Testing Fuzzy Hypotheses with Fuzzy Data and Defuzziﬁcation of the

Fuzzy p-value by the Signed Distance Method

edina Berkachy and Laurent Donz

Applied Statistics and Modelling, Department of Informatics, Faculty of Economics and Social Sciences,

University of Fribourg, Boulevard de P

erolles 90, 1700 Fribourg, Switzerland

Keywords:

Fuzzy P-Value, Fuzzy Statistics, Fuzzy Hypotheses, Fuzzy Data, One-Sided and Two-Sided Tests, Defuzziﬁ-

cation, Signed Distance Method.

Abstract:

We extend the classical approach of hypothesis testing to the fuzzy environment. We propose a method based

on fuzziness of data and on fuzziness of hypotheses at the same time. The fuzzy p-value with its α-cuts

is provided and we show how to defuzzify it by the signed distance method. We illustrate our method by

numerical applications where we treat a one and a two sided test. For the one-sided test, applying our method

to the same data and performing tests on the same signiﬁcance level, we compare the defuzziﬁed p-values

between different cases of null and alternative hypotheses.

1 INTRODUCTION AND

MOTIVATION

The so-called classical approach of the statistical in-

ference is the most used one in statistics. Its exten-

sion to the fuzzy environment was of a big discussion

in many research papers in the last decade. Several

testing methods and approaches in the fuzzy context

were treated. We mention for example (Filzmoser

and Viertl, 2004), (Parchami et al., 2010), (Grze-

gorzewski, 2000) and many others. For instance,

(Grzegorzewski, 2000) proposed a fuzzy test based

on conﬁdence intervals. This test leads to a fuzzy de-

cision, which provides a degree of conviction, mean-

ing a degree of acceptability of the null and alterna-

tive hypotheses. From another side, (Filzmoser and

Viertl, 2004) extended the classical test to a fuzzy test

asserting that the fuzziness is a matter of data. They

proposed a fuzzy p-value and a “three decision” pro-

cedure where no rejection of both null and alternative

hypotheses is considered. (Parchami et al., 2010) rea-

soned similarly to (Filzmoser and Viertl, 2004) but as-

sumed that the fuzziness is coming from the hypothe-

ses instead of the data.

On the other hand, defuzzifying a fuzzy p-value

could be in several situations important to make a

decision. (Grzegorzewski, 2001) presented differ-

ent defuzziﬁcation operators to defuzzify his pro-

posed fuzzy p-value. In the same way, (Berkachy and

Donz

e, 2017), based on the work of (Grzegorzewski,

2000), described how one can defuzzify a fuzzy deci-

sion by the so-called signed distance defuzziﬁcation

method. This method was basically used for instance

by (Berkachy and Donz

e, 2016) in the context of eval-

uating linguistic questionnaires.

In this paper, we reconsider the tests’ procedures

described by (Filzmoser and Viertl, 2004) and (Par-

chami et al., 2010). We propose an inference method

based on both fuzzy data and fuzzy hypotheses. We

also put our attention on fuzzy p-value with its α-cuts,

and show how to apply the signed distance method to

defuzzify this fuzzy p-value. We know that while the

defuzziﬁcation of a fuzzy set reduces some informa-

tions of it, defuzzifying a fuzzy p-value can be useful

in several cases of decision making. At last, we give

two numerical examples of a one-sided and a two-

sided tests. On the same occasion, we play with a

set of different fuzzy null and alternative hypotheses

in order to be able to compare and understand the dif-

ferences between cases.

To summarize, we present in Section 2 some use-

ful deﬁnitions and notations. The Section 3 is devoted

to a brief presentation of the signed distance defuzzi-

ﬁcation method. In Section 4, we recall brieﬂy the

classical testing approach, and we describe the proce-

dure of testing fuzzy hypotheses with fuzzy data. We

ﬁnally end in Section 5 with two numerical examples.

Berkachy R. and DonzÃl’ L.

Testing Fuzzy Hypotheses with Fuzzy Data and Defuzziﬁcation of the Fuzzy p-value by the Signed Distance Method.

DOI: 10.5220/0006500602550264

In Proceedings of the 9th International Joint Conference on Computational Intelligence (IJCCI 2017), pages 255-264

ISBN: 978-989-758-274-5

2 DEFINITIONS AND

NOTATIONS

Let us recall some fundamental deﬁnitions and nota-

tions useful in further sections.

Deﬁnition 2.1 (Fuzzy Set).

If A is a collection of objects denoted generically by x

then a fuzzy set

X in A is a set of ordered pairs:

X = {(x,µ

(x)) : x ∈ A}, (1)

where µ

(x) is the membership function of x in

which maps A to the closed interval [0,1] that char-

acterizes the degree of membership of x in

Deﬁnition 2.2 (Fuzzy Number).

A fuzzy number

X is a convex and normalized fuzzy set

on R, such that its membership function is continuous

and its support is bounded.

Deﬁnition 2.3 (α-cut of a Fuzzy Number).

The α-cut of a fuzzy number

X is a non-fuzzy set de-

ﬁned as:

= {x ∈R : µ

(x) > α}. (2)

The fuzzy number

X can be represented by the family

set {

: α ∈ [0,1]} of its α-cuts.

The α-cut of a fuzzy number

X is the closed in-

terval [

= inf{x ∈ R : µ

(x) > α)} is its

left α-cut and

= sup{x ∈R : µ

(x) > α)} its right

one. We note that the α-cut of a fuzzy number

X is

a union of ﬁnite compact and bounded intervals. Fur-

thermore, the least-upper bound property generalized

to ordered sets and the extension principle induces the

following expression of the membership function of

(see (Viertl, 2011)):

(x) = max{αI

(x) : α ∈ [0,1]}, (3)

where I

(x) is the following indicator function:

(x) = I

{x∈R;µ

≥α}

(x)

(

1 if µ

(x) ≥ α,

0 otherwise.

(4)

Deﬁnition 2.4 (Triangular Fuzzy Number).

A triangular fuzzy number

X is a fuzzy number with

membership function given as follows:

(x) =











x−u

v−u

if u < x ≤v,

x−w

v−w

if v < x ≤w,

0 elsewhere.

(5)

It is common to represent a triangular fuzzy num-

ber by the tuple of three values u, v and w, i.e.

X =

(u,v, w), where u < v < w ∈ R. For a triangular fuzzy

number, the left and right α-cuts

and

are given

respectively by

(

= u + (v −u)α,

= w −(w −v)α.

(6)

3 THE SIGNED DISTANCE

DEFUZZIFICATION METHOD

The signed distance defuzziﬁcation method was

described mainly by (Yao and Wu, 2000) and (Lin

and Lee, 2010). (Berkachy and Donz

e, 2016) use

it extensively in the context of evaluating linguistic

questionnaires. The method appears to have nice

properties and will be implemented in our test proce-

dure for the defuzziﬁcation of the fuzzy p-values. Let

us deﬁne brieﬂy this measure.

Deﬁnition 3.1. The signed distance d

(e,0) mea-

sured from 0 for a real value e in R is e.

Deﬁnition 3.2. Let

X be a fuzzy set on R, such as

X = {(x, µ

(x))|x ∈ R} where µ

(x) is the member-

ship function of x in

X. Suppose that the α-cuts

and

exist, and as a function of α are integrable for

α ∈ [0,1]. The signed distance of

X measured from

the fuzzy origin

0 is:

0) =

[

]dα. (7)

4 TESTING FUZZY

HYPOTHESES WITH FUZZY

DATA

(Filzmoser and Viertl, 2004) and (Parchami et al.,

2010) were ones of many that treated the problem of

testing hypotheses in the fuzzy environment. They

introduced as instance the concept of fuzzy p-value.

Inspired by their methods, we propose a hypotheses

testing method based on fuzzy hypotheses and fuzzy

data at the same time. The signed distance will be

applied in order to defuzzify the fuzzy p-value. But,

ﬁrst, let us recall the main ideas of the classical ap-

proach.

4.1 Testing Hypotheses in the Classical

Approach

We consider a population described by a probability

distribution P

depending on the parameter θ, and be-

longing to a family of distributions P = {P

: θ ∈ Θ}.

Testing hypotheses on a parameter θ in the classical

approach consists on considering a null hypothesis

denoted by H

, H

: θ ∈ Θ

and an alternative one

denoted by H

, H

: θ ∈ Θ

. Θ

and Θ

are

subsets of Θ such that Θ

∩Θ

0. A test statistic

is a function of a random sample Y

,. .. ,Y

used in

testing the null hypothesis against the alternative one.

We call T such a test statistic, where T : R

7→ R. For

this test, two decisions are often treated: not reject

the null hypothesis H

or reject the null hypothesis

. However, the Neyman-Pearson testing approach

(Neyman and Pearson, 1933) could consider the pos-

sibility of having a three decision procedure where

a third case appears: both the null and alternative

hypotheses are neither rejected or not rejected.

The hypothesis testing dilemma is reduced to a

decision problem based on the test statistic T , where

the space of possible values of T is decomposed into

a rejection region R and its complement R

. Three

forms of R are possible depending on the alternative

hypotheses H

Let us suppose the following three tests:

1. H

: θ ≥ θ

vs. H

: θ < θ

; (8)

2. H

: θ ≤ θ

vs. H

: θ > θ

; (9)

3. H

: θ = θ

vs. H

: θ 6= θ

; (10)

where θ is the parameter to test and θ

a particular

value of this parameter.

Then, we would reject the null hypothesis H

if re-

spectively:

1. T ≤ t

(one-sided test); (11)

2. T ≥ t

(one-sided test); (12)

3. T /∈ (t

) (two-sided test); (13)

where t

, t

and t

are quantiles of the distribution

of T .

From another side, we denote by δ the signiﬁcance

level of the test. The quantiles of the distribution t

, t

, and t

are found such that the following proba-

bilities hold:

1. P(T ≤ t

) = δ, (14)

2. P(T ≥ t

) = δ, (15)

3. P(T ≤ t

) = P(T ≥t

) =

. (16)

By this method, we decide to reject the null hypoth-

esis if the value of the test statistic t = T (y

,. .. ,y

)

falls into the rejection region R.

A pratical way to take a decision is to calculate the

p-value in order to decide whether we reject or don’t

the null hypothesis H

. Notice that a p-value depends

on different elements as, for instance, the sample and

its distribution, the boundary of the null hypothesis,

the distribution of the test statistic and its observed

value. We deﬁne a p-value p

θ∗

as function of the

boundary θ∗ of the null-hypothesis. This p-value for

the three cases (11), (12) and (13) can be written re-

spectively as follows:

1. p

∗

= P

∗

(T ≤t), (17)

2. p

∗

= P

∗

(T ≥t), (18)

3. p

∗

= 2min[P

∗

(T ≤t),P

∗

(T ≥t)], (19)

where “P

∗

” means that the probability distribution

depends on the boundary θ

∗

The decision is made by comparing the p-value to

the signiﬁcance level δ: If the p-value is smaller than

δ, we reject the null hypothesis H

. Otherwise, we

don’t reject it.

4.2 Fuzzy Hypotheses

In their paper, (Filzmoser and Viertl, 2004) discussed

the test of hypotheses in the case of fuzzy data.

However, (Parchami et al., 2010) asserted that the

fuzziness is rather coming from the hypothesis. In

the following, we are treating a case inspired by the

above papers but where both data and hypotheses are

fuzzy. First, let us deﬁne a fuzzy hypothesis.

Deﬁnition 4.1 (Fuzzy Hypothesis).

A fuzzy hypothesis

H on the parameter θ, denoted as

“

H : θ is H”, is a fuzzy subset of the parameter space

Θ with its corresponding membership function µ

Remark 4.1. A given fuzzy hypothesis

H reduces to

a crisp hypothesis H when the membership function

= I

It is common practice to postulate as membership

functions of a fuzzy left one-sided hypothesis (an in-

creasing function) or of a fuzzy right one-sided hy-

pothesis (a decreasing function) respectively the fol-

lowing functions:

(x) =











0 if x < u;

x−u

v−u

if u ≤ x < v;

1 if x ≥v.

(20)

(x) =











1 if x ≤u;

x−v

u−v

if u < x ≤ v;

0 if x > v;

(21)

In that case, we simply note these fuzzy hypotheses

= (u,v) and

= (u,v).

A fuzzy two-sided hypothesis

is gener-

ally treated as a triangular fuzzy number, i.e.

= (u,v, w), with membership function (5).

Consider now a crisp random sample Y

,. .. ,Y

with probability distribution P

and a corresponding

fuzzy random sample

X = (

,. .. ,

) where

is a

fuzzy number as described in Deﬁnition 2.2. We de-

note by µ

the membership function of

X, where µ

→[0,1]. We suppose that it exists a given value of

x, x considered as a n-dimensional vector, for which

(x) reaches 1, and that the α-cuts of µ

build a

closed compact and convex subset of R

. On the other

hand, consider furthermore a real valued function φ

such as φ: R

→R. Denote by the fuzzy number

Z the

result of applying the function φ to the fuzzy random

sample, i.e.

Z = φ(

,. .. ,

). Then, by the extension

principle (Zadeh, 1965), the membership function µ

Z is written in the following manner:

(z) =

(

sup{µ

(x) : φ(x) = z} if ∃x : φ(x) = z,

0 if @x : φ(x) = z,

(22)

for all z ∈ R. Furthermore, the α-cuts of

Z are given

by:

= [min

x∈

φ(x),max

x∈

φ(x)], (23)

for all α ∈ (0,1] (Viertl, 2011).

Finally, we have to deﬁne the fuzzy boundaries of

fuzzy hypotheses.

Deﬁnition 4.2 (Boundary of a Hypothesis).

The boundary

∗

of a hypothesis

H : θ is H, is a fuzzy

subset of Θ, with membership function µ

∗

As instance, the fuzzy boundaries corresponding

to the tests (8), (9) and (10) are given respectively by:

∗

= H if θ ≤ θ

, 0 otherwise,

(

H is left one-sided and µ

increasing);

∗

= H if θ ≥ θ

, 0 otherwise,

(

H is right one-sided and µ

decreasing);

∗

= H, (

H is two-sided).

4.3 Fuzzy p-value

Considering hypotheses as fuzzy let us as well see p-

values as fuzzy ones, and in this case taking α-cuts

of the fuzzy p-values can help to evaluate the results

of the tests. Taking in consideration the three possi-

ble rejection regions as deﬁned in (11), (12) and (13),

the following proposition shows how to calculate the

corresponding α-cuts.

Proposition 4.1. Given a test procedure based on

fuzziness of data and hypotheses. Considering the

three rejection regions (11), (12) and (13), the α-cuts

of the fuzzy p-value ˜p are given by:

1. ˜p

= [P

(T ≤

),P

(T ≤

)]; (24)

2. ˜p

= [P

(T ≥

),P

(T ≥

)]; (25)

3. ˜p



[2P

(T ≤

),2P

(T ≤

)] if A

> A

[2P

(T ≥

),2P

(T ≥

)] if A

≤ A

;

(26)

for all α ∈(0, 1], where

and

are the left and right

α-cuts of

t = φ(

,. .. ,

), θ

and θ

are the α-cuts of

the boundary of

, A

is the area under the member-

ship function µ

of the fuzzy number

t on the left side

of the median, and A

is the one on the right side. In

this case, one has to decide on which side the median

is located based on the biggest amount of fuzziness.

Proof 4.1. Since we are treating the case of both

fuzzy data and fuzzy hypotheses, the proof of the

Proposition 4.1 will be done in three steps:

1. According to the above sections, the resulting

fuzzy value

t = T (

,. .. ,

) is fuzzy with its

membership function µ

. We denote by supp(µ

the support of µ

given by supp(µ

) = {x ∈ R :

(x) > 0}. Using the extension principle, (Filz-

moser and Viertl, 2004) wrote the (precise) p-

value p for

T for a one-sided test respectively to

cases (8) and (9) as follows:

1. p = P(T ≤t = max supp(µ

)); (27)

2. p = P(T ≥t = min supp(µ

)). (28)

Our purpose at this moment is to write the α-cuts

of the fuzzy p-value ˜p. Hence, we know that µ

is a membership function and all its α-cuts are

compact and closed on R, then we can deﬁne the

α-cuts of the p-value ˜p

related to (27) and (28)

1. ˜p

= [P(T ≤

),P(T ≤

)]; (29)

2. ˜p

= [P(T ≥

),P(T ≥

)]. (30)

The same procedure can be easily written for the

two-sided test.

2. From another side, we want to extend these for-

mulas to the case of fuzzy hypothesis. (Parchami

et al., 2010) related the fuzziness to the hypothe-

sis and presented the α-cuts of the fuzzy p-value

in the following form:

1. ˜p

= [P

(T ≤t),P

(T ≤t)]; (31)

2. ˜p

= [P

(T ≥t),P

(T ≥t)]; (32)

3. ˜p

(

[2P

(T ≤t),2P

(T ≤t)] if A

> A

[2P

(T ≥t),2P

(T ≥t)] if A

≤ A

;

(33)

Referring to the Equations (17), (18), (29), (30),

to the Deﬁnition 4.1 and to the fuzzy p-value dis-

cussed by (Parchami et al., 2010), we get ˜p

and

˜p

, the left and right α-cuts of the fuzzy p-values

based on fuzziness of data and hypotheses given

by Proposition 4.1.

3. We ﬁnally have to be sure that the properties of

the membership function are fulﬁlled: the facts

that µ

and µ

are membership functions and

the probabilities are restricted to [0, 1] induce

that the resulting membership functions of ˜p

are between 0 and 1 and reach 1 for a given

value. Furthermore, the α-cuts of each case

form a closed ﬁnite interval and thus they are

compact and convex subsets of R for all α ∈(0,1].

For the decision making, (Filzmoser and Viertl,

2004) asserted that a three-decision problem is

adopted based on the left and right α-cuts of ˜p. For a

test with a signiﬁcance level δ, the decisions are made

by the following rules:

• ˜p

< δ: reject the null hypothesis;

• ˜p

> δ: not reject the null hypothesis;

• δ ∈ [ ˜p

, ˜p

]: both null and alternative hypothesis

are neither rejected or not.

4.4 Defuzziﬁcation of the Fuzzy p-value

by the Signed Distance

As mentioned above, the signed distance method is

an attractive one to defuzzify fuzzy numbers. Thus,

we intend to apply this operator in order to defuzzify

the fuzzy p-value and understand whether the deci-

sion made with resulting p-value is similar to the one

in the classical and fuzzy approaches. The idea is to

consider the α-cuts of the fuzzy p-values found in the

previous section. We use the equation (7) to defuzzify

the fuzzy p-value given in equations (24), (25) and

(26). The defuzziﬁed p-values are written as follows:

1. d( ˜p,

0) =

(T ≤

) + P

(T ≤

))dα;

(34)

2. d( ˜p,

0) =

(T ≥

) + P

(T ≥

))dα;

(35)

3. d( ˜p,

0) =











(2P

(T ≥

) + 2P

(T ≥

))dα, if A

> A

(2P

(T ≤

) + 2P

(T ≤

))dα, if A

≤ A

(36)

To interpret, the decision of this testing problem

using the defuzziﬁed p-values is similar to the classi-

cal approach where two main decisions are taken into

account:

• d( ˜p,

0) < δ: reject the null hypothesis;

• d( ˜p,

0) > δ: not reject the null hypothesis;

• d( ˜p,

0) = δ (a rare case): one should decide

whether to reject or not the null hypothesis.

The decision of no rejecting H

or H

doesn’t oc-

cur in this case since the p-value is now on crisp. We

remember that in the fuzzy sense, the no-decision case

is considered. Thus, if one has to defuzzify the p-

value, this case will no longer be possible. Not detect-

ing the no-decision region might be a disadvantage of

defuzzifying the fuzzy p-value.

5 NUMERICAL APPLICATIONS

In this section, we propose two numerical applica-

tions to help the reader to understand the reasoning.

The ﬁrst one treats a one-sided test as (9), and the

second one a two-sided test as (10).

Example 5.1. Consider a random sample of size n =

9 from a normal distribution with an unknown mean

and a standard deviation of 1, N (µ,1). The aim is to

test on the signiﬁcance level δ = 0.05 the following

hypotheses:

: µ is approximately 15,

: µ is approximately bigger than 15.

Suppose that not only the hypotheses are fuzzy but

the sample as well. Let us assume that the fuzzy

null hypothesis is given by the triangular fuzzy num-

ber

= (14.8,15,15.2) and the alternative one by

= (15,16). These hypotheses are shown in Fig-

ure 2. We write the α-cuts of

as:

(

)

(

)

= 14.8 + 0.2α;

(

)

= 15.2 −0.2α.

(37)

We assume furthermore that the membership func-

tion of the observed fuzzy sample mean

X is given by

the following:

(x) =











2x −31 if 15.5 < x ≤16;

−2x + 33 if 16 < x ≤16.5;

0 otherwise;

(38)

with the corresponding α-cuts:

(

= 15.5 + 0.5α;

= 16.5 −0.5α.

(39)

The rejection region in this one-sided test is given

by equation (12). Hence, the expression of the corre-

sponding α-cuts ˜p

of the fuzzy p-value are given by

equation (25). Combining all the previous informa-

tions, ˜p

is as follows:

˜p

= [

∞

(α)

(2π)

−

exp(

−u

)du,

∞

(α)

(2π)

−

exp(

−u

)du], (40)

where θ

(α) et θ

(α) are the following functions of α

based on Equations 37 and 39:

(α) =

−(

)

σ /

√

= 5.1 −2.1 ×α

and

(α) =

−(

)

σ /

√

= 0.9 + 2.1 ×α.

The sum S

of two fuzzy numbers

and

with

The membership function induced by the fuzzy p-

value ˜p is shown in Figure 3. Since the fuzzy p-value

and the signiﬁcance level overlap, we cannot make

any decision visually. In order to make one, we de-

fuzzify the fuzzy p-value. Applying the signed distance

method (equation (35)), we get the following result:

d( ˜p,

0) =

(T ≥

) + P

(T ≥

))dα

(

∞

5.1−2.1×α

(2π)

−

exp(

−u

)du +

∞

0.9+2.1×α

(2π)

−

exp(

−u

)du)dα

= 0.0239.

We remark that the defuzziﬁed p-value (0.0239)

is smaller than the signiﬁcance level (0.05). In this

case, the decision is to reject the null hypothesis at

the level δ = 0.05.

Furthermore, we are interested in understanding

the inﬂuence of the form of the fuzzy hypotheses on

the decision of the test. Let us view some variations of

the fuzzy hypotheses. Table 1 shows the tests where

we change the null or the alternative fuzzy hypothe-

ses of Example 5.1. Figure 1 displays the member-

ship functions of the fuzzy p-values obtained from all

these tests. We can see that although the fuzzy al-

ternative hypothesis determines the rejection region,

then it does not inﬂuence anymore the decision of the

test. Hence, the defuzziﬁed p-value is sensitive to the

fuzzy null hypothesis. From another side, if one has

to compare between defuzziﬁed p-values of the tests

simulated versus the fuzzy p-values illustrated in Fig-

ure 1, we can say that the highest p-value corresponds

to the largest spreaded fuzzy p-value. It is the oppo-

site for the lowest one. Therefore, we can say that

the defuzziﬁed p-value can be a relevant indicator of

fuzziness of the null hypothesis in order to make the

most convenient decision.

Example 5.2. Consider again a random sample of

size n = 49 from a normal distribution with an un-

known mean and a standard deviation of 1, N (µ,1).

their corresponding α-cuts

= [

] and

[

] is written by:

= [

and their difference D

given by:

−

= [

−

Table 1: Testing different hypotheses on the signiﬁcance level δ = 0.05 - Example 5.1.

Test Fuzzy null hypothesis Fuzzy alternative hypothesis Defuzziﬁed p-value Decision

= (14.8, 15.0, 15.2)

= (15, 16) 0.0239 Reject the null hypothesis

= (14.8, 15.0, 15.2)

= (16, 17) 0.0239 Reject the null hypothesis

= (14.8, 15.0, 15.2)

= (16, 16) 0.0239 Reject the null hypothesis

= (15.0, 15.0, 15.0)

= (15, 16) 0.0098 Reject the null hypothesis

= (14.6, 15.0, 15.4)

= (15, 16) 0.0494 Slightly Reject the null hypothesis

= (14.0, 15.0, 16.0)

= (15, 16) 0.1699 Not Reject the null hypothesis

= (14.9, 15.0, 15.1)

= (15, 16) 0.0156 Reject the null hypothesis

0.0 0.2 0.4 0.6 0.8 1.0

Membership functions of the fuzzy p−values corresponding to different tests

Test 1, 2 and 3

Test 4

Test 5

Test 6

Test 7

Significance level

Figure 1: Membership functions of the fuzzy p-values corresponding to the tests in Table 5.1 at the signiﬁcance level δ = 0.05.

The aim is to test on the signiﬁcance level δ = 0.05

the following hypotheses

: µ is near 100,

: µ is away from 100.

Let us suppose that the null and alternative hypothe-

ses are fuzzy and given by the fuzzy triangular num-

bers

= (99.7,100,100.3) and the alternative one

= 1 −

as seen in Figure 4. The α-cuts of

are:

(

)

(

)

= 99.7 + 0.3α;

(

)

= 100.3 −0.3α.

(41)

Furthermore let us assume that the α-cuts of the

observed fuzzy sample mean

X are given by the fol-

lowing:

(

= 100.4 + 0.6α;

= 101.6 −0.6α.

(42)

The rejection region in this two-sided test is given

by the equation (13). We note that in a standardized

normal distribution, the median is equal to the mean,

thus we consider that the fuzzy median is equal to the

one expressed by the null hypothesis. The α-cuts ˜p

the fuzzy p-value can be computed by equation (26).

And since we are in the case of A

≤ A

, we get that

˜p

is as follows:

˜p

= [2

∞

(α)

(2π)

−

exp(

−u

)du,

∞

(α)

(2π)

−

exp(

−u

)du]. (43)

where θ

(α) et θ

(α) are the following functions of

α:

(α) =

−(

)

σ /

√

= 0.9 + 8.1 ×α

and

(α) =

−(

)

σ /

√

= 17.1 −8.1 ×α.

Figure 5 shows the fuzzy p-value at the signiﬁ-

cance level δ = 0.05.

We defuzzify this fuzzy p-value by the signed dis-

tance method using the equation (36) and we obtain

the following distance:

d( ˜p,

0) =

2 ×(P

(T ≥

) + 2 ×P

(T ≥

))dα

(2 ×

∞

0.9+8.1×α

(2π)

−

exp(

−u

)du +

2 ×

∞

17.1−8.1×α

(2π)

−

exp(

−u

)du)dα

= 0.0123989.

The defuzziﬁed p-value (0.0123989) is smaller

than the signiﬁcance level (0.05), then the decision

will be to reject the null hypothesis at the level δ =

0.05.

6 CONCLUSION

In this work, we presented a hypothesis testing proce-

dure when both data and hypotheses are fuzzy. We

introduced as well a fuzzy p-value with its α-cuts.

We discussed after the defuzziﬁcation of this fuzzy p-

value by the so-called ”signed distance method”. We

ﬁnally proposed numerical examples of one-sided and

two-sided tests, in addition to a small comparison be-

tween different null and alternative hypotheses with

the same hypothetical sample at the same signiﬁcance

level. To conclude, despite the fact that the defuzzi-

ﬁcation step reduces the amount of information con-

tained in a fuzzy p-value, we thought that in many

cases defuzziying these p-values with the signed dis-

tance can be of a high relevance. In addition, since

testing hypotheses on linguistic variables is in most

cases complicated and not feasible in classical statis-

tics, proposing such an approach to deal with fuzzi-

ness and obtaining a p-value deserves merit in deci-

sion making. Indeed, we can see the defuzzied p-

value as an ”informal indicator” of rejecting or not a

given null hypothesis. For further researches, we will

be interested in testing the method with many other

statistical distributions.

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14.8 15.0 15.2 15.4 15.6 15.8 16.0

0.0 0.2 0.4 0.6 0.8 1.0

Membership functions of the null and alternative hypotheses

Null hypothesis

Alternative hypothesis

Figure 2: The membership functions of the null and the alternative hypotheses - Example 5.1.

0.0 0.2 0.4 0.6 0.8 1.0

Membership function of the fuzzy p−value and the significance level

Fuzzy p−value

Significance level

Figure 3: The membership function of the fuzzy p-value ˜p

- Example 5.1.

99.6 99.8 100.0 100.2 100.4

0.0 0.2 0.4 0.6 0.8 1.0

Membership functions of the null and alternative hypotheses

Null hypothesis

Alternative hypothesis

Figure 4: The membership functions of the null and the alternative hypotheses - Example 5.2.

0.0 0.2 0.4 0.6 0.8 1.0

Membership function of the fuzzy p−value and the significance level

Fuzzy p−value

Significance level

Figure 5: The membership function of the fuzzy p-value ˜p

- Example 5.2.