the literature on this topic. Kahranam et al. [1] proposed some algorithms for fuzzy non-parametric rank-sum tests based on fuzzy random variables.

Grzegorzewski [2] introduced a method for inference about the median of a population using fuzzy random variables.

Also, the author demonstrated a straightforward generalization of some classical non-parametric tests for fuzzy random variables [3]. The last work relies on the quasi-ordering based on a metric in the space of fuzzy numbers. He studied some non-parametric median tests based on the necessity index of strict dominance suggested by Dubios and Prade [4], for fuzzy observations [5, 6]. In this manner, he obtained a fuzzy test showing a degree of possibility and a degree of necessity for evaluating the underlying hypotheses. Denoeux et al. [7], using a fuzzy partial ordering on closed intervals, extended the non-parametric rank-sum tests based on fuzzy data. For evaluating the hypotheses of interes, they employed the concepts of the fuzzy p-value and degree of rejection of the null hypothesis quantified by a degree of possibility and a degree of necessity, when a given significance level is a crisp number or a fuzzy set. Hryniewicz investigated the fuzzy version of the Goodman-Kruskal γ statistic described by ordered categorical data [8], see also [9]. Grzegorzewski and Szymanowski [10] studied the problem of Goodness-of-fit tests for fuzzy data. Taheri and Hesamian [11] developed the Wilcoxon-rank test for fuzzy data. They also studied some linear rank tests for fuzzy data, by using a p-value-based method [12]. Recently, Taheri et al. [13] studied the statistical inference for contingency tables when the available data are fuzzy rather than crisp.

The present paper aims to develop some non-parametric statistical two-sample tests for fuzzy data, based on extending the concept of classical critical value and comparing it with corresponding test statistic. This paper is organized as follows: In Section 2, we we recall some concepts of fuzzy numbers and fuzzy random variables. In Section 3, based on an index for ranking fuzzy numbers, we introduce a method to construct a version of linear rank tests for fuzzy data. Also, we extend the concept of classical critical value when the given significance level is a fuzzy number, too. In Section 4, we provide the method of decision making to accept or reject the hypothesis of interest. To do this, we use a preference degree to compare the observed fuzzy test statistic and the fuzzy critical value. A numerical example is given in Section 5 to calrify the proposed approach. Finally, a brief conclusion is provided in Section 6. A brief review on some linear rank tests is given in Appendix.

2. Fuzzy Numbers and Fuzzy Random Variables

A fuzzy set à of the universal set X is defined by its membership function μÃ: X → [0, 1], with the set supp(Ã) = {xX: μÃ(x) > 0}, the support of Ã. We say à is a normal fuzzy set if there exists at least one element xX, such that μÃ(x) = 1. In this work, we consider ℝ (the real line) as the universal set. We denote by Ã[α] the α-cut of the fuzzy set Ã, defined for every α ∈ (0, 1], by Ã[α] = {x ∈ ℝ: μÃ(x) ≥ α}, and Ã[0] is the closure of supp(Ã). The sequence {Ã[α]: α ∈ [0, 1]} is a set representation of Ã. à is called convex if μÃx + (1 − λ)y) ≥ min{μÃ(x), μÃ(y)}, ∀λ ∈ [0, 1]. The fuzzy set à of ℝ is called a fuzzy number if it is normal and convex, and for every α ∈ (0, 1], the set Ã[α] is a closed interval. Such an interval is denoted by A˜[α]=[A˜αL,A˜αU], where A˜αL=inf{x:xA˜[α]} and A˜αU=sup{x:xA˜[α]}.

One of the very realistic kind of fuzzy numbers is the triangular fuzzy number denoted by à = (ε1, a,ε2)T, with the following membership function

μA˜(x)={x-ɛ1a-ɛ1ifɛ1x<a,ɛ2-xɛ2-aifaxɛ2,0ifx<ɛ1orx>ɛ2.

We denote by (ℝ) the set of all fuzzy real numbers (for more on fuzzy numbers see [14]).

Note that, given a real number z, we can induce a fuzzy number with membership function μ(r) such that μ(x) = 1 and μ(r) < 1 for rx [15].

Let ℱ(ℝ) ⊆ (ℝ) be the set of all real fuzzy numbers induced by the real set ℝ. We define the relation ~ on ℱ(ℝ) as 1~2 iff 1 and 2 are induced by the same real number z. Then ~ is an equivalence classes [] = {ã: ã ~ }. The quotient set ℱ(ℝ)/~ is the set of all equivalence classes. We call ℱ(ℝ)/~ as the fuzzy real number system. In practice, we take only one element from each equivalence class [] to form the fuzzy real number system (ℱ(ℝ)/~) that is

(F()/~)={z˜:z˜istheonlyelementfrom[z˜]}.

To treat imprecise observations, we use the concept of fuzzy random variable similar to those of Gerzegorzewski [6] and Wu [15].

Definition 2.1

Letbe the sample space of a random variableXdefined on the probability space (Ω, , P). A fuzzy random variable is a mapping: Ω → (ℱ( )/~)if it satisfies the following conditions

  1. (a) For any α ∈ [0, 1] and all ω ∈ Ω, the real valued mapping inf α: Ω → ℝ, satisfying inf α(ω) = inf((ω))α and sup α: Ω → ℝ satisfying sup α(ω) = sup((ω))α are real-valued random variables.

  2. (b) { α(ω): α ∈ [0, 1]} is a set representation of (ω) for all ω ∈ Ω.

Remark 2.1

Note that, in the above definition, (ℱ( )/~) is the support of the fuzzy random variable Therefore, each α-cut of depends on the random variable X. Thus, the crisp random variable X may interpret as the origin of the fuzzy random variable .

Definition 2.2 (see also [2])

We say that1,2, …,m(with observed1,2, …,m) are independent and identically distributed (briefly, say a fuzzy random sample) if their related originsX1,X2, …,Xnare independent and identically distributed crisp random variables.

3. Linear Rrank Tests for Fuzzy Observations

In this section, we are going to extend the statistical linear tests to examine the hypothesis test about the differences in location or variability between two populations based on a set of imprecise (fuzzy) observations. The proposed approach is based on two key concepts of fuzzy test statistic and fuzzy critical value.

3.1 Fuzzy Test Statistic

First, note that the classical linear rank statistic can be rewritten as follows (see Appendix)

TN=i=1na[j=1NI(xizj)]=i=1na[N-j=1NI(zj>xi)],

where, I denotes the indicator functions and zj denotes the jth observation in combined observations x1, x2, …, xn and y1, y2, …, ym.

Now, to perform the non-parametric two-sample tests for location or scale problem based on fuzzy observations 1, 2, …, n and 1, 2, …, m, we need a suitable method of ranking fuzzy numbers. Here, we recall a definition of a common method of ranking fuzzy numbers called the necessity index of strict dominance (NSD index), suggested by Dubois and Prade [4].

Definition 3.1

For two fuzzy numbersÃand, we can evaluate the degree of necessity to which the relationà ≻ B̃is fulfilled by

Nec(A˜B˜)=1-supx,y;xymin{μA˜(x),μB˜(y)}.
Proposition 3.1 [4]

The fuzzy relation Nec is antisymmetric and transitive (i.e. it is a fuzzy partial order on (ℝ)). We use NSD index because of its property and natural interpretation employed in some problems of statistics (for more details, see [7, 16]).

Definition 3.2

Consider the problem of a non-parametric linear rank tests based on combined imprecise observations x̃1, x̃2, …, x̃n and ỹ1, ỹ2, …, ỹm (denoted by z̃1, z̃2 …, z̃N) whose α-cuts are z˜i[α]=[(z˜i)αL,(z˜i)αU]. The fuzzy linear rank test statistic is defined by a fuzzy set with the following membership function

μT˜N(w)=supα[0,1]αI{minβαg(β),,maxβαg(β)}(w),

in which,

g(β)=i=1na[N-j=1NI(Nec(z˜jx˜i)β)].
Remark 3.1

If the available fuzzy observations 1, 2, …, n and 1, 2, …, m reduce to the crisp values x1, x2, …, xn, and y1, y2, …, ym, then the fuzzy linear rank test statistic N reduces to the classical linear rank test statistic TN (see Appendix).

3.2 Fuzzy Critical Value

In the linear rank tests for comparing two statistical populations, the usual approach is to compare the observed test statistic and relative critical value. In this section, by introducing and applying the concept of fuzzy critical value, we are going to generalize this approach for location and scale problem, for fuzzy observations. We establish the approach for location problem. First, we consider the case of testing H0: θ = 0 against H1: θ > 0. At a significance level of δ, the common method rejects H0 if TNwδ, where PH0 (TNwδ) = maxwPH0 (TNw) ≤ δ (Figure 1).

Now, assume that the level of significance is given by a triangular fuzzy number as δ̃ = (ε1, δ,ε2)T [17]. Hence, every value satisfying δ˜αLPH0(TNw)δ˜αU is a candidate to be a critical value with a certain degree. Since the distribution of TN is discrete, the last inequality is equivalent to wδ˜αLLwwδ˜αUU, where PH0(TNwδ˜αLL)=maxwPH0(TNw)δ˜αL and PH0(TNwδ˜αUU)=maxwPH0(TNw)δ˜αU (Figure 2). Based on Appendix, for large sample sizes N, PH0 (.) can be found by normal approximation. Hence, we can define the fuzzy critical as follows.

Definition 3.3

Consider the problem of location hypothesis testsH0: θ = 0 andH1: θ > 0. At the fuzzy significance levelδ̃ = (ε1,δ,ε2)T, the fuzzy critical value is defined to be a fuzzy set with the following membership function

μ(R˜TNc)(y)=supα[0,1]αI{wδ˜αLL,,wδ˜αUU}(y),

in which

wδ˜αLL=max{w:PH0(TNw)δ˜αL},wδ˜αUU=max{w:PH0(TNw)δ˜αU}.
Definition 3.4

Suppose we wish to testH0: θ = 0 againstH1: θ < 0, at the fuzzy significance levelδ̃ = (ε1,δ,ε2)T. The fuzzy critical value is defined to be a fuzzy set with a membership function as follows

μ(R˜TNc)(y)=supα[0,1]αI{wδ˜αLL,,wδ˜αUU}(y),

where

wδ˜αLL=min{w:PH0(TNw)δ˜αU},wδ˜αUU=min{w:PH0(TNw)δ˜αL}.

Now, we consider the case of testing H0: θ = 0 against the alternative H1: θ ≠ 0. In the classical case, at a crisp significance level of δ, we reject the null hypothesis H0 when TNwδ or TNwδ, where wδ and w′ are the critical values such that PH0 (TNwδ) = maxwPH0 (TNw) ≤ δ/2 and PH0(TNwδ)=minwPH0(TNw)δ/2. Now, if the level of significance is a triangular fuzzy number as δ̃ = (ε1,δ,ε2)T, then, using a similar way to that of one-sided cases, we can define the lower and upper fuzzy critical values as follows.

Definition 3.5

In the problem of hypothesis testH0: θ = 0 againstH1: θ ≠ 0, at the fuzzy significance levelδ̃ = (ε1,δ,ε2)T, the lower and upper fuzzy critical values are defined to be fuzzy sets with the following membership functions

Fuzzy lower critical value:

μ(R˜TNcL)(y)=supα[0,1]αI{wδ˜αLL,,wδ˜αUU}(y),

where,

wδ˜αLL=max{w:PH0(TNw)δ˜αL/2},wδ˜αUU=max{w:PH0(TNw)δ˜αU/2}.

Fuzzy upper critical value:

μ(R˜TNcU)(y)=supα[0,1]αI{wδ¯αLL,,wδ¯αUU}(y),

where,

wδ˜αLL=min{w:PH0(TNw)δ˜αU/2},wδ˜αUU=min{w:PH0(TNw)δ˜αL/2}.

Using normal approximation, we can also define the fuzzy critical values, for large sample sizes, in a similar way.

Remark 3.2

Let the probability distribution of TN be symmetric. Then the upper fuzzy critical value for two-sided alternative hypothesis test H1: θ ≠ 0, reduces as follows

μ(R˜TNcU)(y)=supα[0,1]αI{2E(TN)-wδ¯αUU,,2E(TN)-wδ˜αLL}(y).

Hence, based on interval arithmetic, it is easy to see that R˜TNcU=2E(TN)R˜TNcL, where ⊝ denotes the generalized mines operator (for more details see [?]).

4. Method of Decision Making

Consider the problem of linear rank test for location problem with imprecise observations at a given fuzzy significance level. One can expect that, for the case of testing H0: θ = 0 versus H1: θ > 0, if the observed fuzzy test statistic is less than the corresponding fuzzy critical value, then H0 is rejected, otherwise H0 is accepted (the similar argument can be stated for other two cases). To do this, we need a criterion for comparing the observed fuzzy test statistic and fuzzy critical value. The generalized rejection regions may be represented as given in Table 1, in which ⋄ denotes a suitable ranking operator (see, for example, [18, 19]). One of the most commonly used methods for ranking fuzzy sets consists in the definition of the preference degree Pc [19].

Definition 4.1

For two discrete fuzzy sets à and B̃ with the supports {a1, …, an} and {b1, …, bm}, the Pc-index is defined as Pc(A˜,B˜)=S(A˜B˜)+12S(A˜B˜),where, two preference functionsS(à ≻ B̃) andS(Ã) are defined as follows

S(A˜B˜)=x=a1any=b1,y<xbmμA˜(x)μB˜(y)x=a1any=b1bmμA˜(x)μB˜(y),

and

S(A˜B˜)=x=a1any=b1,y=xbmμA˜(x)μB˜(y)x=a1any=b1bmμA˜(x)μB˜(y),

where ⊙ is a t-norm operator that for a > 0 and b > 0 satisfies the condition ab > 0 (see [14, 18]). In the following, we utilize the min-operator as the t-norm operator.

Remark 4.1

The preference degree Pc takes its values in the interval [0, 1]. In addition, Pc(Ã,) = 1 − Pc(,Ã). It is obvious that if Pc(Ã,) = 1, then à is absolutely preferred to , if Pc(Ã,) = 0.5 then à and are equally preferred, and is absolutely preferred to à if Pc(Ã,) = 0.

Definition 4.2

Consider the null hypothesisH0: θ = 0 against an alternative hypothesis given in Table 1 for location problem. Based on Definition 4.1,

  1. in the one-sided case (1), we reject H0 with preference degree η=Pc(T˜N,R˜T˜Nc), and accept H0 with preference degree 1 − η.

  2. in the one-sided (2), we reject H0 with preference degree η=Pc(R˜T˜Nc,T˜N), and accept H0 with preference degree 1 − η.

  3. in the two-sided case (3), we reject H0 with preference degree η=max{Pc(T˜N,R˜T˜NcU),Pc(R˜T˜NcL,T˜N)}, and accept H0 with preference degree 1 − η.

Note that, the comparison approach used in pervious definition is subjective, and so nothing of the main results of the present work will be lost by altering these definitions to ones which fit the demands of the decision makers.

By similar argument as we noted in previous definition, we can define the the preference degree to accept or reject the hypothesis H0: θ = 1 versus an alternative hypothesis given in Table 6 for scale problem.

Remark 4.2

It should be mentioned that, the proposed method of test (which is illustrated in Section 3) and the method of decision making are general. So that, one can use the methods based on non-symmetric fuzzy numbers (observations). Note that, there is not any limitation, in this regard, in the definitions of fuzzy test statistic and fuzzy critical value as well as in the definition of the depreference degree Pc.

5. Numerical Example

To clarify our proposed method, a numerical example is provided in this section.

Example 5.1 ([20], p. 384)

Two potential suppliers of street lighting equipment, A and B, want to present their bids to a city manager. Two independent random samples of size 5 and 4 street lighting equipments were tested from each supplier because the tests are expensive and may take considerable time to complete. Since, under some unexpected situations, we cannot measure the life lengths, precisely, we can just obtain the tire life around a number. The life lengths are reported to be triangular fuzzy numbers as shown in Table 2. We wish to test whether the life length of suppliers A and B have equal variability (i.e. H0: σA = σB).

Before we test for scale, we must determine whether we can assume the locations (medians) can be regarded as equal (i.e. H0: MA = MB). One of the most commonly used tests for the location problem is the Wilcoxon test. Using Definition 3.2, the fuzzy wilcoxon test statistic is obtained as follows

W˜N={129,0.8528,0.7527,0.6926,0.6025,0.5523,0.5022,0.4521,0.3420,0.3019,0.2018,0.1516,0.1015}.

At the fuzzy significance level δ̃ = (0.02, 0.05, 0.08)T, since the probability distribution of WN is symmetric, from Definition 3.5 and Remark 3.2, the fuzzy lower and upper critical values are calculated as follows

R˜WNcL={0.3910,111,0.5312},R˜WNcU={0.5328,129,0.3930}.

By using Definition 4.1, we obtain Pc(W˜N,R˜WNcL)=0 and Pc(R˜WNcU,W˜N)0.06. From Definition 4.2 (part (c)), therefore, we conclude that there is no difference in the locations of the A and B populations with preference degree () 0.94.

Now, we wish to test that whether the life length of suppliers A and B have equal variability. The fuzzy Siegel-Tukey test statistic can be obtained as

S˜N={121,0.8524,0.7525,0.6926,0.5527,0.5030},

since the probability distribution of N is the same as that of the WN, Therefore, the fuzzy critical values are remaind to that of Wilcoxon test. We obtain Pc(T˜N,R˜WNcL)=0 and Pc(R˜WNcU,T˜N)0.07. Therefore, we conclude that, with preference degree () 0.93, there is no basis for difference between variability suppliers A and B.

6. Conclusions

As natural generalizations of the some statistical non-parametric tests, we proposed the corresponding tests based on fuzzy observations, when the given significance level is a fuzzy number, too. To do this, the usual concepts of the test statistic and critical value were extended to the concepts of the fuzzy test statistic and fuzzy critical value. For decision making (about rejection/acceptance the null hypothesis of interest) a method was used to compare the observed fuzzy test statistic and associated fuzzy critical value on the basis of a preference degree for ranking fuzzy sets. The proposed method is general and so, it can be applied for other non-parametric tests such as Mann-Whitney test and Kendall test.

Finally, it should be mentioned that, in many real-world problems, we initially come across with the fuzzy (vague) data, so that the proposed method in this article (generally the methods for fuzzy data) can be applied for such problems. Beside, we may consider the problem of testing fuzzy rather than crisp hypotheses, in which, by using crisp or fuzzy data, we wish to test some vague claims about the underlying statistical polulation(s) (see e.g., [21, 22]). Such extensions are suggested to expand the idea proposed in this article.

Conflict of Interest

No potential conflict of interest relevant to this article was reported.


Figures
Fig. 1.

Classical critical value.


Fig. 2.

The α-cut of fuzzy critical value for a given significance level δ̃.


TABLES

Table 1

Rejection regions

Alternative Rejection region
(1) H1: θ < 0 [T˜N][R˜T˜Nc]
(2) H1: θ > 0 [T˜N][R˜T˜Nc]
(3) H1: θ ≠ 0 [T˜N][R˜T˜NcU]or [T˜N][R˜T˜NcL]

Table 2

Data set in Example 5.1

Supplier A Supplier B
(25, 35, 45)T (36, 46, 56)T
(56, 66, 76)T (46, 56, 66)T
(48, 58, 68)T (50, 60, 70)T
(73, 83, 93)T (39, 49, 59)T
(61, 71, 81)T -

Table 3

Some well-known test statistics for two-sample location problem

Test statistic a[j]
Wilcoxon: WN j
Van-Der Vaerden: VN Φ-1(jN+1)
Terry-Hoeffing: TN E(ζj)

Table 4

Rejection regions for general two-sample location problem

Alternative hypothesis: H0 Rejection region
θ < 0 TNwδ
θ > 0 TNwδ
θ ≠ 0 TNwδ/2 or TNwδ/2

Table 5

Some well-known test statistics for two-sample scale problem

Test Statistic a[j]
Mood: MN (j − (N + 1)/2)2
Ansari-Bradly: AN |j − (N + 1)/2|
Klotz Normal-Scores: KN −1(j/(N + 1)))2
Siegel-Tukey: SN {2jjeven,1<jN/2,2j-1jodd,1jN/2,2(N-j)+2jeven,N/2<jN,2(N-j)+1jodd,N/2<jN.

Table 6

Rejection regions for general two-sample scale problem

Alternative hypothesis: H1 Rejection region
θ > 1 TNwδ
θ < 1 TNwδ
θ ≠ 1 TNwδ/2 or TNwδ/2

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Biographies

S. Mahmoud Taheri received the B.Sc. and the M.Sc. degrees in mathematical statistics from Ferdowsi University of Mashhad, Mashhad, Iran, in 1988 and 1991, respectively, and the Ph.D. degree in statistical inference from Shiraz University, Shiraz, Iran, in 2000. He is a Professor with the College of Engineering, University of Tehran, Tehran, Iran. He spent a one-year research opportunity with the Department of Statistics, University of British Columbia, Vancouver, BC, Canada, in 1999 and a one-year sabbatical with the Institute of Probability and Statistics, Vienna University of Technology, Vienna, Austria, from 2009 to 2010. His research interests include statistical inference, Bayesian statistics, probability and statistics in fuzzy environments, and regression modeling for imprecise data.


G. Hesamian received the B.Sc. and the M.Sc. degrees in mathematical statistics, in 2002 and 2006, respectively. He received his Ph.D. in statistical inference from Isfahan University of Technology in 2012. His research interests include parametric and non-parametric statistical inference in fuzzy and uncertain environments.