Soft Comput (2016) 20:1537–1548DOI 10.1007/s00500-015-1604-x

METHODOLOGIES AND APPLICATION

Parametric testing statistical hypotheses for fuzzy randomvariables

Gholamreza Hesamian · Mehdi Shams

Published online: 5 February 2015© The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract In this paper, amethod is proposed for testing sta-tistical hypotheses about the fuzzy parameter of the underly-ing parametric population. In this approach, using definitionof fuzzy random variables, the concept of the power of testand p value is extended to the fuzzy power and fuzzy p value.To do this, the concepts of fuzzy p value have been definedusing the α-optimistic values of the fuzzy observations andfuzzy parameters. This paper also develop the concepts offuzzy type-I, fuzzy type-II errors and fuzzy power for theproposed hypothesis tests. To make decision as a fuzzy test,a well-known index is employed to compare the observedfuzzy p value and a given significance value. The result pro-vides a fuzzy test function which leads to some degrees toaccept or to reject the null hypothesis. As an application ofthe proposed method, we focus on the normal fuzzy randomvariable to investigate hypotheses about the related fuzzyparameters. An applied example is provided throughout thepaper clarifying the discussions made in this paper.

Keywords α-Optimistic value · Fuzzy random variable ·Fuzzy normal variable · Fuzzy parameter · Fuzzy powertest · Fuzzy type-I error · Fuzzy type-II error

Communicated by V. Loia.

G. Hesamian (B)Department of Statistics, Payame Noor University,19395-3697 Tehran, Irane-mail: gh.hesamian@pnu.ac.ir

M. ShamsDepartment of Mathematical Sciences,University of Kashan, Isfahan, Irane-mail: mehdishams@kashanu.ac.ir

1 Introduction

The purpose of statistical inference is to draw conclusionsabout a population on the basis of data obtained from a sam-ple of that population. Hypothesis testing is the process usedto evaluate the strength of evidence from the sample andprovides a framework for making decisions related to thepopulation, i.e., it provides a method for understanding howreliably one can extrapolate observed findings in a sampleunder study to the larger population from which the samplewas drawn. The investigator formulates a specific hypoth-esis, evaluates data from the sample, and uses these datato decide whether they support the specific hypothesis. Theclassical parametric approaches usually depend on certainbasic assumptions about the underlying population such as:crisp observations, exact parameters, crisp hypotheses, andcrisp possible decisions. In practical studies, however, it isfrequently difficult to assume that the parameter, for whichthe distribution of a random variable is determined, has a pre-cise value or the value of the random variable is recorded asa precise value or the hypotheses of interest are presented asexact relations, and so on. Therefore, to achieve suitable test-ing statistical methods dealing with imprecise information,we need to model the imprecise information and extend theusual approaches to imprecise environments. Since its intro-duction by Zadeh (1965), fuzzy set theory has been devel-oped and applied in some statistical contexts to deal withuncertainty conditions such as above situations. Specially,the topic of testing statistical hypotheses in fuzzy environ-ments has extensively been studied. Below is a brief reviewof some studies relevant to the present work. Arnold (1996,1998) presented an approach for testing fuzzily formulatedhypotheses based on crisp data, in which he proposed andconsidered generalized definitions of the probabilities of theerrors of type-I and type-II. Viertl (2006, 2011) used the

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extension principle to obtain the generalized estimators for acrisp parameter based on fuzzy data. He also developed someother statistical inferences for the crisp parameter, such asgeneralized confidence intervals and p value, based on fuzzydata. Taheri and Behboodian (1999) formulated the prob-lem of testing fuzzy hypotheses when the observations arecrisp. They presented some definitions for the probabilitiesof type-I and type-II errors, and proved an extended ver-sion of the Neyman–Pearson Lemma. Their approach hasbeen extended by Torabi et al. (2006) to the case in whichthe data are fuzzy, too. Taheri and Behboodian (2001) alsostudied the problem of testing hypotheses from a Bayesianpoint of view when the observations are ordinary and thehypotheses are fuzzy. Taheri and Arefi (2009) presented anapproach to the problem of testing fuzzy hypotheses, basedon the so-called fuzzy critical regions. Grzegorzewski (2000)suggested some fuzzy tests for crisp hypotheses concerningan unknown parameter of a population using fuzzy randomvariables (FRVs). Montenegro et al. (2001, 2004), using ageneralized metric for fuzzy numbers, proposed a method totest hypotheses about the fuzzy mean of a FRV in one andtwo populations settings. Gonzalez-Rodríguez et al. (2006)extended a one-sample bootstrap method of testing about themean of a general fuzzy random variable. Gil et al. (2006)introduced a bootstrap approach to the multiple-sample testof means for imprecisely valued sample data. Chachi andTaheri (2011) introduced a new approach to construct fuzzyconfidence intervals for the fuzzy mean of a FRV. Filzmoserand Viertl (2004) and Parchami et al. (2010) presented pvalue-based approaches to the problem of testing hypothe-sis, when the available data or the hypotheses of interest arefuzzy, respectively.Hryniewicz (2006b) investigated the con-cept of p value in a possibilistic context in which the conceptof p value is generalized for the case of imprecisely definedstatistical hypotheses and vague statistical data.

On the other hand, there have been some studieson non-parametric statistical testing hypotheses in fuzzyenvironment. Concerning the purposes of this paper, letus briefly review some of the literature on this topic.Kahraman et al. (2004) proposed some algorithms for fuzzynon-parametric rank-sum tests based on fuzzy random vari-ables.Grzegorzewski (1998) introduced amethod to estimatethe median of a population using fuzzy random variables.He (Grzegorzewski 2004) also demonstrated a straightfor-ward generalization of some classical non-parametric testsfor fuzzy random variables based on a metric in the space offuzzy numbers. He also (Grzegorzewski 2005, 2009) studiedsome non-parametric median fuzzy tests for fuzzy observa-tions showing a degree of possibility and a degree of neces-sity (Dubois and Prade 1983) for evaluating the underlyinghypotheses. In addition, he (Grzegorzewski 2008) proposeda modification of the classical sign test to cope with fuzzydata which was so-called bi-robust test, i.e., a test which is

both distribution free and which does not depend so heavilyon the shape of the membership functions used for modelingfuzzy data. Denœux et al. (2005), using a fuzzy partial order-ing on closed intervals, extended the non-parametric rank-sum tests based on fuzzy data. For evaluating the hypothe-ses of interest at a crisp or a fuzzy significance level, theyemployed the concepts of fuzzy p value and degree of rejec-tion of the null hypothesis quantified by a degree of possi-bility and a degree of necessity. Hryniewicz (2006a) investi-gated the fuzzy version of the Goodman–Kruskal γ -statisticdescribed by ordered categorical data. Lin et al. (2010) con-sidered the problem of two-sample Kolmogorov–Smirnovtest for continuous fuzzy intervals based on a crisp test statis-tic. Taheri andHesamian (2011) introduced a fuzzyversion ofthe Goodman–Kruskal γ -statistic for two-way contingencytables when the observations were crisp, but the categorieswere described by fuzzy sets. In this approach, a method wasalso developed for testing of independence in the two-waycontingency tables. Taheri and Hesamian (2012) extendedthe Wilcoxon signed-rank test to the case where the avail-able observations are imprecise and underlying hypothesesare crisp. Hesamian and Chachi (2013) developed the con-cepts of fuzzy cumulative distribution function and fuzzyempirical cumulative distribution function and investigatedthe large sample property of the classical empirical cumu-lative distribution function for fuzzy empirical cumulativedistribution function. They proposed a method for develop-ing two-sample Kolmogorov–Smirnov test for the case whenthe data are observations of fuzzy random variables, and thehypotheses are imprecise. Formore on fuzzy statistics includ-ing testing hypotheses for imprecise data, see for exampleBertoluzza et al. (2002), Buckley (2006), Kruse and Meyer(1987), Nguyen and Wu (2006), Viertl (2011).

This paper develops an approach to test hypotheses for anunknown fuzzy parameter based on fuzzy random variables.To do this, we extend the concept of fuzzy power functionand fuzzy p value to investigate the hypotheses of interest.Finally, a decision rule is suggested to accept or reject the nulland alternative hypotheses. We also provide a computationalprocedure and an example to express the proposed methodto test statistical hypotheses for a normal FRV.

This paper is organized as follows: Section 2 brieflyreviews the classical parametric testing hypotheses and somedefinitions from fuzzy numbers. In the same section, someresults about α-optimistic values of a fuzzy number arederived to recontract a so-called definition of fuzzy randomvariables introduced in Sect. 3. In Sect. 4, one-sided andtwo-sided hypotheses about a fuzzy parameter of a FRV aredefined. The concept of fuzzy power and fuzzy p value fortesting hypotheses about a fuzzy parameter of a continuousparametric population is also introduced. Then, the proposedmethod is applied for testing hypotheses about the fuzzypara-meters of a normal FRV in Sect. 4.1. A numerical example

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Parametric testing statistical hypotheses 1539

is then provided in Sect. 5 to clarify the discussions made inthis paper. Section 6 compares the proposed method to theother similar existing methods. Finally, a brief conclusion isprovided in Sect. 7. In addition, the proofs of the main resultsin this paper are provided in Appendix.

2 Preliminaries

2.1 Testing statistical hypotheses: the classical approach

Let X = (X1, . . . , Xn) be a random sample, with theobserved value x = (x1, . . . , xn), from a continuous pop-ulation with density function fθ where θ ∈ Θ ⊆ R

p, p ≥ 1.The decision rule to test the null hypothesis H0 : θ ∈ Θ0

versus the alternative H1 : θ ∈ Θ1 is typically denoted by

ϕ(x) ={1 if T (x) ∈ Cδ,

0 if T (x) /∈ Cδ,

whereCδ ⊆ R is the critical region, δ is the significance level,and T (X) is a test statistic. The power function of ϕ(X) isdefined by πϕ(θ) = Eθ [ϕ(X)] = Pθ (T (X) ∈ Cδ). Notethat, if the underlying family of distribution functions hastheMLR property (Monotone Likelihood Ratio property) inT then πϕ(θ) = Eθ [ϕ(X)] is a non-decreasing function ofθ (for more see Lehmann and Romano 2005; Shao 2003).Finally, at a given significance level δ, the hypothesis H0

is completely rejected if and only if p value < δ, wherep value = inf{δ ∈ [0, 1] : T (x) ∈ Cδ} (Shao 2003). So, toaccept or reject the null hypothesis, we have a test as follows

ψ(x) ={1 if p value < δ,

0 if p value ≥ δ.

2.2 Fuzzy numbers

Let R be the set of all real numbers. A fuzzy set of R is amapping μ A : R → [0, 1], which assigns to each x ∈ R adegree of membership 0 ≤ μ A(x) ≤ 1. For each α ∈ (0, 1],the subset {x ∈ R | μ A(x) ≥ α} is called the level set orα-cut of A and is denoted by A[α]. The set A[0] is alsodefined equal to the closure of {x ∈ R | μ A(x) > 0}. Afuzzy set A of R is called a fuzzy number if it satisfies thefollowing three conditions.

1. For each α ∈ [0, 1], the set A[α] is a compact interval,which will be denoted by [ AL

α , AUα ]. Here, AL

α = inf{x ∈R | μ A(x) ≥ α} and AU

α = sup{x ∈ R | μ A(x) ≥ α}.2. For α, β ∈ [0, 1], with α < β, A[β] ⊆ A[α].3. There is a unique real number x∗ = x ∗

A∈ R, such that

A(x∗) = 1. Equivalently, the set A[1] is a singleton.

The set of all fuzzy numbers is denoted by F(R). Moreover,we denote by Fc(R) the set of all fuzzy numbers with con-tinuous membership function.

Lemma 1 (Lee 2005) Let A ∈ F(R). Then, both the mapsα �→ AL

α and α �→ AUα are left continuous on [0, 1].

A LR-fuzzy number A = (a, al , ar )LR where al , ar ≥ 0, isdefined as follows

μ A(x) ={L( a−x

al) a − al ≤ x ≤ a,

R( x−aar ) a < x ≤ a + ar ,

where L and R are continuous and strictly decreasing func-tions with L(0) = R(0) = 1 and L(1) = R(1) = 0(Lee 2005). A special type of LR-fuzzy numbers is the so-called triangular fuzzy numbers with the shape functionsL(x) = R(x) = max{0, 1 − |x |}, x ∈ R.

A well-known ordering of fuzzy numbers, used in the sec-tions below for defining the hypotheses of interest is definedas follows:

Definition 1 (Wu 2005) Let A, B ∈ F(R), then

1. A ( =)B, if ALα = ( =)BL

α and AUα = ( =)BU

α for anyα ∈ [0, 1].

2. A � (≺)B, if ALα ≤ (<)BL

α and AUα ≤ (<)BU

α for anyα ∈ [0, 1].

3. A (�)B, if ALα ≥ (>)BL

α and AUα ≥ (>)BU

α for anyα ∈ [0, 1].

2.3 α-Optimistic values

In this subsection, we drive some results about the α-optimistic values of a fuzzy number.Wewill use these resultsto reconstruct a definition of fuzzy random variable that weuse in the next sections.

For a given fuzzy number A, the credibility of the event{ A ≥ r} is defined by Liu (2004) as follows:

Cr{ A ≥ r} := 1

2

(sup

y∈[r,+∞)

μ A(y)+1 − supy∈(−∞,r)

μ A(y)

),

(1)

It isworth noting that: (1)Cr{ A ≥ r} ∈ [0, 1] and (2)Cr{ A ≥r} = 1 − Cr{ A < r}.

Here, we recall the definition of the α-optimistic, but witha small change in the structure of the original definition. Fora fuzzy number A and the real number α ∈ [0, 1], the α-optimistic value of A, denoted by Aα , is rewritten by

Aα := sup{x ∈ A[0] | Cr{ A ≥ x} ≥ α}. (2)

Remark 1 It is mentioned that, according to the Liu’s defini-tion, the α-optimistic value of the fuzzy number A is definedas follows

Aα := sup{x ∈ R | Cr{ A ≥ x} ≥ α}. (3)

Therefore, we observe that A0 = ∞ and Aα ∈ [ ALα , AR

α ) forα ∈ (0, 1]. While, by Eq. (2), each value of Aα belongs to

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A[0] (which is a compact interval, due to the definition of afuzzy number), for all α ∈ [0, 1]. Moreover, it is clear thatAα is a non-increasing function of α ∈ [0, 1] (see Liu 2004,for more details).

Example 1 For a given LR-fuzzy number A = (a, al , ar ),it is easily seen that

Aα ={a + ar R−1(2α) for 0.0 ≤ α ≤ 0.5,a − al L−1(2(1 − α)) for 0.5 ≤ α ≤ 1.0.

Specially, if A = (a, al , ar )T is a triangular fuzzy number,then

Aα ={ar − 2α(ar − a) for 0.0 ≤ α ≤ 0.5,2a − al − 2α(a − al) for 0.5 ≤ α ≤ 1.0.

For instance, the α-optimistic values of A = (−2, 0, 1)T areobtained as

Aα ={1 − 2α for 0.0 ≤ α ≤ 0.5,2 − 4α for 0.5 ≤ α ≤ 1.0.

In the rest of this section, we drive some properties of α-optimistic values of a fuzzy number A and its relation toα-cuts of A.

Theorem 1 For A ∈ F(R), suppose φ A : [0, 1] → R is thefunction defined by φ A(α) = Aα . Then,

(i) For each α ∈ [0, 1],

φ A(α)

={AU2α 0.0≤α ≤ 0.5,

sup{x ≤ x∗ | μ A(x) ≤ 2(1−α)} 0.5 < α ≤ 1,

where x∗ = x ∗A

∈ R is the unique real number which

satisfies A(x∗) = 1.(ii) φ A is a decreasing and left continuous function.

Proof The proof of Theorem 1 is postponed to theAppendix.

Here, we investigate another useful property of the functionφ A. For a fuzzy number A ∈ F(R), let ψ A : [0, 1] → R bea function defined as follows.

∀α ∈ [0, 1], ψ A(α) ={AU2α α ∈ [0, 0.5],

AL2(1−α) α ∈ (0.5, 1]. (4)

According to Lemma 1, this function is decreasing, left con-tinuous on [0, 0.5] and right continuous on (0.5, 1]. More-over, according to part (i) of the previous theorem, ψ A(α) =φ A(α), for all α ∈ [0, 0.5].Lemma 2 If α0 ∈ (0.5, 1] is a point of continuity of eitherψ A or φ A then the values of the two functions coincide at thispoint.

Proof The proof of Lemma 2 is postponed to the Appendix.

Remark 2 From Lemma 2, note that the α-cuts of a fuzzynumber A ∈ Fc(R) can be rewritten as A[α] = [ A1−α/2,

Aα/2], α ∈ [0, 1]. Therefore, the membership function ofthe fuzzy number A can be obtained from the representationtheorem (e.g., see Lee 2005) as follows

μ A(x) = supα∈[0,1]

I (x ∈ [ A1−α/2, Aα/2]), x ∈ R.

Remark 3 From Lemma 2, note that the ordering of fuzzynumbers A and B introduced in Definition 1 is equivalent to:

1. A ( =)B, if Aα = ( =)Bα for any α ∈ [0, 1].2. A � (≺)B, if Aα ≤ (<)Bα for any α ∈ [0, 1].3. A (�)B, if Aα ≥ (>)Bα for any α ∈ [0, 1].

3 Fuzzy random variables

Let (Ω,A,P) be a probability space, X : Ω → F(R)

be a fuzzy-valued function, X be a random variable hav-ing distribution function fθ with a vector of parametersθ = (θ1, . . . , θp) ∈ Θ in which Θ ⊂ R

p be the parame-ter space, p ≥ 1. Throughout this paper, we assume that allrandom variables have the same probability space (Ω,A,P).

It is mentioned that Kwakernaak (1978, 1979) introduceda notion of FRVs as follows: given a probability space(Ω,A,P), a mapping X : Ω → F(R) is said to be aFRV if for all α ∈ [0, 1], the two real-valued mappingsX L

α : Ω → R and XUα : Ω → R are real-valued random

variables. Here, we rewrite the Kwakernaak’s definition ofFRV using α-optimistic values of a fuzzy number as follows

Definition 2 (Hesamian andChachi 2013)The fuzzy-valuedfunction X : Ω → F(R) is called a FRV if Xα(ω) : Ω → R

is a real-valued random variable for all α ∈ [0, 1].Based onTheorem1 andRemark 2, note that our definition ofa FRV is just a restriction of FRVs defined by Kwakernaak.So, a FRV can be addressed by itsα-optimistic values insteadof its α-cuts which is the reason that we use credibility indexin this paper.

Definition 3 FRVs X and Y are called identically distributedif Xα and Yα are identically distributed, for all α ∈ [0, 1].In addition, they are called independent if each random vari-able Xα is independent of each random variable Yα , for allα ∈ [0, 1]. Similarly, we say that X = (X1, . . . , Xn) is arandom sample of size n if Xi are independent and identi-cally distributed FRVs. The observed fuzzy random sampleis denoted by x = (x1, . . . , xn).

Here, consider a special case of FRVs as the member of afamily of a classical parametric population which is intro-duced by Wu (2005) (see also Chachi and Taheri 2011).

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Parametric testing statistical hypotheses 1541

Definition 4 A FRV X is said to have a (classical) para-metric continuous density function with fuzzy parameterθ : Θ → (F(R))p, denoted by

{fθ : θ ∈ (F(R))p

}, if for

allα ∈ [0, 1], Xα ∼ f(θ)α,where (θ)α = ((θ1)α, . . . , (θp)α).

Remark 4 It is mentioned that a necessary and sufficientcondition to say X ∼ fθ , θ ∈ (F(R))p is that Xα1 is“stochastically greater than Xα2 for any α1 < α2, i.e.,FXα2

(x) ≥ FXα1(x) for all x ∈ R. For instance, let Xα ∼

N (μ1−α, σ 21−α), where μ ∈ F(R) and σ 2 ∈ F((0,∞)).

Then, it is easy to verify that FXα2(x) ≥ FXα1

(x) for

all x ∈ R and for every α1 < α2. So, X is said to benormally distributed with fuzzy parameters μ and σ 2 ifXα ∼ N (μ1−α, σ 2

1−α). As another example, assume thatXα ∼ exp(λ1−α) where λ ∈ F(0,∞). Then, one canobserve that FXα2

(x) ≥ FXα1(x) for all x ∈ R and for every

α1 < α2 where Fλ(x) = 1 − e−λx , x > 0. So, X ∼ exp(λ)

if Xα ∼ exp(λ1−α) for all α ∈ [0, 1].Remark 5 Based on definition of a normal fuzzy randomvariable introduced by Wu (2005), a fuzzy random vari-able X is called normally distributed with fuzzy parame-ters μ and σ 2 if and only if X L

α ∼ N (μLα , (σ 2)Lα ) and

XUα ∼ N (μU

α , (σ 2)Uα ). However, it is easy to verify that “XUα

is not stochastically greater than X Lα ” for any α ∈ [0, 1]. But,

as it noted in Remark 4, we modified the definition of a nor-mal fuzzy randomvariable using the concepts ofα-optimisticvalues.

Now, based on a fuzzy random sample X = (X1, . . . , Xn),we are going to generalize the classical parametric tests forFRVs.

4 Testing hypotheses for FRVs

In this section, based on fuzzy random sample, we develop aprocedure for testing statistical hypothesis. In fact, we wishto test the following hypotheses:

Definition 5 Let X be a FRV from a continuous parametricpopulation with fuzzy parameter τ . Then:

1. Any hypothesis of the form

H : τ τ0 ≡ H : (τ )α = (τ0)α for all α ∈ [0, 1],

is called a simple hypothesis.2. Any hypothesis of the form

H : τ � τ0 ≡ H : (τ )α > (τ0)α for all α ∈ [0, 1],

is called a right one-sided hypothesis.

3. Any hypothesis of the form

H : τ ≺ τ0 ≡ H : (τ )α < (τ0)α for all α ∈ [0, 1],

is called a left one-sided hypothesis.4. Any hypothesis of the form

H : τ = τ0 ≡ H : (τ )α = (τ0)α for all α ∈ [0, 1],

is called to be a two-sided hypothesis.

In the following, by introducing some definitions and lem-mas,wepropose aprocedure to investigate a hypothesis aboutthe fuzzy parameters of the population. To do this, we willdevelop the concepts of fuzzy type-I and fuzzy type-II errors,fuzzy power and fuzzy p value for FRVs.

Definition 6 Let ϕ be the classical test function for testingH0 : τ = τ0 against one-sided H1 : τ > (<)τ0 (or two-sidedH1 : τ = τ0) alternative hypothesis. For testing H0 : τ τ0against the alternative hypothesis introduced in Definition 5,the test statistic, based on a random sample X, is defined to bethe fuzzy set ϕ(X) with α-cuts

[(ϕ)Lα (X), (ϕ)Uα (X)

], where

(ϕ)Lα (X) = infβ≥α

(ϕ(Xβ)), (ϕ)Lα (X) = supβ≥α

(ϕ(Xβ)), (5)

in which Xβ = ((X1)β, (X2)β, . . . , (Xn)β).

Definition 7 Let ϕ(X) be the fuzzy test function for testingH0 : τ τ0 against the alternative hypothesis introduced inDefinition 5. The power of ϕ in τ ∗ is defined to be the fuzzyset πϕ (τ ∗) with α-cuts

[(πϕ (τ ∗)Lα , (πϕ (τ ∗))Uα

], where

(πϕ (τ ∗))Lα = infβ≥α

EHβ :τβ=(τ∗)β [ϕ(Xβ)], (πϕ (τ ∗))Uα

= supβ≥α

EHβ :τβ=(τ∗)β [ϕ(Xβ)],

and Xβ = ((X1)β, (X2)β, . . . , (Xn)β), EH [T ] denotes theexpectation of T given H .

Lemma 3 Assume that X = (X1, . . . , Xn) is a fuzzy randomsample from a continuous family of population { fτ (x); x ∈R, τ ∈ Θ}. If the family of density functions { fτ (x); x ∈R, τ ∈ Θ} has the MLR property in T and ϕ is a non-increasing function of T, then πϕ is a non-increasing functionof τ , i.e., for every τ1, τ2 ∈ Θ where τ1 ≺ τ2, we haveπϕ (τ1) � πϕ (τ2).

Proof The proof is postponed to the Appendix.

Definition 8 Consider testing hypothesis

H0 : τ = τ0 versus H1 : τ = τ1, (6)

in which, it is assumed that τ0 ≺ τ1 or τ0 � τ1. Then, πϕ(τ0)

and 1−πϕ(τ1) are called fuzzy type-I and fuzzy type-II errorsdenoted by αϕ and βϕ , respectively.

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Here, similar to the approaches introduced in Hesamianand Taheri (2013), Hesamian and Chachi (2013), the fuzzyp value for testing hypotheses presented in Definition 5 isgiven as follows.

Definition 9 Consider testing the null hypothesis H0 : τ τ0 versus the alternative one-sided hypothesis H1 : τ (�) ≺τ0 or two-sided hypothesis H1 : τ = τ0. The fuzzy p valueis defined to be the fuzzy set p with α-cuts

[pLα , pUα

], where

pLα = infβ≥α

inf{δ ∈ [0, 1] : T (xβ) ∈ Cβδ },

and

pUα = supβ≥α

inf{δ ∈ [0, 1] : T (xβ) ∈ Cβδ },

where T denotes the classical test statistic and Cβδ is the

related critical region to test the null hypothesis Hβ0 : τβ =

(τ0)β versus the alternative one-sided hypothesis Hβ1 : τβ >

(<)(τ0)β (or two-sided hypothesis Hβ1 : τβ = (τ0)β ) based

on the random sample Xβ .

Finally, in the problem of testing a hypothesis basedon the fuzzy random sample X = (X1, . . . , Xn), wepropose the following definition of a fuzzy test function:

Definition 10 The fuzzy-valued function ϕ : Fn(R) →F({0, 1}) is called a fuzzy test whenever ψ(X), as a fuzzyset on {0, 1}, shows the degrees of rejection and acceptanceof the hypothesis H0 for any X. Here, “0” and “1” stand foracceptance and rejection of the null hypothesis, respectively,and F({0, 1}) is the class of all fuzzy sets on {0, 1}.Definition 11 For the problem of testing hypotheses H0 :τ τ0 where τ0 ∈ F(Θ), based on the fuzzy random sampleX = (X1, . . . , Xn) and at a given significance level δ ∈(0, 1], the fuzzy test is defined to be the following fuzzyset

ψ(X) ={Cr( p ≥ δ)

0,Cr( p < δ)

1

}. (7)

This test accepts the null hypothesis with the credibilitydegree of Cr( p ≥ δ) (degree of acceptance), and rejectsit with the credibility degree of Cr( p < δ) = 1−Cr( p ≥ δ)

(degree of rejection).

4.1 Testing hypotheses about the fuzzy parametersof a normal FRV

A particular class of parametric statistical testing hypothe-ses is the testing hypotheses about the parameters of a nor-mal population. Here, we are going to apply the proposedmethod in the previous section for a normalFRV, i.e., assume

that X = (X1, . . . , Xn) be a random sample from a nor-mal population and Θ = F(Θ) be the class of all fuzzynumbers on the parameter space Θ = {(θ, σ 2) : θ ∈ R, σ

∈ (0,∞)}.Remark 6 Consider testing H0 : θ θ0 for a normal FRVwhere σ 2 is known. Therefore, theα-cuts of the fuzzy p valuedefined in Definition 9 are reduced as follows

1. for testing alternative hypothesis H1 : θ � θ0,

pLα = infβ≥α

⎡⎣1 − Φ

⎛⎝

√n(xβ − (θ0)1−β)√

σ 21−β

⎞⎠

⎤⎦ ,

pUα = supβ≥α

⎡⎣1 − Φ

⎛⎝

√n(xβ − (θ0)1−β)√

σ 21−β

⎞⎠

⎤⎦ .

2. for testing alternative hypothesis H1 : θ ≺ θ0,

pLα = infβ≥α

Φ

⎛⎝

√n(xβ − (θ0)1−β)√

σ 21−β

⎞⎠ ,

pUα = supβ≥α

Φ

⎛⎝

√n(x1−β − (θ0)1−β)√

σ 21−β

⎞⎠ .

3. for testing alternative hypothesis H1 : θ = θ0,

pLα = 2 infβ≥α

Φ

⎛⎝−√

n| xβ − (θ0)1−β√σ 21−β

|⎞⎠ ,

pUα = 2 supβ≥α

Φ

⎛⎝−√

n| xβ − (θ0)1−β√σ 21−β

|⎞⎠ ,

where Φ denotes the standard normal cumulative distri-bution function and xβ = 1/n

∑ni=1 (xi )β .

Now assume that σ 2 is unknown. In this case, the α-cuts ofthe fuzzy p value are given as follows

1. for testing alternative hypothesis H1 : θ � θ0,

pLα = infβ≥α

P

⎛⎝tn−1 ≥

√n(xβ − (θ0)1−β)√

s21−β

⎞⎠ ,

pUα = supβ≥α

P

⎛⎝tn−1 ≥

√n(xβ − (θ0)1−β)√

s21−β

⎞⎠ .

123

Parametric testing statistical hypotheses 1543

2. for testing alternative hypothesis H1 : θ ≺ θ0,

pLα = infβ≥α

P

⎛⎝tn−1 ≤

√n(xβ − (θ0)1−β)√

s21−β

⎞⎠ ,

pUα = supβ≥α

P

⎛⎝tn−1 ≤

√n(xβ − (θ0)1−β)√

s21−β

⎞⎠ .

3. for testing alternative hypothesis H1 : θ = θ0 ,

pLα = 2 infβ≥α

P

⎛⎝tn−1 ≤ −√

n| xβ − (θ0)1−β√s21−β

|⎞⎠ ,

pUα = 2 supβ≥α

P

⎛⎝tn−1 ≤ −√

n| xβ − (θ0)1−β√s21−β

|⎞⎠ ,

where tn−1 denotes the t-student distribution with degreeof freedom n − 1 and s2β = 1

n−1

∑ni=1 ((xi )β − xβ)2.

Remark 7 Here, we describe calculating the fuzzy power totest about μ for a normal FRV. We explain the method forthe case of testing H0 : θ = θ0 versus H1 : θ � θ0 (theother cases can be obtained similarly). First, assume that σ 2

is known. FromDefinition 7, it is easy to verify that theα-cutsof the fuzzy power at θ∗(� θ0) are obtain as follows

(πϕ(θ∗))Lα = infβ≥α

⎡⎣1 − Φ

⎛⎝√

n(cβ − (θ∗)1−β)√

σ 21−β

⎞⎠

⎤⎦ ,

(πϕ(θ∗))Uα = supβ≥α

⎡⎣1 − Φ

⎛⎝√

n(cβ − (θ∗)1−β)√

σ 21−β

⎞⎠

⎤⎦ ,

where cβ = (θ0)1−β + z1− δ2

√σ 21−β√n

.

Now if σ 2 is unknown then, for testing H0 : θ = θ0 versusH1 : θ � θ0, the α-cuts of πϕ at θ∗(� θ0) are given by

(πϕ(θ∗))Lα = infβ≥α

⎡⎣1 − P

⎛⎝tn−1 ≤ √

n(cβ − (θ∗)1−β)√

s21−β

⎞⎠

⎤⎦ ,

(πϕ(θ∗))Uα = supβ≥α

⎡⎣1 − P

⎛⎝tn−1 ≤ √

n(cβ − (θ∗)1−β)√

s21−β

⎞⎠

⎤⎦ ,

where cβ = (θ0)1−β + tn−1, δ2

√s21−β√n

.

Remark 8 Now, consider testing the null hypothesis H0 :σ 2 σ 2

0 . From Definition 9, the α-cuts of the fuzzy p valueare given by

[pLα , pUα

], where

1. for testing alternative hypothesis H1 : σ 2 � σ 20 ,

pLα = infβ≥α

PHβ0 :σ 2

β =(σ 20 )β

(S2β ≥ s2β

)

= infβ≥α

P

(χ2n−1 ≥ (n − 1)s2β

(σ 20 )1−β

),

pUα = supβ≥α

PHβ0 :σ 2

β =(σ 20 )β

(S2β ≥ s2β

),

= supβ≥α

P

(χ2n−1 ≥ (n − 1)s2β

(σ 20 )1−β

).

2. for testing alternative hypothesis H1 : σ 2 ≺ σ 20 ,

pLα = infβ≥α

PHβ0 :σ 2

β =(σ 20 )β

(S2β ≤ s2β

),

= infβ≥α

P

(χ2n−1 ≤ (n − 1)s2β

(σ 20 )1−β

),

pUα = supβ≥α

PHβ0 :σ 2

β =(σ 20 )β

(S2β ≤ s2β

),

= supβ≥α

P

(χ2n−1 ≤ (n − 1)s2β

(σ 20 )1−β

).

3. for testing alternative hypothesis H1 : σ 2 = σ 20 ,

pLα = 2 infβ≥α

min

{PHβ0 :σ 2

β =(σ 20 )β

(S2β ≤ s2β),

PHβ0 :σ 2

β =(σ 20 )β

(S2β ≥ s2β)

},

= 2 infβ≥α

min

{P

(χ2n−1 ≤ (n − 1)s2β

(σ 20 )1−β

),

P

(χ2n−1 ≥ (n − 1)s2β

(σ 20 )1−β

)},

pUα = 2 supβ≥α

min

{PHβ0 :σ 2

β =(σ 20 )β

(S2β ≤ s2β),

PHβ0 :σ 2

β =(σ 20 )β

(S2β ≥ s2β)

},

= 2 supβ≥α

min

{P

(χ2n−1 ≤ (n − 1)s2β

(σ 20 )1−β

),

P

(χ2n−1 ≥ (n − 1)s2β

(σ 20 )1−β

)},

where χ2n−1 denotes the Chi-square distribution with

degree of freedom n − 1.

123

1544 G. Hesamian, M. Shams

By a similar manner described by Remarks 6–8, we canobtain the fuzzy power and fuzzy p value for testing aboutσ 2 for a normal FRV.

Remark 9 It should be mentioned that, based on Remark 5,all definitions of hypotheses given in this paper are just arewritten version of Wu’s ones (Wu 2005).

5 Numerical example

In this section, we provide a numerical example to clarify thediscussions in this paper and give the possible application ofthe proposed method for normal FRVs.

Example 2 (Elsherif et al. 2009) Based on a random sampleof the received signal from a target, assume we have fuzzyobservations x = (x1, x2, . . . , x25) (as shown in Table 1)measured in nanowatt which is approximately normally dis-tributed N (θ , σ 2). We wish to test the following hypothesis:{H0 : θ θ0 (1.35, 1.50, 1.80)T ,

H1 : θ ≺ θ0.

Applying the procedure proposed in Sect. 4.1, usingRemark 6, the fuzzy p value is derived point by point and

its membership function is shown in Fig. 1. This member-ship function can be interpreted as “about 0.075”. Therefore,at level of δ = 0.05, the fuzzy test for testing H0 : θ (1.35, 1.50, 1.80)T is obtained as ψ(X) =

{0.9050 , 0.095

1

}.

So, with respect to the observed fuzzy observations and atsignificance level of δ = 0.05, the null hypothesis H0 : θ (1.35, 1.50, 1.80)T is accepted with a degree of 0.905 andit is rejected with a degree of 0.095. Such a result may beinterpreted as “we absolutely tend to accept H0”. In addition,based on Remark 7, the membership of the fuzzy power atθ∗ (0.50, 0.70, 0.80)T is shown in Fig. 2 which is inter-preted as “about 0.999”.

Now, suppose that we want to test the followinghypotheses about σ 2:{H0 : σ 2 σ 2

0 (0.30, 0.50, 0.70)T ,

H1 : σ 2 � σ 20 .

FromRemark6, themembership functionof the fuzzypvalueis obtained as “about 0.83” whose membership function isshown in Fig. 3. Therefore, the fuzzy test to test H0 : σ 2 (0.30, 0.50, 0.70)T versus H1 : σ 2 � σ 2

0 , at significancelevel of δ = 0.05, is obtained as ψ(X) = { 0.991

0 , 0.0091

}. In

such a case, therefore, we are strongly convinced to accept

Table 1 The observed fuzzyrandom sample in Example 2 x1 = (0.02, 0.10, 0.23)T x2 = (0.08, 0.20, 0.37)T x3 = (0.15, 0.30, 0.46)T

x4 = (0.28, 0.40, 0.56)T x5 = (0.37, 0.50, 0.65)T x6 = (0.42, 0.60, 0.74)Tx7 = (0.54, 0.70, 0.83)T x8 = (0.66, 0.80, 0.93)T x9 = (0.72, 0.90, 1.26)Tx10 = (0.85, 1.00, 1.15)T x11 = (0.98, 1.10, 1.22)T x12 = (1.05, 1.20, 1.32)Tx13 = (1.17, 1.30, 1.48)T x14 = (1.25, 1.40, 1.54)T x15 = (1.32, 1.50, 1.67)Tx16 = (1.37, 1.60, 1.78)T x17 = (1.53, 1.70, 1.88)T x18 = (1.62, 1.80, 1.97)Tx19 = (1.75, 1.90, 2.08)T x20 = (1.76, 2.00, 2.23)T x21 = (1.93, 2.10, 2.27)Tx22 = (2.06, 2.20, 2.35)T x23 = (2.05, 2.30, 2.46)T x24 = (2.21, 2.40, 2.55)Tx25 = (2.32, 2.50, 2.60)T – –

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Deg

ree

of m

embe

rshi

p

Fig. 1 The membership function of the fuzzy p value and the significance level in Example 2 for testing H0 : θ (1.35, 1.50, 1.80)T againstH1 : θ ≺ (1.35, 1.50, 1.80)T

123

Parametric testing statistical hypotheses 1545

0.9994 0.9995 0.9996 0.9997 0.9998 0.9999 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Deg

ree

of m

embe

rshi

p

fuzzy power

Fig. 2 The membership function of the fuzzy power at θ∗ (0.50, 0.70, 0.80)T in Example 2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Deg

ree

of m

embe

rshi

p

Fig. 3 The membership function of the fuzzy p value and the significance level in Example 2 for testing H0 : σ 2 (0.30, 0.50, 0.70)T againstH1 : σ 2 � (0.30, 0.50, 0.70)T

H0 : σ 2 (0.30, 0.50, 0.70)T , at significance level of δ =0.05.

6 A comparison study

It should be mentioned that Akbari and Rezaei (2009,2010) extended the classical approach for testing statisticalhypotheses about the (crisp) mean or variance of a normaldistribution by introducing some fuzzy numbers to interpretthe sentences “larger than”, “smaller than” or “not equal”for population’s parameters based on imprecise observations.Parchami et al. (2010) defined an extension of fuzzy p valuefor a crisp random sample of a normal distribution to testimprecisely hypothesis test about the mean. Arefi and Taheri(2011) extended the classical critical region for testing fuzzyhypothesis about the parameters of the classical normal dis-tributionbasedon fuzzydata.Wu (2005) considered the prob-lem of hypotheses test with fuzzy data. Based on a concept

of fuzzy random variable, first he defined the fuzzy mean ofthe normal distribution. Then, based on a proposed rankingmethod, he defined the hypotheses about the population’sfuzzy mean. He considered two cases: (1) the variance ofthe population is known, (2) the variance of the popula-tion is unknown. Finally, he introduced a fuzzy test basedon the classical critical region. However, for both cases, heassigned a crisp number into the structure of testing hypothe-sis’s method: for the case (1) the variance of the population isconsidered as a crisp number and for the case (2) the varianceof the fuzzy sample is calculated based on core of the fuzzydata as a crisp number. As we observe, in all above methods,the variance of the normal population is considered as a crispnumber. However, for testing fuzzy hypothesis about meanof a normal population, it is reasonable that the variance ofthe population is also considered as a imprecise value. So, itmay be an advantage of our method with respect to above-mentionedmethods. Geyer andMeeden considered the prob-lem of the classical optimal hypothesis tests for the binomial

123

1546 G. Hesamian, M. Shams

distribution (in general for discrete datawith crisp parameter)using an interval estimation of a binomial proportion (GeyerandMeeden 2005). They introduced the notion of fuzzy con-fidence intervals by inverting families of randomized tests.Then, they introduce a notion of fuzzy p value in which it isonly a function of the parameters of the model. Finally, theyprovided a unified description of fuzzy confidence interval,fuzzy p values and fuzzy decision. However, their proposedmethod is not a fuzzy method in general, since this approachdoes not consider the imprecise information about the modelsuch as imprecise observations or imprecise hypotheses andit does not lead to a fuzzy decision. Viertl (2011) and Filz-moser andViertl (2004) also extend a concept of fuzzypvaluewhen the observations are fuzzy and hypotheses are crisp.Arnold (1996, 1998) proposed a method for testing fuzzyhypotheses about the population parameter with crisp data.He provided some definitions for the probability of type-Iand type-II errors and presented the best test for the one-parameter exponential family.

As it is observed, in all above-proposed methods for test-ing statistical hypothesis, at least one of the essential popu-lation’s information such as data, hypothesis or population’sparameters play a crisp role in the structure of testing pro-cedure. While, by restructuring the concept of fuzzy randomvariable, we involved the population’s information includ-ing: fuzzy data and fuzzy parameters into the hypothesis test,type-I error (or type-II error) and power of test. Finally, weobtained a fuzzy test tomake decision for accepting or reject-ing the fuzzy hypothesis of interest about the fuzzy parame-ters.

7 Conclusions

This paper proposes a method for testing hypotheses aboutthe fuzzy parameters of a fuzzy random variable. In thisapproach, we reconstruct a well-known concept of a fuzzyrandom variable using the concept of α-optimistic values.Then, we extended the concepts of the fuzzy power testand fuzzy p value. Finally, based on the credibility index,to compare the observed fuzzy p value and the crisp sig-nificance level, the fuzzy hypothesis of interest can beaccepted or rejected with degrees of conviction between 0and 1.

Although we focused on testing hypotheses about theparameters of a normal fuzzy random variable, the pro-posed method is general and it can be applied for otherkinds of fuzzy random variables as well. Moreover, it can beapplied for other kind of testing hypothesis such as compar-ing two independent fuzzy random variables of two popula-tions, the one-way or two-way analysis of variance and non-parametric approaches including testing hypothesis. How-ever, the proposed method can be applied only for fuzzy

numbers involved in a problem of statistical hypothesis test-ing. Moreover, the topic of testing hypothesis for fuzzy para-meters of a fuzzy randomvariable can be extended to the casewhere the level of significance is given by a fuzzy number,too. To do this, one can easily apply the credibility index forcomparing two fuzzy numbers into the decision making.

OpenAccess This article is distributed under the terms of theCreativeCommons Attribution License which permits any use, distribution, andreproduction in any medium, provided the original author(s) and thesource are credited.

Appendix: Proof of the main results

This appendix provides the mathematical proofs of the the-oretical results in our paper.

Proof of Theorem 1 (i) Let gA : R → [0, 1] be given bygA(x) = Cr( A ≥ x). Then

Aα = sup{x ∈ A[0] | gA(x) ≥ α}.

First, let α ∈ [0, 0.5). Since

gA(AU2α

) = 1

2

⎛⎝ sup

y∈[ AU2α,+∞)

μ A(y) + 1 sup(−∞, AU2α)

μ A(y)

⎞⎠

= 1

2(2α) = α,

wehave Aα ≥ AU2α .On the other hand, for each z > AU

2α ,supy∈[z,+∞) μ A(y) = A(z) and supy∈(−∞,z) μ A(y) =1. Therefore,

gA(z) = Cr( A ≥ z)

= 1

2

(sup

y∈[z,+∞)

μ A(y) + 1 − supy∈(−∞,z)

μ A(y)

)

= 1

2A(z) < α.

Hence, {x ∈ R | gA(x) ≥ α} ∩ ( AU2α,+∞) = ∅, which

by the previous part implies that Aα = AU2α .

Now suppose α ∈ [0.5, 1] and let x0 := sup{x ≤ x∗ |μ A(x) ≤ 2(1 − α)}. Then, μ A(x) ≤ 2(1 − α), for allx < x0. Hence,

supy∈[x0,+∞)

μ A(y)=1 and supy∈(−∞,x0)

μ A(y)≤2(1−α).

Therefore, gA(x0) ≥ α which implies that φ A(α) ≥ x0.On the other hand, for x > x0, since μ A(x) > 2(1−α),

123

Parametric testing statistical hypotheses 1547

we have

gA(x) = 1

2

(sup

y∈[x,+∞)

μ A(y) + 1 − supy∈(−∞,x)

μ A(y)

)

≥ 1

2

(1 + 1 − μ A(x)

)< α.

Hence, φ A(α) = x0. Note that, according to what hasbeen proved, φ A(0.5) = sup{x ≤ x∗ | μ A(x) ≤ 1} =x∗ = AU

2(0.5). Thus, the law of φ A can be written in thefollowing form.

∀α ∈ [0, 1], φ A(α)

={AU2α 0 ≤ α ≤ 0.5,

sup{x ≤ x∗ | μ A(x) ≤ 2(1−α)} 0.5≤α ≤ 1.

(8)

(ii) It is clear that φ A is decreasing on [0, 1]. By the previouspart, since for each α ∈ [0, 0.5], φ A(α) = AU

2α , usingLemma 1, φ A is left continuous on [0, 0.5].

For α ∈ (0.5, 1], suppose {αn}n∈N is an increasingsequence in (0.5, 1] with αn → α. First, suppose thatφ A(α) = x∗. Then, for each x < x∗, μ A(x) ≤ 2(1 − α) ≤2(1 − αn). Hence φ A(αn) ≥ x∗, for all n ∈ N. On the otherhand, since αn > 0.5, we have φ A(αn) ≤ x∗. Combining thetwoarguments,weobtainφ A(αn) = x∗, for alln ∈ N.Hence,in this case φ A(αn) = x∗ → φ A(α) = x∗. Now supposeφ Aα) < x∗. Then, for each y ∈ R with φ A(α) < y < x∗,μ A(y) > 2(1 − α). Therefore, μ A(y) > 2(1 − αn), forlarge values of n ∈ N. This implies that φ A(αn) ≤ y. Henceφ A(αn) → φ A(α). This completes the proof of the left con-tinuity of φ A on (0.5, 1].

Proof of Lemma 2 First suppose α0 ∈ (0.5, 1] is a point ofcontinuity of ψ A. Let x0 := φ A(α0). Then AL

2(1−α0)≤ x0.

If AL2(1−α0)

< x0 then μ A(x) = 2(1 − α0), for all x ∈[ AL

2(1−α0), x0). In this case for each α < α0, since 2(1−α) >

2(1 − α0), we have AL2(1−α) ≥ x0 which contradicts the

assumption of continuity of ψ A at α0. Hence

ψ A(α0) = AL2(1−α0)

= x0 = φ A(α0).

Suppose, on the contrary, thatα0 is a point of discontinuityof ψ A. Then, there ε0 > 0 such that for each n ∈ N, onecan find αn ∈ (α0 − 1

n , α) with ψ A(αn) > ψ A(α0) + ε0.With out loss of generality, we may assume that {αn}n∈N isan increasing sequence. Since ψ A is a monotonic function,there exists an increasing sequence {α′

n}n∈N converging toα0 such that α′

n ≤ αn , for each n ∈ N, and ψ A is continuouson the set {α′

n | n ∈ N}. Let also {βn}n∈N be a decreasingsequence converging to α0 which consist of continuity points

of ψ A. Then

φ A(α′n) = ψ A(α′

n) ≥ ψ A(αn) ≥ ψ A(α0) + ε0 > ψ A(α0)

≥ ψ A(βn) = φ A(βn),

for each n ∈ N. Hence

∀n ∈ N, φ A(α′n) − φ A(βn) > ε0,

which implies that φ A is also discontinuous at this point.

Proof of Lemma 3 Assume τ1 ≺ τ2. Note that, from Defini-tion3, πϕ (τ1) � πϕ (τ2) if andonly if (πϕ (τ1))

Lα ≤ (πϕ (τ2))

Lα

and (πϕ (τ1))Uα ≤ (πϕ (τ2))

Uα for all α ∈ [0, 1]. Now, since

τ1 ≺ τ2, we have (τ1)α < (τ1)α for all α ∈ [0, 1]. Onthe other hand, due to the assumption and MLR propertyof the underlying family of density function, we know thatE(τ∗)β [ϕ(Xβ)] is a non-decreasing function of τβ for everyβ ∈ [0, 1]. So for every α ∈ [0, 1], we have(πϕ (τ1))

Lα = inf

β≥αEHβ :τβ=(τ1)β

(ϕ(Xβ))

≤ infβ≥α

EHβ :τβ=(τ2)β(ϕ(Xβ))

= (πϕ (τ2))Lα ,

and

(πϕ (τ1))Uα = sup

β≥α

EHβ :τβ=(τ1)β(ϕ(Xβ))

≤ supβ≥α

EHβ :τβ=(τ2)β(ϕ(Xβ))

= (πϕ (τ2))Uα ,

which is the desired result.

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