Approximated Fuzzy p-values by Bootstrapped Fuzzy Distributions and

Fuzzy Hypotheses Testing

Julien Rosset

and Laurent Donz

ASAM Group, University of Fribourg, Boulevard de P

erolles 90, 1700 Fribourg, Switzerland

{Julien.Rosset, Laurent.Donz

e}@unifr.ch

Keywords:

Fuzzy Conﬁdence Interval, Fuzzy Hypothesis Testing Procedure, Fuzzy p-value, Fuzzy Logic, Fuzzy Methods

in Data Analysis.

Abstract:

Although we could dispute the use of p-values, it is a standard tool used by many to know if one has to reject

or not a null hypothesis. With the emergence of fuzzy data, fuzzy hypothesis testing procedures appeared.

Among these testing procedures, various methods to compute crisp or fuzzy p-values arising from fuzzy data

and hypotheses were investigated. However, we noticed that, despite calculating a fuzzy test statistic, none

of these approaches assume a fuzzy distribution. Thus, to remedy this, we tackle the problem of ﬁnding

fuzzy p-values in the context of both fuzzy data and hypotheses while considering the fuzzy distribution of

the test statistic. Finding fuzzy p-values alone is not helpful if one does not know how to use them to make a

decision. This is why we also provide a way to interpret fuzzy p-values and present a decision rule to reject

or not the fuzzy null hypothesis. Additionally, we aim to compare this decision rule to fuzzy statistical testing

procedures. We thus offer an empirical application that compares the decisions obtained from fuzzy p-values

to the results given by a fuzzy hypothesis testing procedure.

1 INTRODUCTION AND

NOTATION

It is debatable whether or not the concept of p-value

is a good tool for testing statistical hypotheses. But,

considering that it is widely used in practice, it is

not in vain to broaden this subject in the context of

fuzzy data and fuzzy hypotheses. In this regard, recall

that the classical statistics analysis can be widened by

(1) using fuzzy data, (2) using fuzzy hypotheses, and

(3) assuming a fuzzy distribution for the test statistic.

Several authors already extended the ﬁrst two. To cite

a few, (Viertl, 2011) investigated the problem of ﬁnd-

ing fuzzy p-values in the context of fuzzy hypothe-

ses and crisp data. (Parchami, 2020) and (Berkachy

and Donz

e, 2022) developed a method to ﬁnd fuzzy

p-values with fuzzy data and hypotheses. Many meth-

ods to compute p-values and fuzzy p-values already

exist. Unfortunately, they are all based on the as-

sumption of a crisp and speciﬁc distribution of the test

statistic, principally for computation reasons. Thus,

to our knowledge, no one has contemplated this latter

point, for instance, using fuzzy distributions to com-

https://orcid.org/0000-0002-7883-1512

https://orcid.org/0000-0003-3522-4672

pute fuzzy p-values related to a statistical inference

case.

Consequently, in the case of fuzzy data and hy-

potheses, one major improvement is to consider the

implied fuzzy distribution of the test statistic to com-

pute fuzzy p-values. A reliable procedure should also

ensure that the obtained fuzzy p-values must be re-

duced to classical p-values when the data and hy-

potheses are crisp. At last, the p-values should be

interpretable, and thus, one needs a decision rule to

evaluate the fuzzy p-values

We recall in Section 2 the main recent develop-

ments on the subject of fuzzy inferences. Then, we

explain in Section 3 our method to compute p-values

and propose a decision rule to interpret them. On

other occasions (see e.g. (Rosset and Donz

e, 2024)),

we had the opportunity to present and develop fuzzy

test statistics. Thus, to compare the results of infer-

ences from fuzzy p-values and those from our fuzzy

test statistics, we will brieﬂy recall these approaches

in Section 4. An empirical analysis is provided in Sec-

tion 5 where two case studies are given to compare the

results based on fuzzy p-values to those obtained via a

fuzzy hypothesis testing procedure and discuss them

thoroughly. Finally, we conclude in Section 6.

Rosset, J. and Donzé, L.

Approximated Fuzzy p-values by Bootstrapped Fuzzy Distributions and Fuzzy Hypotheses Testing.

DOI: 10.5220/0012888300003837

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 16th International Joint Conference on Computational Intelligence (IJCCI 2024), pages 387-395

ISBN: 978-989-758-721-4; ISSN: 2184-3236

387

We will use the following notation and conven-

tions below. First, let us deﬁne by ˜x a fuzzy number

and by µ

˜x

(·) its membership function. We consider

also the α-cuts of ˜x denoted by ˜x

or by its equivalent

in interval form by (x

L,α

R,α

). In practice, triangular

fuzzy numbers are often used. We denote them by a

triplet ˜x = (x

,x,x

) with x

≤ x ≤ x

∈ R.

2 LITERATURE REVIEW

p-values are standard elements of statistical infer-

ences. It is, therefore, natural to analyse the p-values

arising from fuzzy testing procedures. Many aspects

could be considered. Indeed, one could just use fuzzy

hypotheses with crisp data or crisp hypotheses with

fuzzy data or both, i.e. have fuzzy data and hy-

potheses. Traditionally, mimicking the classical case,

fuzzy p-values are computed under the assumption of

a given test statistic distribution. Researchers have

thoroughly studied this. But, in the presence of fuzzy

data and, perhaps, fuzzy hypotheses, a fuzzy distribu-

tion of the test statistic should be considered. To our

knowledge, this latter case has not yet been carefully

considered.

Regarding the traditional approach, (Viertl, 2011)

investigated fuzzy p-values from fuzzy data with crisp

hypotheses. To do so, he used a fuzzy test statis-

tic and assumed a crisp distribution for the sample.

This setting allows him to compute a p-value corre-

sponding to each of the left and right alpha-cuts of

the test statistic. This collection of p-values then cor-

responds to the alpha-cuts of a fuzzy p-value. Un-

like Viertl, (Parchami et al., 2010) analysed the case

where hypotheses are fuzzy and data are crisp. They

introduced the notion of fuzzy p-value by apply-

ing Zadeh’s extension principle. Later, (Parchami,

2020) investigated the case where both hypotheses

and data are fuzzy and studied the fuzzy p-values aris-

ing from Zadeh’s extension principle under the as-

sumption that the test statistic has a crisp distribution

despite the fuzziness of the data. Similarly, (Berkachy

and Donz

e, 2017) found fuzzy p-values in the case

of fuzzy data and hypotheses assuming a crisp dis-

tribution of the test statistic. We can also mention

(Hryniewicz, 2018), who developed a procedure to

ﬁnd fuzzy p-values using fuzzy conﬁdence intervals

in the sense of (Kruse and Meyer, 1987). His method

is based on ﬁnding intersection points between the

fuzzy null hypothesis and the fuzzy conﬁdence inter-

vals of the test statistic. Despite being a very interest-

ing method, the procedure has two problems. First,

it doesn’t deal with a fuzzy distribution of the test

statistic. Secondly, when the fuzzy conﬁdence inter-

val becomes crisp, the intersection points (p-values)

take the value 0 or 1, thus not generalising the classi-

cal approach to ﬁnd p-values.

As mentioned above, the study of fuzzy p-values

involves fuzzy hypothesis testing procedures, which

are connected to the notion of fuzzy conﬁdence in-

tervals. Hence, let us brieﬂy review some results in

this context. (Kruse and Meyer, 1987) gave a theoret-

ical deﬁnition of a fuzzy conﬁdence interval. Many

researchers have utilised and reﬁned their deﬁnition

since its introduction in 1987. Indeed, (Viertl and

Yeganeh, 2016) introduced the notion of fuzzy con-

ﬁdence regions. (Kahraman et al., 2016) and (Kahra-

man et al., 2019) studied interval-valued intuitionistic

fuzzy sets (IVIFSs) and hesitant fuzzy sets (HFS) and,

based on them, developed two methods to construct

fuzzy conﬁdence intervals. (Wu, 2009) solved optimi-

sation problems involving a fuzzy Gaussian distribu-

tion to build fuzzy conﬁdence intervals. (Chachi and

Taheri, 2011) proposed fuzzy conﬁdence intervals for

the mean of a fuzzy normal distribution. These ap-

proaches involve a predeﬁned distribution of the test

statistic, which, in practical cases, is not always possi-

ble to know. To bypass this difﬁculty, (Berkachy and

Donz

e, 2022) developed a general procedure to con-

struct fuzzy conﬁdence intervals using the likelihood

ratio method and a bootstrap procedure extended to

the fuzzy environment to estimate the distribution of

this likelihood ratio.

Due to the seminal works of Bradley Efron, boot-

strap techniques are widely known among statisti-

cians. Then, many fuzzy bootstrap approaches were

introduced and used. Among the recent works, we

can point out (Berkachy and Donz

e, 2020), who gave

two algorithms to generate fuzzy bootstrapped sam-

ples with the notions of location and dispersion and

(Grzegorzewski and Romaniuk, 2021), who provided

an epistemic approach to fuzzy bootstrap techniques

for fuzzy data.

Finally, let us mention two concepts that come in

handy in constructing fuzzy conﬁdence intervals. The

ﬁrst one is the fuzzy quantile function proposed by

(Shvedov, 2016), a fuzzy extension of the classical

quantile function. The second one is the construction

of fuzzy distributions where (Areﬁ et al., 2012) de-

scribe how to ﬁnd empirical fuzzy distributions.

3 FUZZY P-VALUES AND

INFERENCES

Statistical inferences are based on samples. Let us de-

note by X

,...,X

a random sample drawn from the

distribution of interest. Assuming the data are fuzzy,

FCTA 2024 - 16th International Conference on Fuzzy Computation Theory and Applications

388

we write as

,...,

, the fuzzy equivalent of the ran-

dom sample. The fuzzy realisations of the fuzzy ran-

dom sample are given by ˜x

,..., ˜x

and their α-cuts

by ˜x

, i = 1,...,n.

The fuzzy random mean is readily computable

X =

∑

i=1

and its realisation is given by

¯x =

∑

i=1

˜x

It’s common practice to express a fuzzy hypothe-

sis using linguistic terms. For example, we could say,

”The value of the parameter θ is more or less equal to

”, or rewritten in terms of differences, i.e. ”The dif-

ference between the value of the parameter θ and θ

is more or less equal to 0”. Thus, we could write, for

instance, the following two-sided fuzzy hypotheses:

: θ =

0 against

: θ ̸=

0. (1)

Of course, in the same way, one-sided fuzzy hypothe-

ses could be written as:

: θ ≤

0 against

: θ >

0, (2)

: θ ≥

0 against

: θ <

0. (3)

Remark that the null fuzzy hypothesis

or the al-

ternative one

can be modelled by triangular num-

bers, i.e. (H

) or (H

). On the other

hand, it is common practice to implement the in-

ference process through a proper test statistic. Let

T = (T

,T,T

) be a fuzzy statistic to test

. For

example, if we want to test the mean of the distribu-

tion, we could use:

T := (

X −θ

) ⊘

¯x

, (4)

where ⊘ is the fuzzy division operator and

¯x

is the

standard-error of

X. We note by

t = (t

,t,t

) the

fuzzy realisation of

T .

In a fuzzy approach, the distribution of the statistic

is difﬁcult to know, even more so if one doesn’t want

to assume strong conditions about the form of the dis-

tribution. As we need this distribution, we propose

using a bootstrapped one instead. One of the many ad-

vantages of using this approach is that it is relatively

easy to generate a fuzzy bootstrapped distribution for

T . Let B be the number of bootstrapped samples and

denote by

⋆

,...,

⋆

,...

⋆

the bootstrapped distribution

of the test statistic. The fuzzy empirical mean is given

∗

= (

∑

⋆

)/B = (

∗L

∗

∗R

We consider, on one hand, the fuzzy observed

statistic

t = (t

,t,t

) and its absolute value:

t| =(t

abs

)



min(|t

|,|t|, |t

|),

|t|,

max(|t

|,|t|, |t



and, on the other hand, the centred bootstrapped dis-

tribution given by

⋆

−

∗

,...,

⋆

−

∗

,...

⋆

−

∗

. We

note this distribution by

∗

= (F

∗,L

∗

∗,R

). Then,

we can compute the following empirical approxima-

tion for the fuzzy p-value ˜p = (p

, p, p

) of the two-

sided hypotheses test (1):

= 2 ·

#(F

⋆,L

> t

abs

) + 1

B + 1

p = 2 ·

#(F

⋆

> t

abs

) + 1

B + 1

= min(2 ·

#(F

⋆,R

> t

abs

) + 1

B + 1

,1),

where # gives the number of cases for which the con-

dition in parentheses is true. For the one-sided hy-

potheses test (2), the fuzzy p-value ˜p = (p

, p, p

) is

calculated as:

#(F

⋆,L

> t

) + 1

B + 1

p =

#(F

⋆

> t) + 1

B + 1

= min(

#(F

⋆,R

> t

) + 1

B + 1

,1),

and for the second one-sided hypotheses test (3), we

have:

#(F

⋆,R

< t

) + 1

B + 1

p =

#(F

⋆

< t) + 1

B + 1

= min(

#(F

⋆,L

< t

) + 1

B + 1

,1).

We must underline that we centred the fuzzy

bootstrapped distribution (left, centre and right parts)

around

∗

. Another possibility would’ve been to

centre it around

∗

by writing it as

⋆

⊖

∗

,...,

⋆

⊖

∗

,...

⋆

⊖

∗

where ⊖ is the fuzzy substraction opera-

tor. Nevertheless, we did not choose it for empirical

reasons. The algorithm 1 summarises the computa-

tion steps to ﬁnd the bootstrapped fuzzy p-values.

Fuzzy p-values ˜p = (p

, p, p

) are not always easy

to interpret, unlike their crisp counterparts. Though

a proper defuzziﬁcation method can help the inter-

pretation, we propose constructing a speciﬁc measure

whose meaning is straightforward. Let δ ∈ (0, 1) be

a chosen signiﬁcance level. We expect δ ∈ [p

, p

but this could not be true. Let us suppose that this

is the case. Furthermore, we assume ﬁrst that δ ≤ p.

In the space [0, 1]

, consider the point (δ, y

) given

by the intersection between the line x = δ and the

triangular fuzzy number (p

, p, p

) and compute the

area A of the new resulting triangle given by the

Approximated Fuzzy p-values by Bootstrapped Fuzzy Distributions and Fuzzy Hypotheses Testing

389

Algorithm 1: Fuzzy p-values by a Bootstrapped Fuzzy Dis-

tribution.

Data: ˜x

,..., ˜x

a fuzzy realisation of the

random sample.

Result: Fuzzy p-values.

begin

Compute: (

t, |

t |) ;

for b = 1 to B do

Draw sample ˜s

= { ˜x

∗

,..., ˜x

∗

} ;

Compute: (

∗

) ;

end

Compute: (

∗

, ˜p) ;

return ˜p ;

end

three points ((p

,0),(δ,y

),(δ,0)). If δ > p, then A

is the area of the polygon given by the four points

((p

,0),(p, 1), (δ, y

),(δ,0)). On the other hand, let

P be the area of the fuzzy p-value. Then, we can pro-

pose the following crisp measure of the p-value:

re j

= A/P. (5)

We set r

re j

= 0 if δ < p

and r

re j

= 1 if δ > p

. The

real number r

re j

is between 0 and 1 and tells us how

much the fuzzy p-value tends to reject the null hy-

pothesis. On the opposite, r

acc

= 1−r

re j

∈ [0,1] gives

us the tendency of not rejecting the null hypothesis.

If r

re j

> 0.5, we will tend to reject the null hypothe-

sis, while if r

re j

< 0.5, we will tend not to reject it.

No decision can be taken for r

re j

= 0.5. r

re j

and r

acc

can be seen as weights to respectively reject or not to

reject the null hypothesis. Figure 1 depicts the ten-

dency to reject the null hypothesis in blue. This area

equals to r

re j

= 0.22. The orange region has an area

of r

acc

= 1 − r

re j

= 0.78 and gives the tendency not to

reject the null hypothesis.

4 FUZZY TESTS AND

CONFIDENCE INTERVALS

(Rosset and Donz

e, 2024) proposed the following in-

ference procedure. They built two real-valued func-

tions of the fuzzy random sample

,...,

deﬁned

rej

: (F

∗

(R))

→ [0,1],

acc

: (F

∗

(R))

→ [0,1].

It can be proven that

rej

and

acc

are two fuzzy

statistical tests on F

∗

(R), the class of the non-empty

compact, convex and normal fuzzy sets on R. Func-

tions of the random sample, their values are given by

the intersection of the fuzzy null hypothesis

and

Figure 1: Example of a fuzzy p-value. Here ˜p =

, p, p

) = (0.01,0.2, 0.4). The tendency to reject the null

hypothesis (in blue) is r = 0.22, and the tendency to not re-

ject the null hypothesis (in orange) is 1 − r = 0.78

a fuzzy conﬁdence interval

⊓ for the parameter to be

tested θ. For technical details, the reader may consult

(Rosset and Donz

e, 2024). The interpretation of these

statistical tests is given in the following deﬁnition:

Deﬁnition 1 (Decision rules).

Let be

rej

[0,1]

rej,α

dα and L

acc

[0,1]

acc,α

dα.

(6)

rej

and L

acc

are called, respectively, the levels of

rejection and acceptance (not rejection) of the (fuzzy)

null hypothesis

1. If L

rej

= 1,

is rejected. In this case L

acc

= 0;

2. If L

rej

= 0,

is not rejected. In this case L

acc

= 1;

3. If 0 < L

rej

< 1, the test shows how strong

rejected. The higher L

rej

is, the stronger the test

rejects

;

4. If 0 < L

acc

< 1, the test shows how strong

not rejected. The higher L

acc

is, the stronger the

test doesn’t reject

;

5. If L

rej

> L

acc

tends to be rejected;

6. If L

rej

< L

acc

tends to be not rejected;

7. If L

rej

= L

acc

, no decision can be taken.

The fuzzy conﬁdence interval

⊓ for the parame-

ter θ of the distribution, deﬁned at a given conﬁdence

level 1 − δ, plays a crucial role in the construction of

the two fuzzy tests

re j

and

acc

. Hence, its construc-

tion is a key point of the inference procedure. Several

methods can be used to build such fuzzy conﬁdence

intervals. We will use the one proposed in (Rosset

and Donz

e, 2024), where we introduced a method

based on bootstrap techniques. This method is attrac-

tive because no speciﬁc (fuzzy) distribution should be

assumed.

FCTA 2024 - 16th International Conference on Fuzzy Computation Theory and Applications

390

5 APPLICATIONS

5.1 Case 1

Let us test the fuzzy mean of a fuzzy distribution at

a signiﬁcance level of 5%. Table 1 depicts the fuzzy

observations taken from (Berkachy and Donz

e, 2020)

that will be used to compute the fuzzy p-values as-

sociated with different fuzzy null hypotheses as illus-

trated in Table 2. The fuzzy mean estimation is the tri-

angular fuzzy number

˜x = (1.8,2.8,3.8) and a fuzzy

conﬁdence interval at a conﬁdence level of 95% is

the fuzzy trapezoidal number (0.933,1.933, 3.6, 4.6).

This fuzzy conﬁdence interval will allow us to test the

null hypotheses and compute their associated levels of

acceptance and rejection as shown in Table 2. Lastly,

we will compare the obtained fuzzy p-values to the re-

sults given by the fuzzy hypothesis testing procedure.

Let us ﬁrst talk about the results given by the fuzzy

hypothesis testing procedure at a 5% signiﬁcant level

in Table 2. One can easily see that in the extreme

cases (tests 1, 2, 3, 13) when the fuzzy null hypoth-

esis is outside the fuzzy conﬁdence interval at 95%

(0.93,1.93,3.6,4.6), the levels of acceptance and re-

jection are respectively 0 and 1. Moreover, when

the fuzzy null hypothesis is entirely contained in the

fuzzy conﬁdence interval’s core (1.93,3.6), e.g. tests

8, 9, 10, the levels of acceptance and rejection are

respectively 1 and 0. These two extreme situations

give the expected binary results one would have found

using a crisp testing procedure. Let us now anal-

yse the cases when the fuzzy null hypothesis overlaps

with the fuzzy conﬁdence interval’s fuzzy regions, i.e.

(0.93,1.93) and (3.6,4.6). With tests 4, 5, 6 and 7 of

Table 2, when the null hypothesis overlaps the inter-

val (0.93, 1.93), we observe that the values of L

acc

and L

re j

change depending on the shape of H

. As

opposed to the crisp case, we see that the values L

acc

and L

re j

are between 0 and 1 and will tend to reject

or not the null hypothesis. Tests 5 and 7 depicted on

Figures 3 and 4 show how the shape of a crispier null

hypothesis (Figure 4) or fuzzier null hypothesis (Fig-

ure 3) affects the values of L

acc

and L

re j

. Lastly, when

the fuzzy conﬁdence interval is fully overlapped (tests

11 and 12), the amount of area outside the fuzzy con-

ﬁdence interval will dictate the value of the rejection

level. Conversely, the amount of area inside the fuzzy

conﬁdence interval will dictate the value of the ac-

ceptance level. This situation is illustrated in Figures

2 and 5.

Let us now discuss the fuzzy p-values obtained by

the method displayed in Section 3. In extreme cases,

as in test 1 and 13, we ﬁnd binary values. For test

1, we ﬁnd the values, r

re j

= 0, r

acc

= 1 and for test

Figure 2: Fuzzy test of the mean θ,

: θ = (0,3.8, 6)

against

: θ ̸= (0,3.8,6). FH0 stands for ”Fuzzy Null Hy-

pothesis”, and FCI stands for ”Fuzzy Conﬁdence Interval”.

The tendency to not reject the null hypothesis (acceptance

area in purple on the left graph) is L

acc

= 0.68. The ten-

dency to reject the null hypothesis (rejection area in purple

on the right graph) is L

re j

= 0.43.

Figure 3: Fuzzy test of the mean θ,

: θ = (1,1.5, 2)

against

: θ ̸= (1,1.5,2). FH0 stands for ”Fuzzy Null Hy-

pothesis”, and FCI stands for ”Fuzzy Conﬁdence Interval”.

The tendency to not reject the null hypothesis (acceptance

area in purple on the left graph) is L

acc

= 0.75. The ten-

dency to reject the null hypothesis (rejection area in purple

on the right graph) is L

re j

= 0.58.

13, the values r

re j

= 1, r

acc

= 0. This shows that

we tend to strongly reject (respectively not reject) the

null hypothesis, which conﬁrms the decisions taken

with L

re j

and L

acc

. However, one quickly notices that

the fuzzy p-values tends to take extreme values for

the left and right parts. Excepted tests 1, 12 and 13,

the left and right parts of every fuzzy p-value are 0

and 1. This makes the interpretation of these fuzzy

p-values quite hard. One can, however, observe that

when the null hypothesis is fully contained in the re-

gion delimited by (1.93,3.6), r

re j

is very low, and r

acc

is almost 1, which is in accordance with the results of

the fuzzy testing procedure. However, in tests 2, 3

Approximated Fuzzy p-values by Bootstrapped Fuzzy Distributions and Fuzzy Hypotheses Testing

391

Figure 4: Fuzzy test of the mean θ,

: θ = (1.7, 1.75,1.78)

against

: θ ̸= (1.7,1.75,1.78). FH0 stands for ”Fuzzy

Null Hypothesis” and FCI stands for ”Fuzzy Conﬁdence

Interval”. The tendency to not reject the null hypothesis

(acceptance area in purple on the left graph) is L

acc

= 0.96.

The tendency to reject the null hypothesis (rejection area in

purple on the right graph) is L

re j

= 0.34.

Figure 5: Fuzzy test of the mean θ,

: θ = (0,4.5, 8)

against

: θ ̸= (0,4.5, 8). The tendency to not reject the

null hypothesis (acceptance area in purple on the left graph)

is L

acc

= 0.44. The tendency to reject the null hypothesis

(rejection area in purple on the right graph) is L

re j

= 0.63.

and 4, when the null hypothesis should be rejected,

the fuzzy p-values give a r

acc

of 0.9025 and a r

re j

0.0975. While it seems surprising, it can be explained

by looking at the meaning behind these two quanti-

ties. On Figure 6,

One can see that although the left and centre parts

of the fuzzy p-value favour rejecting the null hypoth-

esis (both have a value of 0), r

acc

(orange area) is very

large because the right part of the fuzzy p-value is

1. This causes the fuzzy p-value to have a great r

acc

and low r

re j

. This result shows the limits of the use

of fuzzy p-values. However, let us point out the re-

sults for test 12 depicted in Figure 7. Observe that

they are in accordance with the results found with the

Figure 6: Extreme fuzzy p-value ˜p = (0, 0,1). r

acc

0.9025 and r

re j

= 0.0975.

testing procedure. L

acc

is 0.44 and r

acc

is 0.39 which

tends to reject the null hypothesis. This is conﬁrmed

by L

re j

= 0.63 and r

re j

= 0.61. This good result is

a byproduct of having a fuzzy p-value with a small

right part, which can be easily interpreted. Test 12

shows that when the fuzzy p-value is not too extreme,

i.e. does not have a right part close to 1, it holds in-

formation about whether or not to reject the null hy-

pothesis. To conﬁrm this assumption, let us consider

another set of crispier observations that will give us

crispier p-values.

5.2 Case 2

Let X ∼ N(2.8,3), U

∼ Unif(0.001,0.006), U

∼

Unif(0.004,0.009). Let x

, u

1,i

, u

2,i

be realisations of

the random variables X, U

and U

respectively. We

can form fuzzy observations in the following way. Let

= x

− u

1,i

and x

= x

+ u

2,i

. We generate N fuzzy

triangular observations ˜x

= (x

), i = 1,...,N.

By generating N = 500 observations via the above

procedure, we ﬁnd an estimation for the fuzzy mean

¯x = (2.693,2.697, 2.704) and a fuzzy conﬁdence in-

terval at 95% conﬁdence level (2.44,2.45,2.94,2.95).

Table 5 displays fuzzy p-values obtained by a two-

sided test for different null hypotheses for these 500

observations. Table 3 displays fuzzy p-values ob-

tained by a one-sided test for the mean θ,

: θ <

against

: θ ≥

, where

is the conjectured fuzzy

triangular number used to characterise the null hy-

pothesis

. Table 4 shows fuzzy p-values obtained

by a one-sided test for the mean θ,

: θ >

against

: θ ≤

. Now that our fuzzy observations are

crispier, the obtained fuzzy p-values are also crispier.

This allows us to compute r

acc

and r

re j

that hold in-

formation. Indeed, on Table 3, which is about the

one-sided test

: θ <

against

: θ ≥

, we can

see with tests 1 and 2 that the fuzzy p-value strongly

rejects the null hypothesis with a r

re j

of 1. Tests 3,

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392

4, 5 show us how different shapes for the null hy-

pothesis yield to different non binary values r

re j

and

acc

. At last, Table 3 also depicts how shifting null

hypotheses from (2.25, 2.35, 2.40) to (2.5, 2.55, 2.6)

gives values r

acc

gradually going from 0 to 1 and val-

ues r

re j

gradually going from 1 to 0. Table 4 illus-

trates the same results, but this time for the opposite

side,

: θ >

against

: θ ≤

. Finally, for the

two-sided test illustrated by Table 5, we again observe

the same behaviour. Test 6 yields values r

acc

= 0 and

re j

= 1. As opposed to it, test 1 gives values r

acc

= 1

and r

re j

= 0. Tests 2, 3, 4, 5 yield non binary values

for r

acc

and r

re j

. As a whole, we observe that r

acc

gradually goes from 1 to 0 while r

re j

gradually goes

from 0 to 1 when considering the tests 1, 2, 3, 4, 5,

6. This allows us to empirically discover the fuzzy re-

gion in between (2.35,2.4,2.45) and (2.45,2.5,2.5)

where the null hypothesis is starting to be more and

more rejected, respectively less and less rejected de-

pending on the null hypothesis shape.

Figure 7: Crispier fuzzy p-value ˜p = (0, 0,0.132). r

acc

0.39 and r

re j

= 0.61.

6 CONCLUSION

In this paper, we introduced a new procedure to

ﬁnd fuzzy p-values based on precedent works, which

generalises the computation of crisp p-values. Our

method revolves around generating a centred fuzzy

bootstrapped statistic distribution to test and count

how many of these bootstrapped observations are

greater than an observed statistic. Then, we explained

how the obtained fuzzy p-values could be interpreted

as a ratio of a rejection or acceptance area over the to-

tal area formed by the fuzzy p-value. We then enun-

ciated a fuzzy hypothesis testing procedure to be able

to compare fuzzy p-values to results obtained via this

testing procedure. The main takeaway from this com-

parison is that fuzzy p-values tend to be very impre-

cise, with the observations getting fuzzier. However,

it is still a helpful tool when the observations become

crisper. Indeed, in the latter scenario, fuzzy p-values

give us an idea of how much we’re inside the fuzzy

conﬁdence interval or how much we’re outside of it.

In extreme cases, fuzzy p-values give the same binary

results coming from crisp p-values.

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APPENDIX

Table 1: Fuzzy observations from Berkachy and Donz

e (Berkachy and Donz

e, 2020).

˜x

4 (3,4,5) 3 (2,3,4) 3 (2,3,4) 5 (4,5,6) 3 (2,3,4)

1 (0,1,2) 2 (1,2,3) 2 (1,2,3) 2 (1,2,3) 3 (2,3,4)

Note: ¯x = 2.8,

˜x = (1.8,2.8,3.8).

Table 2: Fuzzy Hypothesis testing at a 5% signiﬁcant level and fuzzy p-values.

Tests

acc

re j

Fuzzy p-value r

acc

re j

1 (-0.3,-0.2,-0.1) 0 1 (0,0,0.032) 0 1

2 (0,0.5,0.8) 0 1 (0,0,1) 0.9025 0.0975

3 (0.8,0.85,0.9) 0 1 (0,0,1) 0.9025 0.0975

4 (0.95,1,1.05) 0.12 0.99 (0,0,1) 0.9025 0.0975

5 (1,1.5,2) 0.75 0.58 (0,0,1) 0.9025 0.0975

6 (1.8,1.9,2) 0.99 0.08 (0,0.012,1) 0.914 0.086

7 (1.7,1.75,1.78) 0.96 0.34 (0,0.01,1) 0.912 0.088

8 (2,2.5,3.5) 1 0 (0,0.5,1) 0.995 0.005

9 (2.4,2.45,2.5) 1 0 (0,0.32,1) 0.992 0.008

10 (2.6,2.8,3.2) 1 0 (0,0.87,1) 0.997 0.003

11 (0,3.8,6) 0.68 0.43 (0,0.08,1) 0.91 0.09

12 (0.5,4.5,8) 0.44 0.63 (0,0,0.132) 0.39 0.61

13 (7,8,9) 0 1 (0,0,0) 0 1

This Table shows ﬁrst the levels of acceptance (L

acc

) and rejection (L

re j

) given by the fuzzy testing procedure of

section 4. Then, the levels of acceptance (r

acc

) and rejection (r

re j

) of the associated fuzzy p-values are displayed.

The fuzzy trapezoidal number (0.933,1.933,3.6,4.6) is the computed fuzzy conﬁdence interval at a 95% conﬁ-

dence level for the distribution’s mean. The values shown in red are in contradiction with the results of the testing

procedure of section 4.

Table 3: Fuzzy statistics inferences at a 5% signiﬁcance level. Fuzzy p-values for randomly generated fuzzy observations

(one-sided test for the mean, θ,

: θ <

against

: θ ≥

Tests

Fuzzy p-value r

acc

re j

1 (2.25,2.35,2.40) (0,0.002,0.016) 0 1

2 (2.39,2.42,2.45) (0.002,0.009,0.04) 0 1

3 (2.43,2.46,2.5) (0.006,0.034,0.087) 0.32 0.68

4 (2.44,2.46,2.5) (0.009,0.039,0.088) 0.44 0.56

5 (2.45,2.48,2.52) (0.02,0.048,0.118) 0.68 0.32

6 (2.5,2.55,2.6) (0.051,0.126,0.245) 1 0

This Table shows the levels of acceptance (r

acc

) and rejection (r

re j

) of different fuzzy null hypotheses via the ob-

tained fuzzy p-values. The fuzzy mean of the dataset is (2.693,2.697,2.704) and the associated fuzzy conﬁdence

interval at a 95% conﬁdence level for the distribution’s mean is (2.42, 2.49, 2.94, 3.01).

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394

Table 4: Fuzzy statistics inferences at a 5% signiﬁcance level. Fuzzy p-values for randomly generated fuzzy observations

(one-sided test for the mean, θ,

: θ >

against

: θ ≤

)).

Tests

Fuzzy p-value r

acc

re j

1 (2.7,2.75,2.8) (0.168,0.346,0.511) 1 0

2 (2.75,2.8,2.85) (0.1,0.208,0.354) 1 0

3 (2.8,2.85,2.9) (0.042,0.117,0.221) 0.993 0.007

4 (2.85,2.9,2.95) (0.029,0.057,0.129) 0.85 0.15

5 (2.87,2.92,2.94) (0.026,0.035,0.1) 0.52 0.48

6 (2.9,3,3.1) (0.001,0.004,0.071) 0.1 0.9

7 (3,3.1,3.2) (0,0,0.009) 0 1

This Table shows the levels of acceptance (r

acc

) and rejection (r

re j

) of different fuzzy null hypotheses via the ob-

tained fuzzy p-values. The fuzzy mean of the dataset is (2.693,2.697,2.704) and the associated fuzzy conﬁdence

interval at a 95% conﬁdence level for the distribution’s mean is (2.42, 2.49, 2.94, 3.01).

Table 5: Fuzzy statistics inferences at a 5% signiﬁcance level. Fuzzy p-values for randomly generated fuzzy observations

(one-sided test for the mean, θ,

: θ =

against

: θ ̸=

Tests

Fuzzy p-value r

acc

re j

1 (2.58,2.6,2.62) (0.318,0.466,0.594) 1 0

2 (2.45,2.5,2.55) (0.03,0.112,0.322) 0.984 0.016

3 (2.4,2.45,2.5) (0.01,0.052,0.134) 0.7 0.3

4 (2.37,2.4,2.44) (0.004,0.016,0.062) 0.06 0.94

5 (2.35,2.4,2.45) (0.002,0.026,0.086) 0.74 0.26

6 (2.3,2.35,2.4) (0,0.004,0.02) 0 1

This Table shows the levels of acceptance (r

acc

) and rejection (r

re j

) of different fuzzy null hypotheses through

the obtained fuzzy p-values. The fuzzy mean of the dataset is (2.693,2.697,2.704) and the associated fuzzy

conﬁdence interval at a 95% conﬁdence level for the distribution’s mean is (2.42, 2.49, 2.94, 3.01).

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395