# Estimation and Confidence Intervals of a New PCICNpmc for Logistic-Exponential Process Distribution

Authors: Refah Alotaibi; Sanku Dey; Mahendra Saha
Journal: Journal of Mathematics
 (2022)

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## Abstract

The process capability index has been introduced as an effective tool used in industries to aid in the assessment of process performance as well as to measure how much the product meets the costumer expectations. We are aware that classical process capability indices provide better results when the quality characteristic of the processes follows normal distribution. However, these classical indices may not provide accurate results for evaluating nonnormally distributed process which in turn may hinder the decision-making. In this article, we consider a new process capability indexCNpmc which is based on cost function and is applicable both for normally and nonnormally distributed processes. In order to estimate the process capability index CNpmc when the process follows logistic-exponential distribution, we have used ten classical methods of estimation, and the performances of these classical estimates of the index CNpmc are compared in terms of their mean squared errors through a simulation study. Next, we construct five bootstrap confidence intervals of the process capability index CNpmc and compare them in terms of their average width and coverage probabilities. Finally, two data sets related to electronic industries are reanalyzed to show the applicabilities of the proposed methods.


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## Body

## 1. Introduction

Vannman [1] constructed a superstructure PCI which was referred to as Cpu,v and is defined by
(1)Cpu,v=d−uμ−m3σ2+vμ−T2,u,v>0,where μ and σ are the process mean and standard deviation, T is the target value and d=USL−LSL/2, and m=LSL+USL/2. Let USL and LSL be the upper and lower specification limits. The four basic indices, Cp, Cpk, Cpm, and Cpmk, are special cases of Cpu,v, by letting u=0or1 and v=0or1. Later, in the year 1997, Chen and Pearn [2] generalized the work of Vannman [1] and proposed a new quantile-based PCI superstructure index CNpu,v for any underlying distribution which is defined as
(2)CNpu,v=d−uM−m3F99.865−F0.135/62+vM−T2,where Fα is the αth percentile and M is the median of the distribution. It has been observed that for skewed distributions, process median M is a more robust measure than the process mean μ. Considering the values of u,v=0,0,1,0,0,1,and1,1, we can obtain the following four indices for any underlying distribution, such as
(3)CNp=U−LF99.865−F0.135,(4)CNpk=minU−MF99.865−F0.135/2,M−LF99.865−F0.135/2,(5)CNpm=U−L6F99.865−F0.135/62+M−T2,(6)CNpmk=minU−M3F99.865−F0.135/62+M−T2,M−L3F99.865−F0.135/62+M−T2.Clearly, when the underlying distribution is normal, thenμ=M and σ=F99.865−F0.135/6, and hence, the index CNpu,v reduces to Cpu,v.In this paper, we propose a cost-effective PCI, sayCNpmc, using the tolerance cost function as suggested by Jeang et al. [3] which is obtained by replacing the denominator of CNpm in Equation (3) by sum of quality loss, F99.865−F0.135/62+M−T2, and tolerance cost, CMt, i.e., 6F99.865−F0.135/62+M−T2+CMt which in turn we obtain a new index, defined as
(7)CNpmc=U−L6F99.865−F0.135/62+M−T2+CMt,where CMt=C0+C1exp−C2t, C0,C1,C2 are the coefficients for the tolerance cost function and t is the process tolerance.In order to estimate the parameters of a model, we find multiple papers on various estimation techniques in the literature. But when attempting to estimate a model’s parameters and PCIs, the maximum likelihood (ML) method is frequently utilized as a starting point. Comparative studies of various methods of estimation have been carried out for different models. It has been observed that a particular estimation procedure outperforms the others for a particular model. In the premise of this, in this paper, we consider nine estimators besides ML estimators for estimating the PCI,CNpmc, under logistic-exponential distribution (LED), namely, least squares estimators (LSE), weighted least squares estimators (WLSE), maximum product spacing estimators (MPSE), minimum spacing absolute distance estimators (MDE), minimum spacing absolute-log distance estimators (MLDE), percentile estimators (PCE), Cramèr-von Mises estimators (CME), Anderson-Darling estimators (ADE), and right-tail Anderson-Darling estimators (RADE). Through a Monte-Carlo simulation analysis, the effectiveness of the estimators is evaluated in relation to their respective mean squared errors (MSEs). Point estimation might not offer accurate estimates of the PCIs, nevertheless, because of errors in the estimators. Therefore, the interval estimation methods of PCIs are used to evaluate variability or divergence in the estimates. Many methods, including the bootstrap approach, have recently been developed for building confidence intervals (CIs) for processes with nonnormal distributions. In this regard, readers may refer to the works of Leiva et al. [4], Pearn et al. [5, 6], Kashif et al. [7, 8], Weber et al. [9], Rao et al. [10], and Alomani et al. [11], to name a few. Further, Saha et al. [12] in their studies focussed on parametric estimation, bootstrap confidence interval, and highest posterior density credible interval of the index Cpk using normal distribution. Therefore, based on the aforementioned ten traditional estimation techniques, we consider five bootstrap confidence intervals (BCIs), namely, the standard bootstrap SB, percentile bootstrap PB, Student’s t bootstrap STB, bias-corrected percentile bootstrap BCPB, and bias-corrected accelerated bootstrap BCAB. Estimated coverage probabilities (CPs) and average widths (AWs) are taken into account when assessing BCIs.The goal of this paper is to develop a guideline for the choice of best estimation method that produces better estimates and CI forCNpmc when the processes follow LED. Thus far, we have not come across any report for calculating PCI, CNpmc, where five BCIs based on the ten traditional estimating techniques for the LED are taken into account. It is our endeavour to fill this gap through this work.The remainder of the paper is structured as follows: sensitivity analysis has been carried out in Section2. In Section 3, we introduce the PCI CNpmc for LED. In Section 4, we describe the considered methods of estimation for CNpmc. In Section 5, five BCIs (SB, PB, STB, BCPB, and BCAB) have been discussed for the PCI CNpmc based on considered estimation methods. A simulation study has been conducted and is discussed in Section 6 in order to evaluate the performance of the estimation methods and BCIs under various scenarios. Section 7 presents empirical applications employing data sets connected to the electronic industries. The concluding remarks and future works are provided in Section 8.


## 2. Sensitivity Analysis

According to Flaig [13] and Saha et al. [14], the study of net sensitivity NS using a distribution function for a specific PCI is defined as
(8)NS=1p0limε⟶0FU−FL−FU−ε−FL−εε=fU−fLp0.WhenNS values are positive, the distribution is observed to become less resilient (or more sensitive) at upper specification limit compared to lower specification; conversely, when NS values are negative, the distribution is shown to become less sensitive. The distribution in the context of PCI is less sensitive/more resilient if the NS values are low (in absolute sense). Table 1 has the NS values for the gamma, Weibull, and LE distributions. The mathematical NS values are expressed as defective per million dpm values. Table 1 shows that the Weibull and gamma distributions are more sensitive (or less robust) than LED for L,U=0.50,5.50 and p0=0.95, respectively.Table 1
Analysis of LE distribution’s sensitivity.
ModelfUSLfLSLNSdpmGamma (ϱ=0.25, ϑ=4.00)3.029586×10−110.08878047−93453.13Weibull (ϱ=0.25, ϑ=4.00)3.029586×10−110.08878047−155134.80Logistic-exponential (ϱ=0.25, ϑ=4.00)0.017099950.00267314315186.11


## 3.CNpmc for Logistic-Exponential Distribution

Logistic-exponential distribution (LED) was proposed by Lan and Leemis [15]. This distribution exhibits increasing, decreasing, bathtub (BT), and upside-down bathtub (UBT) shaped hazard rate function, and it is quite useful in product and process control and reliability analysis as all products or items exhibit at least one of the aforementioned characteristics of the hazard functions. The probability density, cumulative distribution, and quantile functions of the LED are
(9)fz∣ϱ,ϑ=ϑ.ϱeϑz−1ϱ−1eϑz1+eϑz−1ϱ2,z>0,ϱ,ϑ>0,(10)Fz∣ϱ,ϑ=eϑz−1ϱ1+eϑz−1ϱ,z>0,ϱ,ϑ>0,(11)ξγ∣ϱ,ϑ=1ϑlog1+γ1−γ1/ϱ,0<γ<1,where ϱ is the shape parameter and ϑ is scale parameter of the two parameter LED, respectively. This distribution is a generalization of exponential distribution, and it can be obtained by taking ϱ=1. This distribution is in the bathtub and upside-down bathtub classes for 0<ϱ<1 and ϱ>1, respectively. Then, the index CNpm (see Chen and Pearn [2]) and the proposed index CNpmc for the two parameter LED are, respectively, given as
(12)CNpm=U−L6ξγ3∣ϱ,ϑ−ξγ1∣ϱ,ϑ/62+ξγ2∣ϱ,ϑ−T2,CNpmc=U−L6ξγ3∣ϱ,ϑ−ξγ1∣ϱ,ϑ/62+ξγ2∣ϱ,ϑ−T2+CMt,where γith is the quantile of the two parameter LED with parameters ϱ,ϑ.


## 4. Estimation ofCNpmc

This section deals with the estimation of unknown parameters of the model using ten methods of estimation, namely, MLE, LSE, WLSE, PCE, CME, MPSE, MDE, MLDE, ADE, and RADE and the corresponding estimator ofCNpmc.


### 4.1. Maximum Likelihood Estimators

Letz1,z2,⋯,zn be a random sample of size n drawn from two parameter LED (9), and then, the likelihood function is given by
(13)Lϱ,ϑ=∏i=1nfzi;ϱ,ϑ=∏i=1nϑ.ρeϑzi−1ϱ−1eϑzi1+eϑzi−1ϱ2.Taking logarithm on both the sides of Equation (13), we have
(14)logLϱ,ϑ=nlogϑ+nlogϱ+ϱ−1∑i=1nlogeϑzi−1+ϑ∑i=1nzi−2∑i=1nlog1+eϑzi−1ϱ.The MLEs ofϱ and ϑ, say ϱ^mle and ϑ^mle, respectively, can be obtained as an iterative solutions of the following two equations:
(15)∂logLϱ,ϑ∂ϱ=nρ+∑i=1nlogeϑzi−1−2∑i=1neϑzi−1ϱlogeϑzi−11+eϑzi−1ϱ=0,(16)∂logLϱ,ϑ∂ϑ=nϑ+∑i=1nϱ−1zi.eϑzieϑzi−1+∑i=1nzi−2∑i=1nϱeϑzi−1ϱ−1xi.eϑzi1+eϑzi−1ϱ=0.Equations (15) and (16) can be solved for ϱ and ϑ using any numerical iterative procedure. Since the MLEs ofϱandϑare not in the closed forms, therefore, we have used nonlinear minimization (NLM) (see Dennis and Schnabel [16]) technique by using some initial guess value for the parameters, sayϱ=0.01andϑ=0.01, and obtaining the estimates ofϱandϑasϱ^mleandϑ^mle, respectively. Consequently, the MLE of CNpmc can be obtained as
(17)C^Npmcmle=U−L6ξγ3∣ϱ∧mle,ϑ∧mle−ξγ1∣ϱ∧,ϑ∧/62+ξγ2∣ϱ∧mle,ϑ∧mle−T2+CMt,where the MLE of the quantile function with parameters ϱ,ϑ is
(18)ξγiϱ^mle,ϑ^mle=1ϑ^mlelog1+log1+γi1−γi1/ϱ∧mle,i=1,2,3.


### 4.2. Least Squares Estimators

We minimize the following function with respect toρ and ϑ for obtaining the LSEs denoted by ϱ^lse and ϑ^lse.
(19)Sϱ,ϑ=∑i=1nFzi:nϱ,ϑ−in+12,where F· is the CDF, given in Equation (10) and zi:n,i=1,⋯,n is the ith order statistic of a random sample z1,z2,⋯,zn. Equivalently, they can be obtained by solving
(20)∑i=1nFzi:nϱ,ϑ−in+1ς1zi:nϱ,ϑ=0,∑i=1nFzi:nϱ,ϑ−in+1ς2zi:nϱ,ϑ=0,where
(21)ς1zi:nϱ,ϑ=1+eϑzi:n−1ϱ−2eϑzi:n−1ϱlogeϑzi:n−1,(22)ς2zi:nϱ,ϑ=1+eϑzi:n−1ϱ−2ρzi:neϑzi:neϑzi:n−1ϱ−1.Substituting the LSEs, we can get the estimator ofCNpmc as
(23)C^Npmclse=U−L6ξγ3∣ϱ∧lse,ϑ∧lse−ξγ1∣ϱ∧,ϑ∧lse/62+ξγ2∣ϱ∧lse,ϑ∧lse−T2+CMt.


### 4.3. Weighted Least Squares Estimators

The WLSEs,ϱ^wlse and ϑ^wlse, can be obtained by minimizing the following function:
(24)Wϱ,ϑ=∑i=1nwiFzi:nϱ,ϑ−in+12.The estimatorsϱ^wlse and ϑ^wlse of the parameters ϱ and ϑ can be obtained by solving the following nonlinear equations:
(25)∑i=1nwiFzi:nϱ,ϑ−in+1ς1zi:nϱ,ϑ=0,∑i=1nwiFzi:nϱ,ϑ−in+1ς2zi:nϱ,ϑ=0,where wi=n+12n+2/in−i+1, ς1zi:nϱ,ϑ, and ς2zi:nϱ,ϑ are defined in Equations (21) and (22), respectively. Substituting the WLSEs, we can get the estimator of CNpmc as
(26)C^Npmcwlse=U−L6ξγ3∣ϱ∧wlse,ϑ∧wlse−ξγ1∣ϱ∧,ϑ∧wlse/62+ξγ2∣ϱ∧wlse,ϑ∧wlse−T2+CMt.


### 4.4. Percentile Estimators

The percentile estimatesϱ^pce and ϑ^pce of the parameters ϱ and ϑ can be obtained by minimizing the following function with respect to ϱ and ϑ:
(27)Pϱ,ϑ=∑j=1nzj:n−1ϑlog1+pj1−pj1/ϱ2.Several estimators ofpj can be used here; see, for example, Mann et al. [17]. In this paper, we have consider pj=j/n+1. Substituting the PCEs, we can get the estimator of CNmpc as
(28)C^Npmcpce=U−L6ξγ3∣ϱ∧pce,ϑ∧pce−ξγ1∣ϱ∧,ϑ∧pce/62+ξγ2∣ϱ∧pce,ϑ∧pce−T2+CMt.


### 4.5. Cramèr-von Mises Estimators

The Cramèr-von Mises estimators ofϱ and ϑ, say ϱ^cme and ϑ^cme, can be obtained by minimizing the following function with respect to ϱ and ϑ.
(29)Cϱ,ϑ=112n+∑i=1nFzi:nϱ,ϑ−2i−12n2.The estimatorsϱ^cme and ϑ^cme of the parameters ϱ and ϑ can be obtained by solving the following nonlinear equations:
(30)∑i=1nFzi:nϱ,ϑ−2i−12nς1zi:nϱ,ϑ=0,∑i=1nFzi:nϱ,ϑ−2i−12nς2zi:nϱ,ϑ=0,where ς1zi:nϱ,ϑ and ς2zi:nϱ,ϑ are given by Equations (21) and (22), respectively. Substituting the CMEs, we can get the estimator of CNpmc as
(31)C^Npmccme=U−L6ξγ3∣ϱ∧cme,ϑ∧cme−ξγ1∣ϱ∧,ϑ∧cme/62+ξγ2∣ϱ∧cme,ϑ∧cme−T2+CMt.


### 4.6. Maximum Product of Spacing Estimators

Maximum product of spacing (MPS) method was proposed by Cheng and Amin [18, 19] which can be used as an alternative to ML method of estimation for estimating parameters of continuous univariate distributions. Define the uniform spacings of a random sample from the LED as
(32)Diϱ,ϑ=Fzi:nϱ,ϑ−Fzi−1:nϱ,ϑ,i=1,2,⋯,n,where zi:n, i=1,2,⋯,n is the ith order statistic of a random sample z1,z2,⋯,zn. Note that z0:n=0 and zn+1:n=1. The MPSEs, ϱ^mpse and ϑ^mpse, of the parameters ϱ and ϑ can be obtained by maximizing the following geometric mean of the spacing function with respect to ϱ and ϑ(33)Gϱ,ϑ=∏i=1n+1Diϱ,ϑ1/n+1,or, equivalently, by maximizing the function
(34)Hϱ,ϑ=1n+1∑i=1n+1logDiϱ,ϑ.The estimatorsϱ^mpse and ϑ^mpse of the parameters ϱ and ϑ can be obtained by solving the nonlinear equations:
(35)∂∂ϱHϱ,ϑ=1n+1∑i=1n+11Diϱ,ϑς1zi:n∣ϱ,ϑ−ς1zi−1:nϱ,ϑ=0,∂∂ϑHϱ,ϑ=1n+1∑i=1n+11Diϱ,ϑς2zi:n∣ϱ,ϑ−ς2zi−1:n∣ϱ,ϑ=0,where ς1zi:nϱ,ϑ and ς2zi:nϱ,ϑ are given by Equations (21) and (22), respectively. Substituting the MPSEs, we can get the estimator of CNpmc as
(36)C^Npmcmpse=U−L6ξγ3∣ϱ∧mpse,ϑ∧mpse−ξγ1∣ϱ∧,ϑ∧mpse/62+ξγ2∣ϱ∧mpse,ϑ∧mpse−T2+CMt.


### 4.7. Minimum Spacing Absolute Distance Estimators

Torabi [20] proposed the minimum spacing absolute distance estimators (MDE) of the parameters of a distribution. Thus, MDE of parameters ϱ and ϑ can be obtained by minimizing the following function:
(37)Tϱ,ϑ=∑i=1n+1Diϱ,ϑ−1n+1,with respect to ϱ and ϑ, respectively. The estimators ϱ^mde and ϑ^mde of the parameters ϱ and ϑ can be obtained by solving the following nonlinear equations:
(38)∂∂ϱTϱ,ϑ=∑i=1n+1Diϱ,ϑ−1/n+1Diϱ,ϑ−1/n+1ς1zi:nϱ,ϑ−ς1zi−1:nϱ,ϑ=0,∂∂ϑTϱ,ϑ=∑i=1n+1Diϱ,ϑ−1/n+1Diϱ,ϑ−1/n+1ς2zi:nϱ,ϑ−ς2zi−1:nϱ,ϑ=0,where Diϱ,ϑ≠1/n+1, ς1·ϱ,ϑ, and ς2·ϱ,ϑ are defined in Equations (21) and (22), respectively. Substituting the MDEs, we can get the estimator of CNpmc as
(39)C^Npmcmde=U−L6ξγ3∣ϱ∧mde,ϑ∧mde−ξγ1∣ϱ∧,ϑ∧mde/62+ξγ2∣ϱ∧mde,ϑ∧mde−T2+CMt.


### 4.8. Minimum Spacing Absolute Log-Distance Estimators

Torabi [20] proposed the minimum spacing absolute-log distance estimators (MLDE). The MLDEs of the parameters ϱ and ϑ can be obtained by minimizing the function:
(40)Tϱ,ϑ=∑i=1n+1logDiϱ,ϑ−log1n+1.The estimatorsϱ^mlde and ϑ^mlde of the parameters ϱ and ϑ can be obtained by solving the following nonlinear equations:
(41)∂∂ϱTϱ,ϑ=∑i=1n+1logDiϱ,ϑ−log1/n+1logDiϱ,ϑ−log1/n+11Diϱ,ϑς1zi:nϱ,ϑ−ς1zi−1:nϱ,ϑ=0,∂∂ϑTϱ,ϑ=∑i=1n+1logDiϱ,ϑ−log1/n+1logDiϱ,ϑ−log1/n+11Diϱ,ϑς2zi:nϱ,ϑ−ς2zi−1:nϱ,ϑ=0,where logDiϱ,ϑ≠log1/n+1, ς1·ϱ,σ, and ς2·ϱ,ϑ are defined in Equations (21) and (22), respectively. Substituting the MLDEs, we can get the estimator of CNpmc as
(42)C^Npmcmlde=U−L6ξγ3ϱ∧mlde,ϑ∧mlde−ξγ1∣ϱ∧,ϑ∧mlde/62+ξγ2∣ϱ∧mlde,ϑ∧mlde−T2+CMt.


### 4.9. Anderson-Darling and Right-Tail Anderson-Darling Estimators

The Anderson-Darling estimator (see Anderson and Darling [21]) is another type of minimum distance estimators. The ADEs ϱ^ade and ϑ^ade of the parameters ϱ and ϑ of L-E distribution are obtained by minimizing the function:
(43)Aϱ,ϑ=−n−1n∑i=1n2i−1logFzi:nϱ,ϑ+logF¯zn+1−i:nϱ,ϑ.These estimators can also be obtained by solving the following nonlinear equations:(44)∑i=1n2i−1ς1zi:nϱ,ϑFzi:nϱ,ϑ−ς1zn+1−i:nϱ,ϑF¯zn+1−i:nϱ,ϑ=0,∑i=1n2i−1ς2zi:nϱ,ϑFzi:nϱ,ϑ−ς2zn+1−i:nϱ,ϑF¯zn+1−i:nϱ,ϑ=0,where ς1·ϱ,ϑ and ς2·ϱ,ϑ are defined in Equations (21) and (22), respectively. Substituting the ADEs, we can get the estimator of CNpmc as
(45)C^Npmcade=U−L6ξγ3∣ϱ∧ade,ϑ∧ade−ξγ1∣ϱ∧,ϑ∧ade/62+ξγ2∣ϱ∧ade,ϑ∧ade−T2+CMt.Similarly, the right-tail Anderson-Darling (RAD) estimatorsϱ^rtade and ϑ^rtade of the parameters ϱ and ϑ are obtained by minimizing the following function:
(46)Rϱ,ϑ=n2−2∑i=1nFzi:nϱ,ϑ−1n∑i=1n2i−1logF¯zn+1−i:nϱ,ϑ.These estimators can also be obtained by solving the nonlinear equations:(47)−2∑i=1nς1xi:nϱ,ϑ+1n∑i=1n2i−1ς1zn+1−i:nϱ,ϑF¯zn+1−i:nϱ,ϑ=0,−2∑i=1nς2xi:nϱ,ϑ+1n∑i=1n2i−1ς2zn+1−i:nϱ,ϑF¯zn+1−i:nϱ,ϑ=0,where ς1·ϱ,ϑ and ς2·ϱ,ϑ are defined in Equations (21) and (22), respectively. Substituting the RADEs, we can get the estimator of CNpmc as
(48)C^Npmcrade=U−L6ξγ3ϱ∧rade,ϑ∧ade−ξγ1∣ϱ∧,ϑ∧rade/62+ξγ2∣ϱ∧rade,ϑ∧rade−T2+CMt.


## 4.1. Maximum Likelihood Estimators

Letz1,z2,⋯,zn be a random sample of size n drawn from two parameter LED (9), and then, the likelihood function is given by
(13)Lϱ,ϑ=∏i=1nfzi;ϱ,ϑ=∏i=1nϑ.ρeϑzi−1ϱ−1eϑzi1+eϑzi−1ϱ2.Taking logarithm on both the sides of Equation (13), we have
(14)logLϱ,ϑ=nlogϑ+nlogϱ+ϱ−1∑i=1nlogeϑzi−1+ϑ∑i=1nzi−2∑i=1nlog1+eϑzi−1ϱ.The MLEs ofϱ and ϑ, say ϱ^mle and ϑ^mle, respectively, can be obtained as an iterative solutions of the following two equations:
(15)∂logLϱ,ϑ∂ϱ=nρ+∑i=1nlogeϑzi−1−2∑i=1neϑzi−1ϱlogeϑzi−11+eϑzi−1ϱ=0,(16)∂logLϱ,ϑ∂ϑ=nϑ+∑i=1nϱ−1zi.eϑzieϑzi−1+∑i=1nzi−2∑i=1nϱeϑzi−1ϱ−1xi.eϑzi1+eϑzi−1ϱ=0.Equations (15) and (16) can be solved for ϱ and ϑ using any numerical iterative procedure. Since the MLEs ofϱandϑare not in the closed forms, therefore, we have used nonlinear minimization (NLM) (see Dennis and Schnabel [16]) technique by using some initial guess value for the parameters, sayϱ=0.01andϑ=0.01, and obtaining the estimates ofϱandϑasϱ^mleandϑ^mle, respectively. Consequently, the MLE of CNpmc can be obtained as
(17)C^Npmcmle=U−L6ξγ3∣ϱ∧mle,ϑ∧mle−ξγ1∣ϱ∧,ϑ∧/62+ξγ2∣ϱ∧mle,ϑ∧mle−T2+CMt,where the MLE of the quantile function with parameters ϱ,ϑ is
(18)ξγiϱ^mle,ϑ^mle=1ϑ^mlelog1+log1+γi1−γi1/ϱ∧mle,i=1,2,3.


## 4.2. Least Squares Estimators

We minimize the following function with respect toρ and ϑ for obtaining the LSEs denoted by ϱ^lse and ϑ^lse.
(19)Sϱ,ϑ=∑i=1nFzi:nϱ,ϑ−in+12,where F· is the CDF, given in Equation (10) and zi:n,i=1,⋯,n is the ith order statistic of a random sample z1,z2,⋯,zn. Equivalently, they can be obtained by solving
(20)∑i=1nFzi:nϱ,ϑ−in+1ς1zi:nϱ,ϑ=0,∑i=1nFzi:nϱ,ϑ−in+1ς2zi:nϱ,ϑ=0,where
(21)ς1zi:nϱ,ϑ=1+eϑzi:n−1ϱ−2eϑzi:n−1ϱlogeϑzi:n−1,(22)ς2zi:nϱ,ϑ=1+eϑzi:n−1ϱ−2ρzi:neϑzi:neϑzi:n−1ϱ−1.Substituting the LSEs, we can get the estimator ofCNpmc as
(23)C^Npmclse=U−L6ξγ3∣ϱ∧lse,ϑ∧lse−ξγ1∣ϱ∧,ϑ∧lse/62+ξγ2∣ϱ∧lse,ϑ∧lse−T2+CMt.


## 4.3. Weighted Least Squares Estimators

The WLSEs,ϱ^wlse and ϑ^wlse, can be obtained by minimizing the following function:
(24)Wϱ,ϑ=∑i=1nwiFzi:nϱ,ϑ−in+12.The estimatorsϱ^wlse and ϑ^wlse of the parameters ϱ and ϑ can be obtained by solving the following nonlinear equations:
(25)∑i=1nwiFzi:nϱ,ϑ−in+1ς1zi:nϱ,ϑ=0,∑i=1nwiFzi:nϱ,ϑ−in+1ς2zi:nϱ,ϑ=0,where wi=n+12n+2/in−i+1, ς1zi:nϱ,ϑ, and ς2zi:nϱ,ϑ are defined in Equations (21) and (22), respectively. Substituting the WLSEs, we can get the estimator of CNpmc as
(26)C^Npmcwlse=U−L6ξγ3∣ϱ∧wlse,ϑ∧wlse−ξγ1∣ϱ∧,ϑ∧wlse/62+ξγ2∣ϱ∧wlse,ϑ∧wlse−T2+CMt.


## 4.4. Percentile Estimators

The percentile estimatesϱ^pce and ϑ^pce of the parameters ϱ and ϑ can be obtained by minimizing the following function with respect to ϱ and ϑ:
(27)Pϱ,ϑ=∑j=1nzj:n−1ϑlog1+pj1−pj1/ϱ2.Several estimators ofpj can be used here; see, for example, Mann et al. [17]. In this paper, we have consider pj=j/n+1. Substituting the PCEs, we can get the estimator of CNmpc as
(28)C^Npmcpce=U−L6ξγ3∣ϱ∧pce,ϑ∧pce−ξγ1∣ϱ∧,ϑ∧pce/62+ξγ2∣ϱ∧pce,ϑ∧pce−T2+CMt.


## 4.5. Cramèr-von Mises Estimators

The Cramèr-von Mises estimators ofϱ and ϑ, say ϱ^cme and ϑ^cme, can be obtained by minimizing the following function with respect to ϱ and ϑ.
(29)Cϱ,ϑ=112n+∑i=1nFzi:nϱ,ϑ−2i−12n2.The estimatorsϱ^cme and ϑ^cme of the parameters ϱ and ϑ can be obtained by solving the following nonlinear equations:
(30)∑i=1nFzi:nϱ,ϑ−2i−12nς1zi:nϱ,ϑ=0,∑i=1nFzi:nϱ,ϑ−2i−12nς2zi:nϱ,ϑ=0,where ς1zi:nϱ,ϑ and ς2zi:nϱ,ϑ are given by Equations (21) and (22), respectively. Substituting the CMEs, we can get the estimator of CNpmc as
(31)C^Npmccme=U−L6ξγ3∣ϱ∧cme,ϑ∧cme−ξγ1∣ϱ∧,ϑ∧cme/62+ξγ2∣ϱ∧cme,ϑ∧cme−T2+CMt.


## 4.6. Maximum Product of Spacing Estimators

Maximum product of spacing (MPS) method was proposed by Cheng and Amin [18, 19] which can be used as an alternative to ML method of estimation for estimating parameters of continuous univariate distributions. Define the uniform spacings of a random sample from the LED as
(32)Diϱ,ϑ=Fzi:nϱ,ϑ−Fzi−1:nϱ,ϑ,i=1,2,⋯,n,where zi:n, i=1,2,⋯,n is the ith order statistic of a random sample z1,z2,⋯,zn. Note that z0:n=0 and zn+1:n=1. The MPSEs, ϱ^mpse and ϑ^mpse, of the parameters ϱ and ϑ can be obtained by maximizing the following geometric mean of the spacing function with respect to ϱ and ϑ(33)Gϱ,ϑ=∏i=1n+1Diϱ,ϑ1/n+1,or, equivalently, by maximizing the function
(34)Hϱ,ϑ=1n+1∑i=1n+1logDiϱ,ϑ.The estimatorsϱ^mpse and ϑ^mpse of the parameters ϱ and ϑ can be obtained by solving the nonlinear equations:
(35)∂∂ϱHϱ,ϑ=1n+1∑i=1n+11Diϱ,ϑς1zi:n∣ϱ,ϑ−ς1zi−1:nϱ,ϑ=0,∂∂ϑHϱ,ϑ=1n+1∑i=1n+11Diϱ,ϑς2zi:n∣ϱ,ϑ−ς2zi−1:n∣ϱ,ϑ=0,where ς1zi:nϱ,ϑ and ς2zi:nϱ,ϑ are given by Equations (21) and (22), respectively. Substituting the MPSEs, we can get the estimator of CNpmc as
(36)C^Npmcmpse=U−L6ξγ3∣ϱ∧mpse,ϑ∧mpse−ξγ1∣ϱ∧,ϑ∧mpse/62+ξγ2∣ϱ∧mpse,ϑ∧mpse−T2+CMt.


## 4.7. Minimum Spacing Absolute Distance Estimators

Torabi [20] proposed the minimum spacing absolute distance estimators (MDE) of the parameters of a distribution. Thus, MDE of parameters ϱ and ϑ can be obtained by minimizing the following function:
(37)Tϱ,ϑ=∑i=1n+1Diϱ,ϑ−1n+1,with respect to ϱ and ϑ, respectively. The estimators ϱ^mde and ϑ^mde of the parameters ϱ and ϑ can be obtained by solving the following nonlinear equations:
(38)∂∂ϱTϱ,ϑ=∑i=1n+1Diϱ,ϑ−1/n+1Diϱ,ϑ−1/n+1ς1zi:nϱ,ϑ−ς1zi−1:nϱ,ϑ=0,∂∂ϑTϱ,ϑ=∑i=1n+1Diϱ,ϑ−1/n+1Diϱ,ϑ−1/n+1ς2zi:nϱ,ϑ−ς2zi−1:nϱ,ϑ=0,where Diϱ,ϑ≠1/n+1, ς1·ϱ,ϑ, and ς2·ϱ,ϑ are defined in Equations (21) and (22), respectively. Substituting the MDEs, we can get the estimator of CNpmc as
(39)C^Npmcmde=U−L6ξγ3∣ϱ∧mde,ϑ∧mde−ξγ1∣ϱ∧,ϑ∧mde/62+ξγ2∣ϱ∧mde,ϑ∧mde−T2+CMt.


## 4.8. Minimum Spacing Absolute Log-Distance Estimators

Torabi [20] proposed the minimum spacing absolute-log distance estimators (MLDE). The MLDEs of the parameters ϱ and ϑ can be obtained by minimizing the function:
(40)Tϱ,ϑ=∑i=1n+1logDiϱ,ϑ−log1n+1.The estimatorsϱ^mlde and ϑ^mlde of the parameters ϱ and ϑ can be obtained by solving the following nonlinear equations:
(41)∂∂ϱTϱ,ϑ=∑i=1n+1logDiϱ,ϑ−log1/n+1logDiϱ,ϑ−log1/n+11Diϱ,ϑς1zi:nϱ,ϑ−ς1zi−1:nϱ,ϑ=0,∂∂ϑTϱ,ϑ=∑i=1n+1logDiϱ,ϑ−log1/n+1logDiϱ,ϑ−log1/n+11Diϱ,ϑς2zi:nϱ,ϑ−ς2zi−1:nϱ,ϑ=0,where logDiϱ,ϑ≠log1/n+1, ς1·ϱ,σ, and ς2·ϱ,ϑ are defined in Equations (21) and (22), respectively. Substituting the MLDEs, we can get the estimator of CNpmc as
(42)C^Npmcmlde=U−L6ξγ3ϱ∧mlde,ϑ∧mlde−ξγ1∣ϱ∧,ϑ∧mlde/62+ξγ2∣ϱ∧mlde,ϑ∧mlde−T2+CMt.


## 4.9. Anderson-Darling and Right-Tail Anderson-Darling Estimators

The Anderson-Darling estimator (see Anderson and Darling [21]) is another type of minimum distance estimators. The ADEs ϱ^ade and ϑ^ade of the parameters ϱ and ϑ of L-E distribution are obtained by minimizing the function:
(43)Aϱ,ϑ=−n−1n∑i=1n2i−1logFzi:nϱ,ϑ+logF¯zn+1−i:nϱ,ϑ.These estimators can also be obtained by solving the following nonlinear equations:(44)∑i=1n2i−1ς1zi:nϱ,ϑFzi:nϱ,ϑ−ς1zn+1−i:nϱ,ϑF¯zn+1−i:nϱ,ϑ=0,∑i=1n2i−1ς2zi:nϱ,ϑFzi:nϱ,ϑ−ς2zn+1−i:nϱ,ϑF¯zn+1−i:nϱ,ϑ=0,where ς1·ϱ,ϑ and ς2·ϱ,ϑ are defined in Equations (21) and (22), respectively. Substituting the ADEs, we can get the estimator of CNpmc as
(45)C^Npmcade=U−L6ξγ3∣ϱ∧ade,ϑ∧ade−ξγ1∣ϱ∧,ϑ∧ade/62+ξγ2∣ϱ∧ade,ϑ∧ade−T2+CMt.Similarly, the right-tail Anderson-Darling (RAD) estimatorsϱ^rtade and ϑ^rtade of the parameters ϱ and ϑ are obtained by minimizing the following function:
(46)Rϱ,ϑ=n2−2∑i=1nFzi:nϱ,ϑ−1n∑i=1n2i−1logF¯zn+1−i:nϱ,ϑ.These estimators can also be obtained by solving the nonlinear equations:(47)−2∑i=1nς1xi:nϱ,ϑ+1n∑i=1n2i−1ς1zn+1−i:nϱ,ϑF¯zn+1−i:nϱ,ϑ=0,−2∑i=1nς2xi:nϱ,ϑ+1n∑i=1n2i−1ς2zn+1−i:nϱ,ϑF¯zn+1−i:nϱ,ϑ=0,where ς1·ϱ,ϑ and ς2·ϱ,ϑ are defined in Equations (21) and (22), respectively. Substituting the RADEs, we can get the estimator of CNpmc as
(48)C^Npmcrade=U−L6ξγ3ϱ∧rade,ϑ∧ade−ξγ1∣ϱ∧,ϑ∧rade/62+ξγ2∣ϱ∧rade,ϑ∧rade−T2+CMt.


## 5. Bootstrap Confidence Intervals

In this section, we use bootstrap technique to construct confidence intervals for the PCICNpmc using all considered methods of estimation. Here, we consider five methods of bootstrap CIs, namely, (i) SB, (ii) PB, (iii) STB, (iv) BCPB, and (v) BCAB. We discuss the algorithm for the bootstrap methods based on maximum likelihood only.
(1)
Let (Z1,Z2,⋯,Zn) be a random sample of size n drawn from LED (ϱ,ϑ). Compute MLEs (ϱ^mle,ϑ^mle) of (ϱ,ϑ). A bootstrap sample (Z1∗,Z2∗,⋯,Zn∗) is obtained by multiplying 1/n as mass at each point from the original sample(2)
Compute the MLEs (ϱ^mle∗,ϑ^mle∗) of (ϱ,ϑ) as well as C^Npmc∗mle of CNpmc. The Mth bootstrap estimator of CNpmc is computed as C^Npmc∗mleM=C^NpmcmleZ1∗M,Z2∗M,⋯,Zn∗M(3)
There are total number ofnn resamples. From these resamples, the entire collection of R values of C^Npmc∗mle from smallest to largest would constitute an empirical bootstrap distribution as C^Npmc∗mleI,I=11R


### 5.1.SB

LetAM∗mle and SD∗mle be the sample mean and standard deviation of C^Npmc∗mleI,I=11R, i.e.,
(49)AM∗mle=1R∑I=1RC^Npmc∗mleI,SD∗mle=1R−1∑I=1RC∧Npmc∗mleI−AM′∗mle2,respectively. A 1001−δ%SB confidence interval of CNpmc is given as
(50)AM∗mle−Zδ/2×SD∗mle,AM∗mle+Zδ/2×SD∗mle,where Zδ/2 is obtained by using upper δ/2th point of the standard normal deviate.


### 5.2.PB

LetC^Npmc∗mleξ be the ξ percentile of C^Npmc∗mleI,I=11R, i.e., C^Npmc∗mleξ is such that
(51)1R∑I=1RInC^Npmc∗I≤C^Npmc∗ξ=ξ,0<ξ<1,where In· is an indicator function. Then, a 1001−δ%PB confidence interval of CNpmc is given as
(52)C^Npmc∗mleR×δ/2,C^Npmc∗mleR×1−δ/2.


### 5.3.STB

Lett^Npmc∗mleξ be the ξ percentile of C^Npmc∗mleI−C^Npmc/SD∗mle,I=1,2,⋯,R, i.e., t^Npmc∗mleξ is such that
(53)1B∑I=1RInC^Npmc∗mleI−C^NpmcSD∗mle≤t^Npmc∗mleξ=τ,0<τ<1,where In. is defined above. A 1001−γ%TB confidence interval of Cpc is given by
(54)AM∗mle−t^Npmc∗mleξ×SD∗mle,AM∗mle+t^Npmc∗mleξ×SD∗mle.


### 5.4.BCPB

At first, locate the observedC^Npmc in the order statistics C^Npmc∗mleI,I=11R. Next, we compute G0=1/R∑I=1RInC^Npmc∗mleI≤C^Npmc and Ψ0=Φ−1G0 to calculate ψl and ψu where
(55)ψl=Φ2Ψ0−ξ1−δ/2,ψu=Φ2Ψ0+ξ1−δ/2,respectively. Then, 1001−δ%BCPB confidence interval of CNpmc is
(56)C^Npmc∗mleR×ψl,C^Npmc∗mleR×ψu.


### 5.5.BCAB

Calculate(57)Φ0=∑i=1nC∧Npmcmle·−C∧NpmcmleI36∑i=1nC∧Npmcmle·−C∧NpmcmleI23/2,where Φ0 is called the acceleration factor and C^NpmcmleI is the MLE of CNpmc based on n−1 observations after excluding the Ith observation.
(58)C^Nmlepmc·=1n∑i=1nC^NpmcmleI.Then, a1001−δ%BCAB confidence interval of CNpmc is given as
(59)C^Npmc∗mleV1,C^Npmc∗mleV2,where V1=ΦΨ0+Ψ0+Zδ/2/1−Φ0Ψ0+Zδ/2 and V2=ΦΨ0+Ψ0+Z1−δ/2/1−Φ0Ψ0+Z1−δ/2, respectively.


## 5.1.SB

LetAM∗mle and SD∗mle be the sample mean and standard deviation of C^Npmc∗mleI,I=11R, i.e.,
(49)AM∗mle=1R∑I=1RC^Npmc∗mleI,SD∗mle=1R−1∑I=1RC∧Npmc∗mleI−AM′∗mle2,respectively. A 1001−δ%SB confidence interval of CNpmc is given as
(50)AM∗mle−Zδ/2×SD∗mle,AM∗mle+Zδ/2×SD∗mle,where Zδ/2 is obtained by using upper δ/2th point of the standard normal deviate.


## 5.2.PB

LetC^Npmc∗mleξ be the ξ percentile of C^Npmc∗mleI,I=11R, i.e., C^Npmc∗mleξ is such that
(51)1R∑I=1RInC^Npmc∗I≤C^Npmc∗ξ=ξ,0<ξ<1,where In· is an indicator function. Then, a 1001−δ%PB confidence interval of CNpmc is given as
(52)C^Npmc∗mleR×δ/2,C^Npmc∗mleR×1−δ/2.


## 5.3.STB

Lett^Npmc∗mleξ be the ξ percentile of C^Npmc∗mleI−C^Npmc/SD∗mle,I=1,2,⋯,R, i.e., t^Npmc∗mleξ is such that
(53)1B∑I=1RInC^Npmc∗mleI−C^NpmcSD∗mle≤t^Npmc∗mleξ=τ,0<τ<1,where In. is defined above. A 1001−γ%TB confidence interval of Cpc is given by
(54)AM∗mle−t^Npmc∗mleξ×SD∗mle,AM∗mle+t^Npmc∗mleξ×SD∗mle.


## 5.4.BCPB

At first, locate the observedC^Npmc in the order statistics C^Npmc∗mleI,I=11R. Next, we compute G0=1/R∑I=1RInC^Npmc∗mleI≤C^Npmc and Ψ0=Φ−1G0 to calculate ψl and ψu where
(55)ψl=Φ2Ψ0−ξ1−δ/2,ψu=Φ2Ψ0+ξ1−δ/2,respectively. Then, 1001−δ%BCPB confidence interval of CNpmc is
(56)C^Npmc∗mleR×ψl,C^Npmc∗mleR×ψu.


## 5.5.BCAB

Calculate(57)Φ0=∑i=1nC∧Npmcmle·−C∧NpmcmleI36∑i=1nC∧Npmcmle·−C∧NpmcmleI23/2,where Φ0 is called the acceleration factor and C^NpmcmleI is the MLE of CNpmc based on n−1 observations after excluding the Ith observation.
(58)C^Nmlepmc·=1n∑i=1nC^NpmcmleI.Then, a1001−δ%BCAB confidence interval of CNpmc is given as
(59)C^Npmc∗mleV1,C^Npmc∗mleV2,where V1=ΦΨ0+Ψ0+Zδ/2/1−Φ0Ψ0+Zδ/2 and V2=ΦΨ0+Ψ0+Z1−δ/2/1−Φ0Ψ0+Z1−δ/2, respectively.


## 6. Simulation and Discussion

Here, we conduct a simulation research to evaluate the behaviour of the different PCICNpmc estimators as described in Section 3. The performance of the estimators is compared in terms of their respective MSEs. The BCIs are compared in terms of AW and CP. The sample sizes are 20, 50, and 100 each. Additionally, we set the target value, T=2.50, and the lower and upper specification limitations as 0.50 and 9.50 as well as the ϱ,ϑ values of (8.0,0.25), (8.0,0.75), (12.0,0.25), and (12.0,0.75), respectively. The tolerance cost function’s coefficients are given as C0=1.0, C1=3.0, and C2=2.0. The values of CMt, CNpm, and CNpmc are then determined at t=0.75. The following stages are used to present the method for obtaining the average estimations of the index CNpmc and the accompanying MSEs:
(1)
Draw a random sample of sizen from LEDϱ,ϑ(2)
Estimate the parametersϱ, ϑ using MLE(3)
Estimate the PCICNpmc using the estimates of parameters ϱ and ϑ(4)
Repeat steps 1-3;K=1,000 times(5)
Calculate the average estimate ofCNpmc and MSEs based on K repetitionsIn a similar way, we can find the estimates ofCNpmc and the corresponding MSEs by using LSE, WLSE, CME, PCE, MPSE, MDE, MLDE, ADE, and RADE wherein MLEs are used as initial values. The simulation results are reported in Table 2. The step-by-step procedure to obtain the considered BCIs (SB, PB, BCPB, TB, and BCAB) is discussed in details in Section 4. For each design, R=1,000 bootstrap samples with each of size n are drawn from the original sample using the estimates of the parameters and replicated K=5,000 times. The results of the 95% BCIs, viz., SB, PB, BCPB, STB, and BCAB, are constructed by each of the classical methods of estimation for CNpmc and are reported in Tables 3–12, respectively. The R codes for calculation of point estimate, corresponding to MSEs and AWs and CPs of BCIs of CNpmc, are given as Appendix at last.Table 2
Average estimates of different methods of estimation and the corresponding MSEs for LE distribution with true value ofCNpmc.
Estimates ofCnpmc and corresponding MSEsSample sizeMLEMSELSEMSEWLSEMSEPCEMSECMEMSEMPSEMSEMDEMSEMLDEMSEADEMSERADEMSEϱ=8.0, ϑ=0.25, CNpm=2.441746, CNpmc=1.048471201.0258811.0340551.0360511.0233611.0467931.0252161.0367681.0360821.0388671.0387750.0016600.0026510.0018950.0022590.0026960.0015810.0021650.0019460.0029210.002699501.0261421.0425981.0451201.0354201.0484581.0375921.0455341.0451861.0459111.0458220.0006390.0009130.0006910.0010110.0007860.0005670.0007480.0007210.0011640.0011241001.0267481.0465191.0477011.0408511.0489841.0489841.0477691.0477461.0478681.0478240.0003840.0006860.0004470.0007140.0006770.0002410.0005130.0004940.0008870.000834ϱ=8.0, ϑ=0.75, CNpm=0.945505, CNpmc=0.733128200.7418860.7328090.7329330.7325430.7333370.7325120.7338890.7336720.7423440.7421327.97×e−058.84×e−057.99×e−058.93×e−058.37×e−057.87×e−058.24×e−058.04×e−059.13×e−059.05×e−05500.7416260.73291170.7330040.73271420.7331060.7327250.7379780.7378620.7488420.7467323.22×e−053.69×e−053.27×e−053.26×e−053.64×e−053.10×e−053.47×e−053.39×e−053.96×e−053.81×e−051000.7416290.7440740.7431150.7441390.7432680.7329400.7432870.7432360.7441740.7441641.62×e−051.93×e−051.69×e−051.99×e−051.89×e−051.53×e−051.84×e−051.77×e−052.11×e−052.07×e−05ϱ=12.0, ϑ=0.25, CNpm=3.281044, CNpmc=1.094454201.0847771.0874691.0855211.0880171.0870781.0832221.0859631.0858841.0882641.0881550.0005040.0006820.0005290.0007340.0006270.0004890.0006040.0005660.0008140.000786501.0866021.0881701.0866901.0884521.0878961.0856621.0874421.0872121.0882841.0882240.0002010.0004170.0002560.0004490.0003850.0001610.0003140.0002880.0005640.0005121001.0919651.0935891.0925041.0937511.0933741.0915641.0928961.0927641.0942641.0940777.77×e−059.53×e−058.24×e−059.66×e−059.24×e−057.16×e−058.98×e−058.76×e−059.97×e−059.82×e−05ϱ=12.0, ϑ=0.75, CNpm=0.949039, CNpmc=0.734772200.7346190.7347460.7346410.7347630.7347030.7344290.7346790.7346680.7347850.7347783.56×e−053.91×e−053.94×e−053.66×e−053.88×e−053.42×e−053.81×e−053.74×e−053.99×e−053.97×e−05500.7346660.7348440.7347740.7348760.7348120.7346570.7347960.7347810.7348840.7348491.52×e−051.81×e−051.59×e−051.84×e−051.76×e−051.41×e−051.69×e−051.64×e−051.95×e−051.89×e−051000.7385050.7346330.7345670.7346880.7345850.7344870.7345410.7345080.7347590.7347227.47×e−067.81×e−067.55×e−067.88×e−067.75×e−067.39×e−067.69×e−067.62×e−067.98×e−067.93×e−06Table 3
AW and CPs of BCIs ofCNpmc by using MLEs of the parameters along with true values of CNpmc.
PCISBPBSTBBCPBBCABParametersnAWCPAWCPAWCPAWCPAWCPCNpmc=1.048471200.9387870.9200.9201820.9220.9142830.9190.8899940.9180.8867890.917ϱ=8.0500.8265690.9250.7987850.9260.7911240.9240.7721340.9220.7685120.921ϑ=0.251000.4565660.9310.4244370.9330.4167540.9290.4013260.9290.3988450.928CNpmc=0.733128200.7511140.9210.7156340.9240.7025610.9200.6734320.9190.6680250.918ϱ=8.0500.6811260.9260.6522380.9280.6467240.9240.6069720.9220.5977740.920ϑ=0.751000.3488420.9330.3178790.9340.3111740.9320.2845320.9290.2799730.927CNpmc=1.094454200.9699670.9210.9343710.9230.9288840.9200.9011260.9180.8977690.917ϱ=12.0500.8501380.9260.8178920.9290.8078540.9270.7778780.9260.7692380.924ϑ=0.251000.4825410.9330.4512140.9340.4421560.9320.4199920.9300.4155370.929CNpmc=0.734772200.7486920.9220.7145620.9230.7003640.9200.6776490.9190.6698980.918ϱ=12.0500.6877790.9240.6578850.9260.6421140.9230.6012420.9230.5999860.922ϑ=0.751000.3501270.9350.3188560.9360.3143260.9310.2978590.9300.2854370.929Table 4
AW and CPs of BCIs ofCNpmc by using LSEs of the parameters along with true values of CNpmc.
PCISBPBSTBBCPBBCABParametersnAWCPAWCPAWCPAWCPAWCPCNpmc=1.048471200.9467860.9200.9277640.9210.9198680.9190.8988880.9190.8976810.917ϱ=8.0500.8344270.9240.8023410.9250.7989980.9230.7811120.9220.7709830.921ϑ=0.251000.4677890.9330.4376750.9350.4224570.9300.4167430.9290.4040750.928CNpmc=0.733128200.7567540.9230.7277990.9240.7154540.9210.6889750.9210.6789640.918ϱ=8.0500.6989880.9250.6678930.9270.6489890.9240.6145640.9230.6088790.922ϑ=0.751000.3611460.9330.3479860.9350.3232470.9310.2988870.9290.2829870.927CNpmc=1.094454200.9892330.9210.9499880.9220.9357640.9200.9100680.9180.9076120.917ϱ=12.0500.8589670.9250.8276820.9270.8167790.9260.7865250.9250.7777790.922ϑ=0.251000.4888970.9340.4609870.9360.4477770.9310.4233590.9300.4187560.929CNpmc=0.734772200.7500020.9210.7200980.9220.7145630.9200.6897970.9180.6766670.916ϱ=12.0500.6908930.9240.6688770.9250.6456570.9230.6188770.9230.6167820.922ϑ=0.751000.3677770.9350.3249820.9360.3188770.9310.3030670.9290.2888980.927Table 5
AW and CPs of BCIs ofCNpmc by using WLSEs of the parameters along with true values of CNpmc.
PCISBPBSTBBCPBBCABParametersnAWCPAWCPAWCPAWCPAWCPCNpmc=1.048471200.9398970.9200.9224360.9210.9177650.9180.8912320.9180.8879840.916ϱ=8.0500.8354210.9250.8011370.9260.7965790.9240.7733650.9220.7686240.921ϑ=0.251000.4599930.9300.4278930.9320.4243110.9290.4078630.9290.4011470.927CNpmc=0.733128200.7547620.9210.7243170.9230.7092180.9200.6744560.9180.6692360.917ϱ=8.0500.6867820.9250.6576840.9270.6488790.9230.6176170.9220.5988340.920ϑ=0.751000.3507820.9330.3233470.9350.3178890.9310.2897960.9290.2811240.928CNpmc=1.094454200.9711240.9210.9367540.9220.9292270.9190.9023410.9180.8989540.917ϱ=12.0500.8521360.9250.8197970.9280.8132540.9270.7782410.9250.7697760.923ϑ=0.251000.4844870.9330.4527830.9350.4433650.9320.4257430.9290.4167840.928CNpmc=0.734772200.7511260.9220.7176980.9240.7048720.9210.6789890.9190.6706340.918ϱ=12.0500.6935610.9250.6598890.9270.6488440.9240.6106780.9230.6024710.922ϑ=0.751000.3544890.9330.3256780.9350.3199820.9320.2999670.9310.2878640.929Table 6
AW and CPs of BCIs ofCNpmc by using PC PCEs of the parameters along with true values of CNpmc.
PCISBPBSTBBCPBBCABParametersnAWCPAWCPAWCPAWCPAWCPCNpmc=1.048471200.9511120.9200.9278750.9230.9199990.9200.8956840.9180.8899880.917ϱ=8.0500.8343210.9240.8126730.9250.8007420.9230.7789870.9220.7722440.921ϑ=0.251000.4587890.9330.4325310.9350.4188890.9300.4156430.9300.4078250.929CNpmc=0.733128200.7566670.9220.7287670.9240.7056750.9210.6788870.9200.6754330.918ϱ=8.0500.6877790.9250.6623120.9260.6499910.9240.6167450.9220.6088660.921ϑ=0.751000.3423170.9330.3577690.9350.3197980.9310.2986040.9290.2809940.928CNpmc=1.094454200.9789760.9210.9478930.9230.9321250.9200.9089960.9190.9007810.917ϱ=12.0500.8588880.9260.8298790.9280.8177660.9250.7809570.9240.7745320.922ϑ=0.251000.4878870.9330.4657810.9360.4499990.9310.4234520.9290.4187880.927CNpmc=0.734772200.7508970.9210.7234110.9220.7088890.9200.6898780.9190.6756430.917ϱ=12.0500.6978650.9250.6745320.9260.6488870.9230.6176840.9220.6078930.921ϑ=0.751000.3666790.9340.3322140.9350.3188870.9300.3079840.9290.2888670.927Table 7
AW and CPs of BCIs ofCNpmc by using CMEs of the parameters along with true values of CNpmc.
PCISBPBSTBBCPBBCABParametersnAWCPAWCPAWCPAWCPAWCPCNpmc=1.048471200.9423170.9200.9267540.9210.9178960.9190.8976830.9180.8898970.917ϱ=8.0500.8453280.9240.8123170.9260.8067540.9240.7776570.9220.7698970.921ϑ=0.251000.4609870.9320.4288890.9340.4297970.9290.4167430.9290.4067540.928CNpmc=0.733128200.7643510.9210.7278990.9230.7100680.9190.6799880.9180.6708940.916ϱ=8.0500.6907140.9250.6623240.9270.6467450.9230.6198780.9220.5988340.921ϑ=0.751000.3607920.9330.3276770.9360.3198780.9310.2911140.9290.2837840.928CNpmc=1.094454200.9734340.9210.9477650.9220.9307140.9190.9078860.9180.9012620.917ϱ=12.0500.8588670.9250.8265130.9270.8176560.9250.7867450.9240.7711350.922ϑ=0.251000.4887940.9320.4565790.9350.4489670.9310.4277740.9290.4177980.928CNpmc=0.734772200.7687940.9220.7188890.9230.7087980.9210.6822370.9190.6777840.918ϱ=12.0500.6977780.9250.6644320.9270.6507980.9240.6145320.9240.6067560.923ϑ=0.751000.3602240.9330.3354290.9360.3221340.9320.3007850.9300.2897980.928Table 8
AW and CPs of BCIs ofCNpmc by using MPSEs of the parameters along with true values of CNpmc.
PCISBPBSTBBCPBBCABParametersnAWCPAWCPAWCPAWCPAWCPCNpmc=1.048471200.9344610.9210.9187920.9220.9132620.9190.8898790.9170.8865730.916ϱ=8.0500.8236170.9250.7923140.9260.7894360.9230.7699810.9230.7684350.921ϑ=0.251000.4511290.9300.4210270.9320.4107820.9290.3995610.9290.3978140.927CNpmc=0.733128200.7451370.9210.7073480.9230.6933730.9200.6698940.9190.6671240.917ϱ=8.0500.6788560.9240.6412760.9250.6358910.9230.5987860.9220.5967820.921ϑ=0.751000.3458720.9330.3133960.9340.3074820.9320.2799570.9300.2783460.928CNpmc=1.094454200.9672370.9220.9308670.9230.9256430.9190.8989780.9190.8956410.917ϱ=12.0500.8456320.9250.8134250.9260.8056730.9240.7732170.9240.7688910.923ϑ=0.251000.4785670.9330.4442470.9350.4371240.9320.4187930.9310.4141680.929CNpmc=0.734772200.7478620.9220.7127520.9240.6989850.9200.6732510.9190.6687690.918ϱ=12.0500.6833470.9250.6547610.9260.6377740.9240.5989840.9230.5968880.922ϑ=0.751000.3467670.9340.3177740.9360.3099690.9310.2922350.9310.2831290.929Table 9
AW and CPs of BCIs ofCNpmc by using MDEs of the parameters along with true values of CNpmc.
PCISBPBSTBBCPBBCABParametersnAWCPAWCPAWCPAWCPAWCPCNpmc=1.048471200.9456740.9200.9267840.9220.9188670.9190.8912680.9190.8896740.917ϱ=8.0500.8287950.9240.7999990.9260.7965480.9230.7785670.9220.7699780.921ϑ=0.251000.4569530.9320.4288840.9340.4178780.9300.4076590.9290.3999990.929CNpmc=0.733128200.7545430.9230.7233210.9250.7034340.9220.6778890.9210.6699880.919ϱ=8.0500.6845630.9250.6587940.9270.6477640.9240.6112460.9220.6067840.921ϑ=0.751000.3499990.9330.3567430.9340.3177860.9320.2976340.9290.2806730.927CNpmc=1.094454200.9768530.9210.9458790.9230.9304270.9200.9045630.9180.8999780.917ϱ=12.0500.8544590.9260.8211140.9280.8125460.9260.7788890.9250.7709890.923ϑ=0.251000.4856570.9330.4591280.9350.4467680.9320.4208930.9290.4177750.928CNpmc=0.734772200.7499940.9210.7178760.9230.7097850.9200.6867530.9190.6711420.917ϱ=12.0500.6888840.9250.6654320.9260.6444570.9230.6137630.9220.6060970.922ϑ=0.751000.3622480.9340.3254760.9360.3155790.9310.2998670.9300.2867820.928Table 10
AW and CPs of BCIs ofCNpmc by using MLDEs of the parameters along with true values of CNpmc.
PCISBPBSTBBCPBBCABParametersnAWCPAWCPAWCPAWCPAWCPCNpmc=1.048471200.9376840.9210.9199810.9220.9167670.9210.8933470.9180.8866660.917ϱ=8.0500.8286740.9240.7967540.9260.7898990.9240.7722360.9230.7687780.921ϑ=0.251000.4589750.9310.4267450.9330.4167850.9290.4026720.9290.3989730.928CNpmc=0.733128200.7489780.9210.7126350.9230.6978940.9200.6722340.9200.6687950.919ϱ=8.0500.6801260.9250.6467560.9270.6378800.9240.6099940.9230.5978960.922ϑ=0.751000.3478940.9330.3144570.9350.3104770.9320.2811460.9300.2756740.929CNpmc=1.094454200.9756430.9210.9411340.9230.9207840.9200.9011140.9190.8977690.917ϱ=12.0500.8587630.9250.8322560.9270.8177640.9250.7803450.9250.7745320.924ϑ=0.251000.4767850.9330.4499980.9360.4379820.9320.4199440.9310.4189760.929CNpmc=0.734772200.7487980.9220.7200670.9230.7056210.9200.6777890.9190.6699870.917ϱ=12.0500.6911140.9250.6577650.9260.6389890.9250.6129340.9230.6007890.922ϑ=0.751000.3508910.9340.3254610.9360.3122220.9320.2978960.9310.2899910.929Table 11
AW and CPs of BCIs ofCNpmc by using ADEs of the parameters along with true values of CNpmc.
PCISBPBSTBBCPBBCABParametersnAWCPAWCPAWCPAWCPAWCPCNpmc=1.048471200.9576120.9200.9288890.9220.9207650.9190.8999940.9180.8987560.917ϱ=8.0500.8377860.9240.8076830.9260.7998980.9230.7845680.9230.7732540.922ϑ=0.251000.4699390.9330.4398560.9350.4256730.9300.4187560.9290.4065720.927CNpmc=0.733128200.7623150.9230.7288830.9240.7167770.9210.6899960.9200.6797980.918ϱ=8.0500.6988790.9250.6688670.9260.6563450.9240.6177560.9230.6211460.921ϑ=0.751000.3645340.9340.3489870.9360.3265470.9310.3007560.9290.2876480.928CNpmc=1.094454200.9961120.9210.9476580.9220.9398670.9200.9144360.9180.9089780.917ϱ=12.0500.8589880.9250.8299940.9270.8177780.9240.7876450.9230.7789870.921ϑ=0.251000.4900640.9340.4623160.9360.4500670.9310.4233640.9300.4197540.928CNpmc=0.734772200.7500020.9210.7200980.9230.7145630.9200.6897970.9180.6766670.917ϱ=12.0500.6922350.9240.6691380.9250.6487990.9240.6191620.9230.6177890.922ϑ=0.751000.3691930.9340.3266740.9360.3209780.9310.3065470.9290.2900670.928Table 12
AW and CPs of BCIs ofCNpmc by using RADEs of the parameters along with true values of CNpmc.
PCISBPBSTBBCPBBCABParametersnAWCPAWCPAWCPAWCPAWCPCNpmc=1.048471200.9545620.9210.9287870.9230.9231260.9200.8988650.9190.8907830.918ϱ=8.0500.8387980.9240.8147650.9260.8065340.9240.7799330.9220.7754560.921ϑ=0.251000.4604730.9330.4367580.9360.4199890.9310.4176580.9300.4097860.928CNpmc=0.733128200.7608910.9220.7293370.9240.7076780.9210.6795680.9200.6787960.918ϱ=8.0500.6888890.9240.6676590.9260.6523110.9240.6188670.9220.6111240.921ϑ=0.751000.3455590.9340.3589780.9370.3206780.9310.2993930.9290.2834260.927CNpmc=1.094454200.9799260.9210.9498980.9230.9376580.9200.9122230.9180.9045360.917ϱ=12.0500.8588880.9240.8298790.9270.8177660.9250.7809570.9230.7745320.922ϑ=0.251000.4933210.9340.4679830.9360.4502170.9310.4278650.9290.4204070.927CNpmc=0.734772200.7543270.9210.7278650.9220.7215430.9200.6921570.9180.6777680.917ϱ=12.0500.6999930.9250.6779680.9260.6511240.9240.6184840.9230.6098940.922ϑ=0.751000.3723140.9350.3423160.9370.3192270.9310.3088780.9290.2890980.928Simulated outcomes of considered estimators for the indexCNpmc are listed in Table 2. From Table 2, we observe that as the sample sizes increases, MSEs decrease in all the cases which eventually proves the consistency of the considered methods of estimation for our study. Simulation results of the configurations examined in our study indicate that MPSE outperform other estimators, while second best estimator is MLE followed by WLSE using MSEs as the criteria. Thus, the order of performance of the methods of estimation in terms of MSE is MPSE < MLE < WLSE < MLDE < MDE < CME < LSE < PCE < RADE < ADE. Further, we observe that when the parameter values of ϱ increases, the value of the index CNpmc increases. Results of the estimated AW and CPs of BCIs of the index CNpmc using all the considered methods of estimation (MLE, LSE, WLSE, CME, PCE, MPSE, MDE, MLDE, ADE, and RADE) are listed in Tables 3–5, respectively. The comparisons of BCIs are made on the basis of lower average width and higher coverage probabilities. We take into account the nominal value as 95% for comparing coverage probabilities. Results in Tables 3–12 indicate that CI of BCAB provides smaller AW, while CI of PB provides higher CPs for all configurations and for all methods of estimation considered in the study. Moreover, we can say that, for almost all sample sizes, among the five methods of BCIs, the simulation results show the following order from the least in terms of the AW: BCAB < BCPB < STB < PB < SB for all settings considered in this study. Therefore, we conclude that BCAB method is superior to all other considered BCIs for LE distribution. Also, it has been observed that in most of the situations in simulation study, the AW of BCIs are small by using MPSE method than the other considered methods.


## 7. Applications

In this section, two electronic industry-related data sets are reanalyzed for illustrative purposes. In order to check the validity of the proposed model, one sample Kolmogorov-Smirnov (K-S) statistic along with itsp values and two information theoretic criteria such as AIC and BIC are used. The associated unknown parameters of the model are estimated using the likelihood method. The steps listed below are used to determine the K-S statistic’s p values:
(i)
Fit the chosen distribution to the data(ii)
Compute the corresponding K-S statistic(iii)
10,000 identical samples to the size of the data from the fitted distribution are simulated(iv)
Calculate the K-S statistic using the associated 10,000 values(v)
Using step (iv), create a histogram of the 10,000 values(vi)
By contrasting the histogram with the recorded statistic from step (ii), one can determine thep value


### 7.1. Data Set I: Electronic Telecommunication Amplifier Data

The data set I relates to the quality of the electronic communication amplifiers. The data was collected by Juran Institute [22] and reanalyzed by Peng [23]. The key quality characteristic of the data set is the gain of decibels with production specification limits L,T,U=7.75,10,12.25. In case of L-E distribution, MLEs of the parameters ϱ and ϑ are ϱ^=0.07745682 and ϑ^=13.33916572. The model fitting summary (viz., log-likelihood, AIC, BIC, empirical and theoretical densities and CDFs, P-P plot, and Q-Q plot; see Figure 1) is reported in Table 13. The data set is given below:Figure 1
Histogram-density, P-P plot, Q-Q plot, and theoretical and empirical CDFs of data set I.Table 13
Model fitting summary of the considered data sets I and II.
Data setModelLog-likelihoodAICBICK-S statisticK-Sp valueIL-E distribution-151.0645306.129311.70400.0765640.4826IIL-E distribution-184.7552373.5105378.72080.0421230.99431,10.4,8.8,9.7,7.8,9.9,11.7,8.0,9.3,9.0,8.2,8.9,10.1,9.4,9.2,7.9,9.5,10.9,7.8,8.3,9.1,8.4,9.6,11.1,7.9,8.5,8.7,7.8,10.5,8.5,11.5,8.0,7.9,8.3,8.7,10.0,9.4,9.0,9.2,10.7,9.3,9.7,8.7,8.2,8.9,8.6,9.5,9.4,8.8,8.3,8.4,9.1,10.1,7.8,8.1,8.8,8.0,9.2,8.4,7.8,7.9,8.5,9.2,8.7,10.2,7.9,9.8,8.3,9.0,9.6,9.9,10.6,8.6,9.4,8.8,8.2,10.5,9.7,9.1,8.0,8.7,9.8,8.5,8.9,9.1,8.4,8.1,9.5,8.7,9.3,8.1,10.1,9.6,8.3,8.0,9.8,9.0,8.9,8.1,9.7,8.5,8.2,9.0,10.2,9.5,8.3,8.9,9.1,10.3,8.4,8.6,9.2,8.5,9.6,9.0,10.7,8.6,10.0,8.8,8.6.


### 7.2. Data Set II: Data Set Relates to Electronic Industry

The data set II is taken from Leiva et al. [4] which represents the ball size of wire bonding for an electronic connection from the integrated circuit apparatus to the lead frame. Here, the process was monitored with LSL=0.5 mil and USL=8.0 mil (1mil=1/1000 in =0.0254 mm). Here, we have considered the target value T=3.0 mil. In case of L-E distribution, MLEs of the parameters ϱ and ϑ are ϱ^=0.2561459 and ϑ^=2.0632703. The model fitting summary (viz., log-likelihood, AIC, BIC, empirical and theoretical densities and CDFs, P-P plot, and Q-Q plot; see Figure 2) is reported in Table 13. The data set is given below:Figure 2
Histogram-density, P-P plot, Q-Q plot, and theoretical and empirical CDFs of data set II.2.891,4.035,4.495,2.890,2.312,3.158,5.228,3.334,5.896,5.639,3.842,1.590,1.954,1.842,0.680,2.752,1.301,2.260,0.889,2.381,0.619,2.788,1.050,3.750,3.508,6.123,6.549,5.954,2.207,4.417,4.805,1.516,2.227,2.797,1.636,1.066,0.940,4.101,4.542,1.295,1.770,3.492,5.706,3.722,6.644,2.472,1.383,4.494,1.694,2.892,2.111,3.591,2.093,3.222,2.891,2.582,0.665,3.234,1.102,1.083,1.508,1.811,2.803,6.659,0.923,6.229,3.177,2.333,1.311,4.419,2.495,0.921,4.061,9.725,1.600,4.281,3.360,1.131,1.618,4.489,3.696,1.982,2.413,5.480,1.992,2.573,1.845,4.620,6.221,1.694,4.882,1.380,3.982,2.260,2.366,2.899,3.782,2.336,1.175,3.055.In Table14, we report the point estimates of CNpmc and CNpm and the width of BCIs based on different methods of estimation. For both the data sets, we observe that the width of BCAB interval is minimum as compared to width of other BCIs. Similar trend is exhibited in the simulation study. Further, we observe that among all methods of estimation, MPSE performs better than other methods of estimation for data sets I and II, respectively.Table 14
Classical estimates ofCNpmc and CNpm (MLE, LSE, WLSE, PCE, CME, MPSE, MDE, MLDE, ADE, and RADE) along with the widths of BCIs (SB, PB, STB, BCPB, and BCAB) based on considered data sets I and II (C0=1, C1=3, C2=2, and t=0.50).
Data setMethods of estimationClassical estimates ofCNpm and CNpmcWidths of BCIsSBPBSTBBCPBBCABIMLEC^Npmmle=0.5288830.7649420.7343250.6846890.6676030.661133C^Npmcmle=0.3798260.6645230.6478950.6245330.6187940.615676ILSEC^Npmlse=0.5213930.7649540.7343380.6846970.6676150.661142C^Npmclse=0.3809230.6645270.6478950.6245350.6187940.61565IWLSEC^Npmwlse=0.5294760.7649480.7343330.6846970.6676110.661138C^Npmcwlse=0.3808770.6645240.6478960.6245330.6187950.615676IPCEC^Npmpce=0.5294890.7649550.7343380.6846960.6676150.661143C^Npmcpce=0.3809060.6645280.6478940.6245350.6187950.61566ICMEC^Npmcme=0.5213790.7649530.7343380.6846950.6676140.661140C^Npmccme=0.3721480.6645280.6478950.6245340.6187960.61566IMPSEC^Npmmpse=0.5213620.7648940.7325630.6845320.6675870.661127C^Npmcmpse=0.3721370.6645210.6478940.6245320.6187950.615672IMDEC^Npmmde=0.5297740.7649520.7343390.6846960.6676130.661141C^Npmcmde=0.3809440.6645270.6478960.6245340.6187950.615677IMLDEC^Npmmlde=0.5296840.7649480.7343360.6846960.6676100.661139C^Npmcmlde=0.3809130.6645270.6478970.6245340.6187950.615676IADEC^Npmade=0.5313410.7649560.7343410.6846960.6676160.661144C^Npmcade=0.3812370.6645290.6478950.6245350.6187970.615679IRADEC^Npmrade=0.5312110.7649540.7343390.6846950.6676140.661144C^Npmcrade=0.3811550.6645280.6478950.6245350.6187960.615677IIMLEC^Npmmle=0.5941090.7798680.75086720.7034610.6967730.692257C^Npmcmle=0.4891470.6887140.6645320.6411780.6378610.635642IILSEC^Npmlse=0.5941310.7798680.75086720.7034630.6967730.692256C^Npmclse=0.4891590.6887290.6645460.6411780.6378640.635643IIWLSEC^Npmwlse=0.5941260.7798670.75086730.7034600.6967720.692256C^Npmcwlse=0.4891470.6887160.6645330.6411760.6378620.635641IIPCEC^Npmpce=0.5941330.7798690.75086750.7034640.6967720.692258C^Npmcpce=0.4891640.6887330.6645490.6411780.6378650.635645IICMEC^Npmcme=0.5941300.7798670.75086720.7034610.6967720.692256C^Npmccme=0.4891570.6887240.6645430.6411790.6378640.635644IIMPSEC^Npmmpse=0.5941070.7798660.75086720.7034600.6967710.692255C^Npmcmpse=0.4891450.6887120.6645310.6411760.6378610.635641IIMDEC^Npmmde=0.5941330.7798690.75086770.7034600.6967740.692258C^Npmcmde=0.4891540.6887220.6645390.6411780.6378640.635642IIMLDEC^Npmmlde=0.5941290.7798660.75086720.7034600.6967720.692257C^Npmcmlde=0.4891510.6887180.6645370.6411770.6378620.635642IIADEC^Npmade=0.5941380.7798690.75086770.7034640.6967750.692259C^Npmcade=0.4891770.6887520.6645590.6411870.6378720.635649IIRADEC^Npmrade=0.5941360.7798680.75086760.7034640.6967740.692259C^Npmcrade=0.4891710.6887460.6645550.6411820.6378680.635647


## 7.1. Data Set I: Electronic Telecommunication Amplifier Data

The data set I relates to the quality of the electronic communication amplifiers. The data was collected by Juran Institute [22] and reanalyzed by Peng [23]. The key quality characteristic of the data set is the gain of decibels with production specification limits L,T,U=7.75,10,12.25. In case of L-E distribution, MLEs of the parameters ϱ and ϑ are ϱ^=0.07745682 and ϑ^=13.33916572. The model fitting summary (viz., log-likelihood, AIC, BIC, empirical and theoretical densities and CDFs, P-P plot, and Q-Q plot; see Figure 1) is reported in Table 13. The data set is given below:Figure 1
Histogram-density, P-P plot, Q-Q plot, and theoretical and empirical CDFs of data set I.Table 13
Model fitting summary of the considered data sets I and II.
Data setModelLog-likelihoodAICBICK-S statisticK-Sp valueIL-E distribution-151.0645306.129311.70400.0765640.4826IIL-E distribution-184.7552373.5105378.72080.0421230.99431,10.4,8.8,9.7,7.8,9.9,11.7,8.0,9.3,9.0,8.2,8.9,10.1,9.4,9.2,7.9,9.5,10.9,7.8,8.3,9.1,8.4,9.6,11.1,7.9,8.5,8.7,7.8,10.5,8.5,11.5,8.0,7.9,8.3,8.7,10.0,9.4,9.0,9.2,10.7,9.3,9.7,8.7,8.2,8.9,8.6,9.5,9.4,8.8,8.3,8.4,9.1,10.1,7.8,8.1,8.8,8.0,9.2,8.4,7.8,7.9,8.5,9.2,8.7,10.2,7.9,9.8,8.3,9.0,9.6,9.9,10.6,8.6,9.4,8.8,8.2,10.5,9.7,9.1,8.0,8.7,9.8,8.5,8.9,9.1,8.4,8.1,9.5,8.7,9.3,8.1,10.1,9.6,8.3,8.0,9.8,9.0,8.9,8.1,9.7,8.5,8.2,9.0,10.2,9.5,8.3,8.9,9.1,10.3,8.4,8.6,9.2,8.5,9.6,9.0,10.7,8.6,10.0,8.8,8.6.


## 7.2. Data Set II: Data Set Relates to Electronic Industry

The data set II is taken from Leiva et al. [4] which represents the ball size of wire bonding for an electronic connection from the integrated circuit apparatus to the lead frame. Here, the process was monitored with LSL=0.5 mil and USL=8.0 mil (1mil=1/1000 in =0.0254 mm). Here, we have considered the target value T=3.0 mil. In case of L-E distribution, MLEs of the parameters ϱ and ϑ are ϱ^=0.2561459 and ϑ^=2.0632703. The model fitting summary (viz., log-likelihood, AIC, BIC, empirical and theoretical densities and CDFs, P-P plot, and Q-Q plot; see Figure 2) is reported in Table 13. The data set is given below:Figure 2
Histogram-density, P-P plot, Q-Q plot, and theoretical and empirical CDFs of data set II.2.891,4.035,4.495,2.890,2.312,3.158,5.228,3.334,5.896,5.639,3.842,1.590,1.954,1.842,0.680,2.752,1.301,2.260,0.889,2.381,0.619,2.788,1.050,3.750,3.508,6.123,6.549,5.954,2.207,4.417,4.805,1.516,2.227,2.797,1.636,1.066,0.940,4.101,4.542,1.295,1.770,3.492,5.706,3.722,6.644,2.472,1.383,4.494,1.694,2.892,2.111,3.591,2.093,3.222,2.891,2.582,0.665,3.234,1.102,1.083,1.508,1.811,2.803,6.659,0.923,6.229,3.177,2.333,1.311,4.419,2.495,0.921,4.061,9.725,1.600,4.281,3.360,1.131,1.618,4.489,3.696,1.982,2.413,5.480,1.992,2.573,1.845,4.620,6.221,1.694,4.882,1.380,3.982,2.260,2.366,2.899,3.782,2.336,1.175,3.055.In Table14, we report the point estimates of CNpmc and CNpm and the width of BCIs based on different methods of estimation. For both the data sets, we observe that the width of BCAB interval is minimum as compared to width of other BCIs. Similar trend is exhibited in the simulation study. Further, we observe that among all methods of estimation, MPSE performs better than other methods of estimation for data sets I and II, respectively.Table 14
Classical estimates ofCNpmc and CNpm (MLE, LSE, WLSE, PCE, CME, MPSE, MDE, MLDE, ADE, and RADE) along with the widths of BCIs (SB, PB, STB, BCPB, and BCAB) based on considered data sets I and II (C0=1, C1=3, C2=2, and t=0.50).
Data setMethods of estimationClassical estimates ofCNpm and CNpmcWidths of BCIsSBPBSTBBCPBBCABIMLEC^Npmmle=0.5288830.7649420.7343250.6846890.6676030.661133C^Npmcmle=0.3798260.6645230.6478950.6245330.6187940.615676ILSEC^Npmlse=0.5213930.7649540.7343380.6846970.6676150.661142C^Npmclse=0.3809230.6645270.6478950.6245350.6187940.61565IWLSEC^Npmwlse=0.5294760.7649480.7343330.6846970.6676110.661138C^Npmcwlse=0.3808770.6645240.6478960.6245330.6187950.615676IPCEC^Npmpce=0.5294890.7649550.7343380.6846960.6676150.661143C^Npmcpce=0.3809060.6645280.6478940.6245350.6187950.61566ICMEC^Npmcme=0.5213790.7649530.7343380.6846950.6676140.661140C^Npmccme=0.3721480.6645280.6478950.6245340.6187960.61566IMPSEC^Npmmpse=0.5213620.7648940.7325630.6845320.6675870.661127C^Npmcmpse=0.3721370.6645210.6478940.6245320.6187950.615672IMDEC^Npmmde=0.5297740.7649520.7343390.6846960.6676130.661141C^Npmcmde=0.3809440.6645270.6478960.6245340.6187950.615677IMLDEC^Npmmlde=0.5296840.7649480.7343360.6846960.6676100.661139C^Npmcmlde=0.3809130.6645270.6478970.6245340.6187950.615676IADEC^Npmade=0.5313410.7649560.7343410.6846960.6676160.661144C^Npmcade=0.3812370.6645290.6478950.6245350.6187970.615679IRADEC^Npmrade=0.5312110.7649540.7343390.6846950.6676140.661144C^Npmcrade=0.3811550.6645280.6478950.6245350.6187960.615677IIMLEC^Npmmle=0.5941090.7798680.75086720.7034610.6967730.692257C^Npmcmle=0.4891470.6887140.6645320.6411780.6378610.635642IILSEC^Npmlse=0.5941310.7798680.75086720.7034630.6967730.692256C^Npmclse=0.4891590.6887290.6645460.6411780.6378640.635643IIWLSEC^Npmwlse=0.5941260.7798670.75086730.7034600.6967720.692256C^Npmcwlse=0.4891470.6887160.6645330.6411760.6378620.635641IIPCEC^Npmpce=0.5941330.7798690.75086750.7034640.6967720.692258C^Npmcpce=0.4891640.6887330.6645490.6411780.6378650.635645IICMEC^Npmcme=0.5941300.7798670.75086720.7034610.6967720.692256C^Npmccme=0.4891570.6887240.6645430.6411790.6378640.635644IIMPSEC^Npmmpse=0.5941070.7798660.75086720.7034600.6967710.692255C^Npmcmpse=0.4891450.6887120.6645310.6411760.6378610.635641IIMDEC^Npmmde=0.5941330.7798690.75086770.7034600.6967740.692258C^Npmcmde=0.4891540.6887220.6645390.6411780.6378640.635642IIMLDEC^Npmmlde=0.5941290.7798660.75086720.7034600.6967720.692257C^Npmcmlde=0.4891510.6887180.6645370.6411770.6378620.635642IIADEC^Npmade=0.5941380.7798690.75086770.7034640.6967750.692259C^Npmcade=0.4891770.6887520.6645590.6411870.6378720.635649IIRADEC^Npmrade=0.5941360.7798680.75086760.7034640.6967740.692259C^Npmcrade=0.4891710.6887460.6645550.6411820.6378680.635647


## 8. Conclusions and Future Works

In this work, we evaluate five BCIs of the PCICNpmc using MLEs, LSEs, WLSEs, CMEs, MPEs, MDEs, MLDEs, PCEs, ADEs, and RADEs. Theoretical comparisons of the cited methods will be tedious; therefore, in order to compare the performance of estimators, we undertake simulation study using different sample sizes and varied combination of parameters. We compare the performance of the estimators in respect of MSE. The performance of BCIs for the index CNpmc is compared in respect of AW and CPs. Results from the simulation study indicate that MPSEs outperform other estimators while the second best estimator is MLEs followed by WLSEs. Further, CIs of BCAB outperform other CIs with respect to AW and CPs for considered methods of estimation. The results of the data analysis portray similar trend as in case of simulation study. Further, the considered index may be applicable to other areas associated with quality control such as process monitoring and acceptance sampling. Over and above, results and methods discussed in this study may be utilized by industries for decision-making. The present study can also be extended to neutrosophic statistics when the data comes from the production process or when a product lot is incomplete, incredible, and indeterminate (see Aslam and Albassam [24] and Aslam et al. [25]).



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*Source: 3135264-2022-09-23.xml*