# Generalized Estimation for Two-Parameter Life Time Distributions Based on Fuzzy Life Times

Authors: Syed Habib Shah; Muhammad Shafiq; Qamruz Zaman
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Ongoing developments of the measurement sciences say that measurements based on continuous phenomena are no more precise observations but more or less fuzzy. Therefore, it is necessary to utilize this imprecision of observations to obtain such estimators, which are based on all the available information that is given in the form of randomness and fuzziness. Objective of this research was to get such parameter estimation procedure that utilizes all the available information for some well-known two-parameter life time distributions. Therefore, the estimators need to be generalized in such a way to cover both uncertainties. For this purpose, based onδ-cuts of the life time observations, the generalized estimators are developed in such manner to cover stochastic variation in addition to fuzziness. The proposed generalized estimators are much preferred over classical estimators for life time analysis as these are based on all the available information present in the form of fuzziness of single observations and random variation among the observations to make suitable inferences.


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## Body

## 1. Introduction

Statistics is the science to make inference about the population from the obtained data. The obtained data are usually presented in the form of numbers, vectors, or functions, generally containing precise measurements of some phenomena. Countless techniques (stochastic models) are available to model or to draw inference from these obtained measurements.Survival analysis or reliability analysis can generally be defined as the collection of techniques for analyzing so-called life time data.In broad sense, one can say life time is “the time to the occurrence of a specified event.”Life time is also called survival time, event time, or failure time and is usually measured in hours, days, weeks, months, or years.The prominence of survival analysis is to predict the probability of response, average survival time, identifying the important investigative factors associated to the life time of units, and to compare the survival distributions. Models used for survival times are usually termed as “time to event models” [1].The analysis techniques of life time data can be traced back centuries, but the rapid development started about few decades ago, especially World War II stimulated interest in the reliability of military equipment [2].Nowadays, life time analysis is used in almost every of field of life like biomedical sciences, industrial reliability, social sciences, and business. In the time to event modeling, the event of interest may be failure, death, recovery time, or change of address, in engineering, medical and social sciences, etc. Therefore, there are a number of reasons to say that specialized methods are required to model life time data in the best possible way [1].Exponential, Weibull, log-logistic, and Birnbaum–Saunders distributions are considered in most applied distributions in life time analyses.Exponential distribution has a vital role in life time analysis analogous to normal distribution in other fields. It is purely based on random failure pattern because of its “memoryless property.” A two-parameter function is a more generalized form with probability density function of exponential distribution:(1)fyλ,θ=λe−λy−θ,y≥θ,1,y<θ.Forn precise life time observations y1,y2,…,yn, their classical parameter estimates, i.e., maximum likelihood estimates, are given as(2)θ^=miny1,y2,…,yn,and(3)λ^=n∑i=1nyi−θ^.For details, see [1].For the nonconstant hazard rate, Weibull distribution is among the top most distributions for the life time analysis. Its density is defined by(4)fy|τ,η=ητyτη−1exp−yτη∀y>0,τ>0,η>0,τ: scale parameteralso called characteristic life time,η: shape parameter.According to [3], let CV denote the coefficient of variation for the data defined as the ratio of standard deviation and mean, i.e., σ/y¯.For the parameter estimation of Weibull distribution, the moment method estimators are defined as(5)CV=Γ1+2/η^−Γ21+1/η^Γ1+1/η^.Solve the above equation for the value ofη^ to get an estimate.(6)τ^=y¯Γ1+1/η^1η^.The log-logistic distribution is the extension of logistic distribution, for which it has been observed that it can be decreasing, right-skewed, or unimodal. Because of its flexibility in shapes, it is very useful to fit data from many different fields, including engineering, economics, hydrology, and survival analysis.Its pdf is defined as(7)fy|α,β=βαyαβ−11+yαβ2,y>0.For the parameter estimation, maximum likelihood estimators are obtained through the following equations:(8)nβ−nlogα+∑i=1nlogyi−2∑i=1nyiαβlogyiα1+yiαβ−1=0,−nβα+2βα∑i=1nyiαβ1+yiαβ−1=0.For details, see [4].Birnbaum–Saunders life time distribution was first proposed in [5], for fatigue failures caused under cyclic loading, having the density function given below:(9)fy|μ,γ=122πμγγy12+γy32exp−12μ2yγ+γy−2,y,μ,γ>0.Forn precise life time observations y1,y2,…,yn, the corresponding modified moment estimators are obtained as follows.LetA and H be arithmetic mean and harmonic mean of the life times, respectively:(10)A=∑i=1nyinandH=n∑i=1n1/yi.Then,(11)μ^=2AH12−112,(12)γ^=A·H12.For the proof, see [6].For a life time random variable, Gamma distribution is defined by the density(13)fy|ϕ,ν=1Γνϕyν−1e−yϕwithϕ>0,ν>0.Lety¯ and s2 be mean and variance of the data y1,y2,…,yn, respectively; then, the moment estimators of the parameters are defined as(14)ν^=y¯2s2,(15)ϕ^=s2y¯.For the proof, see [7].The emergence of technological advancement augments the increase in life time of units. Therefore, the researchers with only few observations draw inference about the aggregate of units. Hence, it is pertinent to utilize all the available information in the best possible manner.According to [8], in the modern science of measurements, it is not possible to get exact measurement of a continuous real variable, and stochastic models are used to model variation among the precise observations.In addition to that, in practical situations, especially dealing with continuous variables, the measurements have two kinds of uncertainties, the first is variation among the observations and second is imprecision of single observations, called fuzziness [9].Realizing the importance of fuzziness in the life time observations, some work has been done like ([10–18]); [19, 20]. Yet, most of the times, the information available in the form of fuzziness is ignored in the publications, which may cause misleading results.Therefore, the very up-to-date fuzzy number approaches are more realistic and suitable for the inferences of life time observations [21].In this research work, some the generalized estimators for well-known distributions are presented to accommodate fuzziness along with random variation.


## 2. Preliminary Concepts of Fuzzy Set Theory

### 2.1. Fuzzy Number

Lety∗ denote a fuzzy number and is a special subsets of ℝ; it is determined by a real-valued function, so-called characterizing function (CF) χ·, with conditions:(1)
0≤χ≤1(2)
Support ofχ· is bounded:      suppχ·≔y∈ℝ:χy>0⊆ℝa,ℝb with −∞<ℝa<ℝb<∞.(3)
The so-calledδ -cut, i.e., Cδy∗≔y∈ℝ:χy≥δ∀δ∈0,1, is a finite union of nonempty compact intervals, i.e., Cδy∗=∪j=1Jδy¯j,δ,y¯j,δ≠∅.In case of fuzzy number for which all theδ-cuts are closed bounded intervals, is called a fuzzy interval.


### 2.2. Lemma

According to [9], for a set A⊆ℝ, where 1A· is denoting the indicator function for set A, then to obtain the characterizing function for a generating fuzzy number, the given lemma holds:(16)χy=maxδ·1Cδy∗y:δ∈0,1∀y∈ℝ.


### 2.3. Nested Interval

LetIδ;δ∈0,1 be a family of intervals, called nested if Iδ1⊆Iδ2 for all δ1>δ2.


### 2.4. Remark

IfAδ;δ∈0,1 is denoting a nested family of finite unions of compact intervals, it is not necessary that all nested families are the δ-cuts of a fuzzy number. Then, the characterizing function of the generated fuzzy number is obtained by the given lemma.


### 2.5. Construction Lemma

According to [22], let Aδ=∪j=1Jδy¯δ,j,y¯δ,j∀δ∈0,1 be a nested family; then, the characterizing function (CF) of the generated fuzzy number is obtained by χy=supδ·1Aδy:δ∈0,1∀y∈ℝ.


### 2.6. Extension Principle

Consider an arbitrary functionℋ: ℵ⟶ℜ, where ℵ and ℜ are two spaces.Letm∗ be a fuzzy element of ℵ, with corresponding membership function ψ:ℵ⟶0,1; then, the fuzzy value y∗=ℋm∗ is defined to be the corresponding fuzzy element in ℜ for which the membership function Ψ· is defined by(17)Ψy≔supψm:m∈ℵ,ℋω=yif∃m:ℋm=y0if∄m:ℋm=y∀y∈ℜ.For details, see [23].


### 2.7. Minimum and Maximum of Fuzzy Numbers

If there aren fuzzy intervals, i.e., y1∗,y2∗,…,yn∗ with corresponding characterizing functions χ1·,χ2·,…,χn·, respectively, then its δ-cuts are denoted as Cδyi∗=y¯i,δ,y¯i,δ∀δ∈0,1andi=11n. Then, the minimum ymin∗ of the fuzzy numbers is fuzzy interval, with δ-cuts Cδymin∗. These are defined by(18)Cδymin∗≔miny¯i,δ,miny¯i,δ∀δ∈0,1.Furthermore, the maximumymax∗ of the fuzzy numbers is fuzzy interval, with δ-cuts Cδymax∗ defined by(19)Cδymax∗≔maxy¯i,δ,maxy¯i,δ∀δ∈0,1.Figure1 shows CF of minimum and maximum fuzzy observations from the above sample of fuzzy observations mentioned in Figure 2.Figure 1
Minimum and maximum fuzzy observations from above sample.Figure 2
Sample of fuzzy observations.


## 2.1. Fuzzy Number

Lety∗ denote a fuzzy number and is a special subsets of ℝ; it is determined by a real-valued function, so-called characterizing function (CF) χ·, with conditions:(1)
0≤χ≤1(2)
Support ofχ· is bounded:      suppχ·≔y∈ℝ:χy>0⊆ℝa,ℝb with −∞<ℝa<ℝb<∞.(3)
The so-calledδ -cut, i.e., Cδy∗≔y∈ℝ:χy≥δ∀δ∈0,1, is a finite union of nonempty compact intervals, i.e., Cδy∗=∪j=1Jδy¯j,δ,y¯j,δ≠∅.In case of fuzzy number for which all theδ-cuts are closed bounded intervals, is called a fuzzy interval.


## 2.2. Lemma

According to [9], for a set A⊆ℝ, where 1A· is denoting the indicator function for set A, then to obtain the characterizing function for a generating fuzzy number, the given lemma holds:(16)χy=maxδ·1Cδy∗y:δ∈0,1∀y∈ℝ.


## 2.3. Nested Interval

LetIδ;δ∈0,1 be a family of intervals, called nested if Iδ1⊆Iδ2 for all δ1>δ2.


## 2.4. Remark

IfAδ;δ∈0,1 is denoting a nested family of finite unions of compact intervals, it is not necessary that all nested families are the δ-cuts of a fuzzy number. Then, the characterizing function of the generated fuzzy number is obtained by the given lemma.


## 2.5. Construction Lemma

According to [22], let Aδ=∪j=1Jδy¯δ,j,y¯δ,j∀δ∈0,1 be a nested family; then, the characterizing function (CF) of the generated fuzzy number is obtained by χy=supδ·1Aδy:δ∈0,1∀y∈ℝ.


## 2.6. Extension Principle

Consider an arbitrary functionℋ: ℵ⟶ℜ, where ℵ and ℜ are two spaces.Letm∗ be a fuzzy element of ℵ, with corresponding membership function ψ:ℵ⟶0,1; then, the fuzzy value y∗=ℋm∗ is defined to be the corresponding fuzzy element in ℜ for which the membership function Ψ· is defined by(17)Ψy≔supψm:m∈ℵ,ℋω=yif∃m:ℋm=y0if∄m:ℋm=y∀y∈ℜ.For details, see [23].


## 2.7. Minimum and Maximum of Fuzzy Numbers

If there aren fuzzy intervals, i.e., y1∗,y2∗,…,yn∗ with corresponding characterizing functions χ1·,χ2·,…,χn·, respectively, then its δ-cuts are denoted as Cδyi∗=y¯i,δ,y¯i,δ∀δ∈0,1andi=11n. Then, the minimum ymin∗ of the fuzzy numbers is fuzzy interval, with δ-cuts Cδymin∗. These are defined by(18)Cδymin∗≔miny¯i,δ,miny¯i,δ∀δ∈0,1.Furthermore, the maximumymax∗ of the fuzzy numbers is fuzzy interval, with δ-cuts Cδymax∗ defined by(19)Cδymax∗≔maxy¯i,δ,maxy¯i,δ∀δ∈0,1.Figure1 shows CF of minimum and maximum fuzzy observations from the above sample of fuzzy observations mentioned in Figure 2.Figure 1
Minimum and maximum fuzzy observations from above sample.Figure 2
Sample of fuzzy observations.


## 3. Generalized Estimation for Fuzzy Data

In Figure3, the frame diagram explains the steps for obtaining the generalized estimators for the two-parameter life time distributions given below.Figure 3
A frame diagram for the analysis.


### 3.1. Exponential Distribution

Lety1∗,y2∗,…,yn∗1/2 represent fuzzy life time intervals having δ-cuts:(20)Cδyi∗=y¯i,δ,y¯i,δ,i=1,2,…,n,∀δ∈0,1.where y¯i,δ=infy∈ℝ:χy≥δ and y¯i,δ=supy∈ℝ:χy≥δ are lower and upper ends of the corresponding δ-cuts.Based on fuzzy life times, the fuzzy (generalized) estimators of the two-parameter exponential distribution are denoted byθ^∗ and λ^∗.Based on lower and upper ends of theδ-cuts of fuzzy life times, the estimator presented in (2) can be generalized in the following way:(21)θ∗^=miny1∗,y2∗,…,yn∗.For the fuzzy parameter estimatorθ∗^, θ¯δ and θ¯δ are denoting lower and upper ends of the corresponding generating family of intervals, and these are obtained in the following way:(22)θ¯δ=miny¯i,δ,i=11n,∀δ∈0,1,and(23)θ¯δ=miny¯i,δ,i=11n,∀δ∈0,1.LetAδθ^∗=θ¯δ,θ¯δ∀δ∈0,1 be the generating family of intervals; using construction lemma, the CF of the fuzzy estimate θ^∗ is obtained.Example 1.
Let us consider 12 fuzzy life time observations for two-parameter exponential distribution, i.e.,y1∗,y2∗,…,y12∗=1,2,3,4,4,5,6,7,8,9,10,11,(24)11,12,13,14,12,13,14,15,14,15,16,17,17,18,19,20,19,20,21,22,21,22,23,24.23,24,25,26,24,25,26,27,26,27,28,29 whose characterizing functions are given in Figure 4.
Based on the given fuzzy life time observations, the CF of the fuzzy parameter estimate obtained through (22) and (23) is depicted in Figure 5.
This parameter estimate is more suitable for realistic life time observations, as it covers both types of uncertainties.
In the same way, the fuzzy (generalized) estimator for the parameterλ is denoted by λ∗^ having lower and upper ends λ¯δ and λ¯δ of the δ-cuts, respectively, where(25)λ¯δ=n∑i=1ny¯i,δ−θ¯δ∀δ∈0,1,and(26)λ¯δ=n∑i=1ny¯i,δ−θ¯δ∀δ∈0,1.
LetAδλ^∗=λ¯δ,λ¯δ∀δ∈0,1 be the desired generating family of intervals; using construction lemma, the CF of the generated fuzzy estimate λ^∗ is obtained.
In Figure6, CF of the fuzzy estimate is obtained through (25) and (26) based on all the available information which is given in the form of fuzziness and stochastic variation; these make it more suitable in real-life applications.Figure 4
CF of a fuzzy sample.Figure 5
CF of the fuzzy estimatorθ^∗.Figure 6
CF of the fuzzy estimatorλ^∗.


### 3.2. Weibull Distribution

Based on (5), the fuzzy (generalized) estimates of the Weibull shape parameter can be obtained in the following way:(27)η¯δ≔miny¯∈×i=1nCδyi∗CVy¯=Γ1+2/η^−Γ21+1/η^Γ1+1/η^∀δ∈0,1,and(28)η¯δ≔maxy¯∈×i=1nCδyi∗CVy¯=Γ1+2/η^−Γ21+1/η^Γ1+1/η^∀δ∈0,1.Similarly, based on (6), fuzzy (generalized) estimates of the Weibull scale parameter can be obtained in the following way:(29)τ¯δ=y¯¯δΓ1+1/η^¯δ1η^¯δ,and(30)τ¯δ=y¯¯δΓ1+1/η^¯δ1η^¯δ.Example 2.
Consider fuzzy life timesy1∗,y2∗,…,y8∗=0,1,2,3,1,2,3,4,3,4,5,6, 5,6,7,8,7,8,9,10,9,9.5,10.5,11,10,10.5,11.5,12,11,11.5,12.5,13 for the Weibull distribution with characterizing functions in Figure 7.
Using (27) and (28), let Aδη^∗=η¯δ,η¯δ∀δ∈0,1 be the desired generating family of intervals; using the construction lemma, CF of the fuzzy estimate η^∗ is obtained as shown in Figure 8.
LetAδτ^∗=τ¯δ,τ¯δ∀δ∈0,1 be the desired generating family of intervals through which the CF of the fuzzy estimate τ^∗ mentioned in Figure 9 is obtained by using the construction lemma.
The above CF of the fuzzy estimate obtained through (29) and (30) is based on all the available information which is given in the form of fuzziness and stochastic variation; these kinds of additional information make it more suitable in real-life applications.Figure 7
CF of the fuzzy life times.Figure 8
CF of the fuzzy estimatorη^∗.Figure 9
CF of the fuzzy estimatorτ^∗.


### 3.3. Log-Logistic Distribution

For the log-logistic distribution, the corresponding fuzzy estimators are denoted byα^∗ and β^∗. Denoting y¯=y1,y2,…,yn, the corresponding lower and upper ends of the generating family can be obtained through the following equations:(31)α¯δ≔miny¯∈×i=1nCδyi∗−nβα+2βα∑i=1nyiαβ1+yiαβ−1=0,nβ−nlogα+∑i=1nlogyi−2∑i=1nyiαβlogyiα1+yiαβ−1=0∀δ∈0,1,and(32)α¯δ≔maxy¯∈×i=1nCδyi∗−nβα+2βα∑i=1nyiαβ1+yiαβ−1=0,nβ−nlogα+∑i=1nlogyi−2∑i=1nyiαβlogyiα1+yiαβ−1=0∀δ∈0,1.Also,(33)β¯δ≔miny¯∈×i=1nCδyi∗−nβα+2βα∑i=1nyiαβ1+yiαβ−1=0,nβ−nlogα+∑i=1nlogyi−2∑i=1nyiαβlogyiα1+yiαβ−1=0∀δ∈0,1,and(34)β¯δ≔maxy¯∈×i=1nCδyi∗−nβα+2βα∑i=1nyiαβ1+yiαβ−1=0,nβ−nlogα+∑i=1nlogyi−2∑i=1nyiαβlogyiα1+yiαβ−1=0∀δ∈0,1.LetAδα^∗=α¯δ,α¯δ∀δ∈0,1 and Aδβ^∗=β¯δ,β¯δ∀δ∈0,1 be the desired generating families of intervals of the fuzzy parameter estimators through which the CF of the fuzzy estimates α^∗ and β^∗ is obtained using the construction lemma.Example 3.
Consider fuzzy life timesy1∗,y2∗,…,y6∗=1,2,3,4,4,5,6,7,6,7,8,9, 8,9,10,11,13,14,15,16,17,18,19,20 for the log-logistic distribution with characterizing functions in Figure 10.
From the above fuzzy life time observations using (31) and (32), the CF of the fuzzy parameter estimate is depicted in Figure 11.
This estimate is based on both uncertainties, i.e., fuzziness and random variation, which make it more representative for the corresponding parameter.
From the above fuzzy life time observations shown in Figure9, using (33) and (34), the CF of the fuzzy parameter estimate β^∗ is depicted in Figure 12.
The above CF is the fuzzy estimate of the parameterβ, which incorporates all the available information in the inference. The above CF for the fuzzy parameter estimates is based on fuzzy life time observations which holds both uncertainties, i.e., stochastic variation and fuzziness of the single observations, which make these more suitable in the real-life applications.Figure 10
CF of a fuzzy sample.Figure 11
CF of the fuzzy estimatorα^∗.Figure 12
CF of the fuzzy estimatorβ^∗.


### 3.4. Birnbaum–Saunders Distribution

For fuzzy life timesy1∗,y2∗,…,yn∗, the fuzzy parameter estimators of the Birnbaum–Saunders distribution are denoted by μ^∗ and γ^∗, having δ-cuts:(35)Cδμ^∗=μ¯δ,μ¯δ∀δ∈0,1,and(36)Cδγ^∗=γ¯δ,γ¯δ∀δ∈0,1.LetA∗ and H∗ be fuzzy arithmetic mean and fuzzy harmonic mean, respectively, and their δ-cuts are denoted as(37)CδA^∗=A¯δ,A¯δ∀δ∈0,1,and(38)CδH^∗=H¯δ,H¯δ,∀δ∈0,1.Using (10), the corresponding lower and upper ends of the δ-cuts of A^∗ are obtained in the following way:(39)A¯δ=∑i=1ny¯i,δnandA¯δ=∑i=1ny¯i,δn,∀δ∈0,1.Example 4.
Based on fuzzy life times presented in Figure1, characterizing functions of the fuzzy estimates of the Birnbaum–Saunders distribution are given in Figures 13 and 14.
Figure15 shows the CF of the fuzzy estimate of the arithmetic mean based on fuzzy life times. Similarly, using (10), the corresponding lower and upper ends of the δ-cuts of H^∗ are obtained in the following way:(40)H¯δ=n∑i=1n1/y¯i,δandH¯δ=n∑i=1n1/y¯i,δ,∀δ∈0,1.
Figure16 shows the CF of the fuzzy estimate of the harmonic mean based on fuzzy life times. Using (11), lower and upper ends of the corresponding fuzzy parameter estimators μ^∗ are obtained in the following way:(41)μ¯δ=2A¯δH¯δ12−112,∀δ∈0,1,and(42)μ¯δ=2A¯δH¯δ12−112,∀δ∈0,1.
Denoting byAδμ^∗=μ¯δ,μ¯δ∀δ∈0,1 the desired generating family of intervals of the fuzzy parameter estimator and using the construction lemma, the CF of the fuzzy estimator μ^∗ is obtained.
In Figure13, the CF of the fuzzy estimate based on fuzzy life times is depicted. Using (12), lower and upper ends of the corresponding fuzzy parameter estimator γ^∗ are obtained in the following way:(43)γ¯δ=A¯δ·H¯δ12andγ¯δ=A¯δ·H¯δ12,∀δ∈0,1.
Denoting byAδγ^∗=γ¯δ,γ¯δ∀δ∈0,1 the desired generating family of intervals of the fuzzy parameter estimator and using the construction lemma, the CF of the fuzzy estimator μ^∗ is obtained.
Figure14 shows the CF of the fuzzy estimator of the parameter γ, which utilized all the available information in the form of fuzziness and random variation.
The fuzzy estimation of the parameter indicates that the value ofγ is about 9.7 to 15.2 in the sence of the function in Figure 13. It means that it is completely possible that γ is 9.7 or 15.2. In addition, it is not possible that γ is less than 11.7 or greater than 14.11, with possibility degree of 0.8.Figure 13
CF of the fuzzy estimatorμ^∗.Figure 14
CF of the fuzzy estimatorγ^∗.Figure 15
CF of the fuzzy estimatorA^∗.Figure 16
CF of the fuzzy estimatorH^∗.


### 3.5. Gamma Distribution

Lety1∗,y2∗,…,yn∗ represent fuzzy life time intervals having δ-cuts:(44)Cδyi∗=y¯i,δ,y¯i,δ,i=11n,∀δ∈0,1.Then, the corresponding lower and upper ends of the generating family of the mean can be obtained through the following equations:(45)y¯¯δ=∑i=1ny¯i,δnandy¯¯δ=∑i=1ny¯i,δn,∀δ∈0,1.Denotingy¯=y1,y2,…,yn, then the corresponding lower and upper ends of the generating family of the variance can be obtained through the following equations:(46)s¯δ2≔miny¯∈×i=1nCδyi∗s2∀δ∈0,1,and(47)s¯δ2≔maxy¯∈×i=1nCδyi∗s2∀δ∈0,1.The fuzzy parameter estimators of the gamma distribution are denoted byν^∗ and ϕ^∗, having δ-cuts:(48)Cδν^∗=ν¯δ,ν¯δ,∀δ∈0,1,and(49)Cδϕ^∗=ϕ¯δ,ϕ¯δ,∀δ∈0,1.Using lower and upper ends of the generating family of the fuzzy estimates of mean and variance and (14), lower and upper ends of the corresponding fuzzy parameter estimators ν^∗ are obtained in the following way:(50)ν¯δ=y¯¯δ2s¯δ2andν¯δ=y¯¯δ2s¯δ2,∀δ∈0,1.In the same way, using lower and upper ends of the generating family of the fuzzy estimates of mean and variance and (15), lower and upper ends of the corresponding fuzzy parameter estimators ϕ^∗ are obtained in the following way:(51)ϕ¯δ=s¯δ2y¯¯δandϕ¯δ=s¯δ2y¯¯δ,∀δ∈0,1.Example 5.
Characterizing functions of fuzzy life timesy1∗,y2∗,…,yn∗ for the gamma distribution are given in Figure 17.
From (50), denoting by Aδν^∗=ν¯δ,ϕ¯δ∀δ∈0,1 the desired generating family of intervals of the fuzzy parameter estimator and using the construction lemma, the CF of the fuzzy estimator ν^∗ is obtained and depicted in Figure 18.
From (51), denoting by Aδϕ^∗=ϕ¯δ,ϕ¯δ∀δ∈0,1 the desired generating family of intervals of the fuzzy parameter estimator and using the construction lemma, the CF of the fuzzy estimator ϕ^∗ is obtained.
The fuzzy estimation of the parameter indicates that the value ofϕ is about 1.6 to 2.7 in the sence of the function in Figure 19. It means that it is completely possible that ϕ is 1.6 or 2.7. In addition, it is not possible that ϕ is less than 1.9 or greater than 2.3, with possibility degree of 0.8.Figure 17
CF of the fuzzy life times.Figure 18
CF of the fuzzy estimatorν^∗.Figure 19
CF of the fuzzy estimatorϕ^∗.


## 3.1. Exponential Distribution

Lety1∗,y2∗,…,yn∗1/2 represent fuzzy life time intervals having δ-cuts:(20)Cδyi∗=y¯i,δ,y¯i,δ,i=1,2,…,n,∀δ∈0,1.where y¯i,δ=infy∈ℝ:χy≥δ and y¯i,δ=supy∈ℝ:χy≥δ are lower and upper ends of the corresponding δ-cuts.Based on fuzzy life times, the fuzzy (generalized) estimators of the two-parameter exponential distribution are denoted byθ^∗ and λ^∗.Based on lower and upper ends of theδ-cuts of fuzzy life times, the estimator presented in (2) can be generalized in the following way:(21)θ∗^=miny1∗,y2∗,…,yn∗.For the fuzzy parameter estimatorθ∗^, θ¯δ and θ¯δ are denoting lower and upper ends of the corresponding generating family of intervals, and these are obtained in the following way:(22)θ¯δ=miny¯i,δ,i=11n,∀δ∈0,1,and(23)θ¯δ=miny¯i,δ,i=11n,∀δ∈0,1.LetAδθ^∗=θ¯δ,θ¯δ∀δ∈0,1 be the generating family of intervals; using construction lemma, the CF of the fuzzy estimate θ^∗ is obtained.Example 1.
Let us consider 12 fuzzy life time observations for two-parameter exponential distribution, i.e.,y1∗,y2∗,…,y12∗=1,2,3,4,4,5,6,7,8,9,10,11,(24)11,12,13,14,12,13,14,15,14,15,16,17,17,18,19,20,19,20,21,22,21,22,23,24.23,24,25,26,24,25,26,27,26,27,28,29 whose characterizing functions are given in Figure 4.
Based on the given fuzzy life time observations, the CF of the fuzzy parameter estimate obtained through (22) and (23) is depicted in Figure 5.
This parameter estimate is more suitable for realistic life time observations, as it covers both types of uncertainties.
In the same way, the fuzzy (generalized) estimator for the parameterλ is denoted by λ∗^ having lower and upper ends λ¯δ and λ¯δ of the δ-cuts, respectively, where(25)λ¯δ=n∑i=1ny¯i,δ−θ¯δ∀δ∈0,1,and(26)λ¯δ=n∑i=1ny¯i,δ−θ¯δ∀δ∈0,1.
LetAδλ^∗=λ¯δ,λ¯δ∀δ∈0,1 be the desired generating family of intervals; using construction lemma, the CF of the generated fuzzy estimate λ^∗ is obtained.
In Figure6, CF of the fuzzy estimate is obtained through (25) and (26) based on all the available information which is given in the form of fuzziness and stochastic variation; these make it more suitable in real-life applications.Figure 4
CF of a fuzzy sample.Figure 5
CF of the fuzzy estimatorθ^∗.Figure 6
CF of the fuzzy estimatorλ^∗.


## 3.2. Weibull Distribution

Based on (5), the fuzzy (generalized) estimates of the Weibull shape parameter can be obtained in the following way:(27)η¯δ≔miny¯∈×i=1nCδyi∗CVy¯=Γ1+2/η^−Γ21+1/η^Γ1+1/η^∀δ∈0,1,and(28)η¯δ≔maxy¯∈×i=1nCδyi∗CVy¯=Γ1+2/η^−Γ21+1/η^Γ1+1/η^∀δ∈0,1.Similarly, based on (6), fuzzy (generalized) estimates of the Weibull scale parameter can be obtained in the following way:(29)τ¯δ=y¯¯δΓ1+1/η^¯δ1η^¯δ,and(30)τ¯δ=y¯¯δΓ1+1/η^¯δ1η^¯δ.Example 2.
Consider fuzzy life timesy1∗,y2∗,…,y8∗=0,1,2,3,1,2,3,4,3,4,5,6, 5,6,7,8,7,8,9,10,9,9.5,10.5,11,10,10.5,11.5,12,11,11.5,12.5,13 for the Weibull distribution with characterizing functions in Figure 7.
Using (27) and (28), let Aδη^∗=η¯δ,η¯δ∀δ∈0,1 be the desired generating family of intervals; using the construction lemma, CF of the fuzzy estimate η^∗ is obtained as shown in Figure 8.
LetAδτ^∗=τ¯δ,τ¯δ∀δ∈0,1 be the desired generating family of intervals through which the CF of the fuzzy estimate τ^∗ mentioned in Figure 9 is obtained by using the construction lemma.
The above CF of the fuzzy estimate obtained through (29) and (30) is based on all the available information which is given in the form of fuzziness and stochastic variation; these kinds of additional information make it more suitable in real-life applications.Figure 7
CF of the fuzzy life times.Figure 8
CF of the fuzzy estimatorη^∗.Figure 9
CF of the fuzzy estimatorτ^∗.


## 3.3. Log-Logistic Distribution

For the log-logistic distribution, the corresponding fuzzy estimators are denoted byα^∗ and β^∗. Denoting y¯=y1,y2,…,yn, the corresponding lower and upper ends of the generating family can be obtained through the following equations:(31)α¯δ≔miny¯∈×i=1nCδyi∗−nβα+2βα∑i=1nyiαβ1+yiαβ−1=0,nβ−nlogα+∑i=1nlogyi−2∑i=1nyiαβlogyiα1+yiαβ−1=0∀δ∈0,1,and(32)α¯δ≔maxy¯∈×i=1nCδyi∗−nβα+2βα∑i=1nyiαβ1+yiαβ−1=0,nβ−nlogα+∑i=1nlogyi−2∑i=1nyiαβlogyiα1+yiαβ−1=0∀δ∈0,1.Also,(33)β¯δ≔miny¯∈×i=1nCδyi∗−nβα+2βα∑i=1nyiαβ1+yiαβ−1=0,nβ−nlogα+∑i=1nlogyi−2∑i=1nyiαβlogyiα1+yiαβ−1=0∀δ∈0,1,and(34)β¯δ≔maxy¯∈×i=1nCδyi∗−nβα+2βα∑i=1nyiαβ1+yiαβ−1=0,nβ−nlogα+∑i=1nlogyi−2∑i=1nyiαβlogyiα1+yiαβ−1=0∀δ∈0,1.LetAδα^∗=α¯δ,α¯δ∀δ∈0,1 and Aδβ^∗=β¯δ,β¯δ∀δ∈0,1 be the desired generating families of intervals of the fuzzy parameter estimators through which the CF of the fuzzy estimates α^∗ and β^∗ is obtained using the construction lemma.Example 3.
Consider fuzzy life timesy1∗,y2∗,…,y6∗=1,2,3,4,4,5,6,7,6,7,8,9, 8,9,10,11,13,14,15,16,17,18,19,20 for the log-logistic distribution with characterizing functions in Figure 10.
From the above fuzzy life time observations using (31) and (32), the CF of the fuzzy parameter estimate is depicted in Figure 11.
This estimate is based on both uncertainties, i.e., fuzziness and random variation, which make it more representative for the corresponding parameter.
From the above fuzzy life time observations shown in Figure9, using (33) and (34), the CF of the fuzzy parameter estimate β^∗ is depicted in Figure 12.
The above CF is the fuzzy estimate of the parameterβ, which incorporates all the available information in the inference. The above CF for the fuzzy parameter estimates is based on fuzzy life time observations which holds both uncertainties, i.e., stochastic variation and fuzziness of the single observations, which make these more suitable in the real-life applications.Figure 10
CF of a fuzzy sample.Figure 11
CF of the fuzzy estimatorα^∗.Figure 12
CF of the fuzzy estimatorβ^∗.


## 3.4. Birnbaum–Saunders Distribution

For fuzzy life timesy1∗,y2∗,…,yn∗, the fuzzy parameter estimators of the Birnbaum–Saunders distribution are denoted by μ^∗ and γ^∗, having δ-cuts:(35)Cδμ^∗=μ¯δ,μ¯δ∀δ∈0,1,and(36)Cδγ^∗=γ¯δ,γ¯δ∀δ∈0,1.LetA∗ and H∗ be fuzzy arithmetic mean and fuzzy harmonic mean, respectively, and their δ-cuts are denoted as(37)CδA^∗=A¯δ,A¯δ∀δ∈0,1,and(38)CδH^∗=H¯δ,H¯δ,∀δ∈0,1.Using (10), the corresponding lower and upper ends of the δ-cuts of A^∗ are obtained in the following way:(39)A¯δ=∑i=1ny¯i,δnandA¯δ=∑i=1ny¯i,δn,∀δ∈0,1.Example 4.
Based on fuzzy life times presented in Figure1, characterizing functions of the fuzzy estimates of the Birnbaum–Saunders distribution are given in Figures 13 and 14.
Figure15 shows the CF of the fuzzy estimate of the arithmetic mean based on fuzzy life times. Similarly, using (10), the corresponding lower and upper ends of the δ-cuts of H^∗ are obtained in the following way:(40)H¯δ=n∑i=1n1/y¯i,δandH¯δ=n∑i=1n1/y¯i,δ,∀δ∈0,1.
Figure16 shows the CF of the fuzzy estimate of the harmonic mean based on fuzzy life times. Using (11), lower and upper ends of the corresponding fuzzy parameter estimators μ^∗ are obtained in the following way:(41)μ¯δ=2A¯δH¯δ12−112,∀δ∈0,1,and(42)μ¯δ=2A¯δH¯δ12−112,∀δ∈0,1.
Denoting byAδμ^∗=μ¯δ,μ¯δ∀δ∈0,1 the desired generating family of intervals of the fuzzy parameter estimator and using the construction lemma, the CF of the fuzzy estimator μ^∗ is obtained.
In Figure13, the CF of the fuzzy estimate based on fuzzy life times is depicted. Using (12), lower and upper ends of the corresponding fuzzy parameter estimator γ^∗ are obtained in the following way:(43)γ¯δ=A¯δ·H¯δ12andγ¯δ=A¯δ·H¯δ12,∀δ∈0,1.
Denoting byAδγ^∗=γ¯δ,γ¯δ∀δ∈0,1 the desired generating family of intervals of the fuzzy parameter estimator and using the construction lemma, the CF of the fuzzy estimator μ^∗ is obtained.
Figure14 shows the CF of the fuzzy estimator of the parameter γ, which utilized all the available information in the form of fuzziness and random variation.
The fuzzy estimation of the parameter indicates that the value ofγ is about 9.7 to 15.2 in the sence of the function in Figure 13. It means that it is completely possible that γ is 9.7 or 15.2. In addition, it is not possible that γ is less than 11.7 or greater than 14.11, with possibility degree of 0.8.Figure 13
CF of the fuzzy estimatorμ^∗.Figure 14
CF of the fuzzy estimatorγ^∗.Figure 15
CF of the fuzzy estimatorA^∗.Figure 16
CF of the fuzzy estimatorH^∗.


## 3.5. Gamma Distribution

Lety1∗,y2∗,…,yn∗ represent fuzzy life time intervals having δ-cuts:(44)Cδyi∗=y¯i,δ,y¯i,δ,i=11n,∀δ∈0,1.Then, the corresponding lower and upper ends of the generating family of the mean can be obtained through the following equations:(45)y¯¯δ=∑i=1ny¯i,δnandy¯¯δ=∑i=1ny¯i,δn,∀δ∈0,1.Denotingy¯=y1,y2,…,yn, then the corresponding lower and upper ends of the generating family of the variance can be obtained through the following equations:(46)s¯δ2≔miny¯∈×i=1nCδyi∗s2∀δ∈0,1,and(47)s¯δ2≔maxy¯∈×i=1nCδyi∗s2∀δ∈0,1.The fuzzy parameter estimators of the gamma distribution are denoted byν^∗ and ϕ^∗, having δ-cuts:(48)Cδν^∗=ν¯δ,ν¯δ,∀δ∈0,1,and(49)Cδϕ^∗=ϕ¯δ,ϕ¯δ,∀δ∈0,1.Using lower and upper ends of the generating family of the fuzzy estimates of mean and variance and (14), lower and upper ends of the corresponding fuzzy parameter estimators ν^∗ are obtained in the following way:(50)ν¯δ=y¯¯δ2s¯δ2andν¯δ=y¯¯δ2s¯δ2,∀δ∈0,1.In the same way, using lower and upper ends of the generating family of the fuzzy estimates of mean and variance and (15), lower and upper ends of the corresponding fuzzy parameter estimators ϕ^∗ are obtained in the following way:(51)ϕ¯δ=s¯δ2y¯¯δandϕ¯δ=s¯δ2y¯¯δ,∀δ∈0,1.Example 5.
Characterizing functions of fuzzy life timesy1∗,y2∗,…,yn∗ for the gamma distribution are given in Figure 17.
From (50), denoting by Aδν^∗=ν¯δ,ϕ¯δ∀δ∈0,1 the desired generating family of intervals of the fuzzy parameter estimator and using the construction lemma, the CF of the fuzzy estimator ν^∗ is obtained and depicted in Figure 18.
From (51), denoting by Aδϕ^∗=ϕ¯δ,ϕ¯δ∀δ∈0,1 the desired generating family of intervals of the fuzzy parameter estimator and using the construction lemma, the CF of the fuzzy estimator ϕ^∗ is obtained.
The fuzzy estimation of the parameter indicates that the value ofϕ is about 1.6 to 2.7 in the sence of the function in Figure 19. It means that it is completely possible that ϕ is 1.6 or 2.7. In addition, it is not possible that ϕ is less than 1.9 or greater than 2.3, with possibility degree of 0.8.Figure 17
CF of the fuzzy life times.Figure 18
CF of the fuzzy estimatorν^∗.Figure 19
CF of the fuzzy estimatorϕ^∗.


## 4. Conclusion

According to recent developments of the measurement sciences, it is very easy to say that measurements based on continuous phenomena are no more precise observations but more or less fuzzy.Life time is a continuous phenomenon, and in the development of life time distributions, it has been noted that life time observations are recorded as precise numbers. But as discussed, in real-life applications, life times are no more precise observations but fuzzy numbers.In order to get more suitable and realistic results, this imprecision needs to be addressed; therefore, in this study, generalized estimators are proposed so that fuzziness of life time observations is integrated in the inference.Since the proposed estimators utilize all the available information, i.e., fuzziness as well as random variation of the life time observations to cover all the available information. The proposed estimators are based on random variation like other classical approaches, but in addition to that, these estimators also utilize the fuzziness of the observations. This integration of fuzziness in the estimates make it more realistic in real-life applications. The characterizing functions for the generalized estimators are obtained and explained to cover both the uncertainties. On the other hand, the classical approaches are only based on random variations and have nothing to do with other kinds of variation.Therefore, the results based on the proposed estimators are more suitable and realistic to real-life applications.


## 5. Limitation and Future Work of the Study

The study is limited to the complete observations, and this can be extended to the censored observation; in addition to that, this can be further extended to Pythagorean fuzzy uncertainty mentioned in [24].



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