# Classical Estimation of the IndexSpmk and Its Confidence Intervals for Power Lindley Distributed Quality Characteristics

Authors: Ghadah Alomani; Refah Alotaibi; Sanku Dey; Mahendra Saha
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The process capability index (PCI) has been introduced as a tool to aid in the assessment of process performance. Usually, conventional PCIs perform well under normally distributed quality characteristics. However, when these PCIs are employed to evaluate nonnormally distributed process, they often provide inaccurate results. In this article, in order to estimate the PCISpmk when the process follows power Lindley distribution, first, seven classical methods of estimation, namely, maximum likelihood method of estimation, ordinary and weighted least squares methods of estimation, Cramèr–von Mises method of estimation, maximum product of spacings method of estimation, Anderson–Darling, and right-tail Anderson–Darling methods of estimation, are considered and the performance of these estimation methods based on their mean squared error is compared. Next, three bootstrap confidence intervals (BCIs) of the PCI Spmk, namely, standard bootstrap, percentile bootstrap, and bias-corrected percentile bootstrap, are considered and compared in terms of their average width, coverage probability, and relative coverage. Besides, a new cost-effective PCI, namely, Spmkc is introduced by incorporating tolerance cost function in the index Spmk. To evaluate the performance of the methods of estimation and BCIs, a simulation study is carried out. Simulation results showed that the maximum likelihood method of estimation performs better than their counterparts in terms of mean squared error, while bias-corrected percentile bootstrap provides smaller confidence length (width) and higher relative coverage than standard bootstrap and percentile bootstrap across sample sizes. Finally, two real data examples are provided to investigate the performance of the proposed procedures.


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## Body

## 1. Introduction

Any process performance can be conveniently measured by process capability indices (PCIs). In recent years, great attention has been devoted to PCIs in the statistical quality control literature. To evaluate whether the production process is conforming to predefined specifications and to analyze the quality process and productivity, one can adopt PCIs. The fundamental assumption on process capability analysis was that the process was stable and the process quality characteristic of interest was normally distributed, while in practice, it has been observed that very often manufacturing processes follow nonnormally distributed quality characteristics. Examples of these are taper, flatness, concentricity, perpendicularity, roundness, and hole location. When any process is analyzed which is not normally distributed, one can employ various methods to define the PCI. One popular methodology is to convert a nonnormal process into a normal process by using transformation methods such as the Johnson transformation system and Box–Cox power transformation. Another approach is to modify a PCI so that it can be used suitably both for normal as well as nonnormal families of distributions. Details of the latter approach can be found in the works of [1, 2].For handling nonnormality distributed quality characteristics, Chen and Ding [3] first reviewed the PCIs Cp, Cpk, Cpm, and Cpmk and their generalizations by taking into account the process variability, departure of process mean from the target value, and proportion of nonconformity and then gave a new PCI Spmk as follows:(1)Spmk=Φ−11+FUSL−FLSL/231+μ−T/σ2=Φ−11−p/231+μ−T/σ2,where p=1−FUSL−FLSL is the exact proportion of nonconformity, CDF of the process distribution is denoted by F., and the CDF of the standard normal distribution is denoted by Φ.. Note that Spmk=CNpmk=Cpmk=Cpm, when μ=m and the underlying process distribution is normal.In the product life cycle, quality values will vary under different circumstances. In this regard, Jeang et al.[4] pointed out that a good quality (low quality loss) implies a tight tolerance (high production cost), while a poor quality (high quality loss) indicates a loose tolerance (low production cost). Therefore, it is important to consider the cost tolerance in evaluating PCIs. In this paper, we adopt the tolerance cost function as CMt=C0+C1exp−C2t (see [4]), where C0,C1,andC2 are the coefficients for the tolerance cost function and t is the process tolerance. Replacing the denominator of Spmk in equation (1) by 31+μ−T/σ2+CMt, and the cost effective PCI, say, Spmkc can be written as(2)Spmkc=Φ−11−p/231+μ−T/σ2+CMt.Usually, two basic estimation techniques called point estimation and confidence intervals (CIs) (see [5]) are used to evaluate a PCI. Of the two techniques, researchers have found that variability of the estimator can be easily assessed using CIs while process performance can be assessed by the point estimator. For more details, see [6]. Hsiang and Taguchi [7] first introduced the concept of construction of confidence limits for PCIs. Since then, several researchers have developed numerous techniques for constructing CIs. In this regard, readers may refer to [8–19] and the references cited therein.The objective of this paper is three fold: First, the estimates ofSpmk using seven different classical methods of estimation are obtained. The estimators used in this paper are: maximum likelihood estimator (MLE), least squares estimator (LSE), weighted least squares estimator (WLSE), Cramèr–von Mises estimator (CME), maximum product of spacing estimator (MPSE), Anderson–Darling estimator (ADE), and right-tail Anderson–Darling estimator (RTADE). For several well-known distributions and in varied contexts, several authors have used different classical methods of estimation to estimate the parameters of the model (see [20–25] and many others). The performance of different estimation methods is demonstrated in terms of their mean squared errors (MSEs) using a simulation study. Second, three BCIs of Spmk based on the above-cited classical methods of estimation are constructed. The performance of the BCIs is demonstrated in terms of their estimated average widths (AWs), coverage probabilities (CPs), and relative coverages (RCs). Third, a new cost-effective PCI Spmkc incorporating a tolerance cost function in the index Spmk is developed. These two PCIs are then compared using different methods of estimation and BCIs. Finally, our aim is to select the best estimation method among the seven different frequentist methods of the indices Spmk and Spmkc which would be of significance to quality control engineers, where the item/subgroup quality characteristic follows power Lindley distribution (PLD).The rest of the paper is organized as follows: In Section2, description of the PCIs Spmk and Spmkc for PLD is provided. In Section 3, different frequentist methods of estimation (MLE, LSE, WLSE, CME, MPSE, ADE, and RTADE) for the indices Spmk and Spmkc are described. In Section 4, three BCIs, namely, standard bootstrap (S-boot), percentile bootstrap (P-boot), and bias-corrected percentile bootstrap (BCp-boot) based on aforementioned methods of estimation of the PCIs Spmk and Spmkc are discussed. In Section 5, a simulation study has been performed to assess the performance of the above-cited classical methods of estimation of the indices Spmk and Spmkc in terms of their MSEs. Also, the performance of different BCIs under the considered methods of estimation in terms of AWs, CPs, and RCs are assessed. Two data sets are presented and discussed in Section 6. The article ends with a brief conclusion in Section 7.


## 2. IndicesSpmk and Spmkc for the PLD

The power Lindley distribution (PLD) was recently proposed in [26]. The PDF and CDF of the PLD are given by(3)Fzα,β=1−1+ββ+1zαe−βzα;z>0,α,β>0,fzα,β=αβ2β+11+zαzα−1e−βzα,where α and β are the shape and scale parameters, respectively. Various authors have attempted to build a statistical inference for PLD using classical and Bayesian approaches. In this regard, readers may refer to the works of [27–32] and many others.Ther-th order raw moment (about origin) of the PLD is given by (see [26])(4)μr′=rΓr/ααβ+1+rα2βr/αβ+1.Therefore, mean (μ) and variance (σ2) of the PLD, respectively, are(5)μ=Γ1/ααβ+1+1α2β1/αβ+1,σ2=2Γ2/ααβ+1+2α2β+1−Γ21/ααβ+1+12α4β2/αβ+12.Now, the PCIsSpmk and Spmkc, where the quality characteristic T which follows the PLD, are, respectively, given by(6)Spmk=Φ−11−p/231+μ−T/σ2,(7)Spmkc=Φ−11−p/231+μ−T/σ2+CMt,where(8)p=1−1+ββ+1LSLαe−βLSLα−1+ββ+1USLαe−βUSLα,is the exact proportion of nonconformity and μ and σ are the process mean and process standard deviation, respectively.


## 3. Estimation of the IndicesSpmk and Spmkc

Here, we describe different classical estimators for the indicesSpmk and Spmkc, respectively.


### 3.1. Maximum Likelihood Estimators

Letz1,z2,…,zn be random samples of size n taken from the PLD with parameter (α,β). The likelihood function is given by(9)ℒα,βz=αβ21+βne−β∑i=1nziα∏i=1n1+ziαziα−1.Taking logarithm on both the sides of equation (9), we get(10)logℒα,βz=nlogα+2nlogβ−nlog1+β−β∑i=1nziα+∑i=1nlog1+ziα+α−1∑i=1nlogzi.The resulting partial derivatives with respect toα and β of the log-likelihood function and equating to zero yields the solutions of α and β and are given by(11)nα−β∑i=1nziαlogzi+∑i=1nlogzi+∑i=1nziαlogzi1+ziα=0,n2+ββ1+β−∑i=1nziα=0.The MLEs ofα and β denoted by α^mle and β^mle can be obtained by solving the above equations simultaneously or by maximizing the log-likelihood function given in equation (10) with respect to α and β. Substituting the MLEs (α^mle,β^mle) of (α,β) and by using the invariance property of MLE, we can get the estimators of Spmk and Spmkc as(12)S^pmkmle=Φ−11−p^mle/231+z¯−T/s2,S^pmkcmle=Φ−11−p^mle/231+z¯−T/s2+CMt,where z¯, s, and p^ are the sample mean, sample standard deviation, and sample proportion of nonconformity, respectively.


### 3.2. Least Squares and Weighted Least Squares Estimators

Here, we compute least square and weighted least square estimators of the indicesSpmk and Spmkc. This method was initially studied in [33] for estimation problems related to beta distribution. However, this method can be applied to some other distributions as well. Let z1:n<z2:n<⋯<zn:n be the order observations obtained from a sample of size n from Fα,β. Therefore, the least square estimators of α and β, say, α^lse and β^lse, can be obtained by minimizing the following function:(13)Sα,β=∑i=1nFzi:nα,β−in+12,with respect to α and β, respectively. Equivalently, they can be obtained by solving the following equations:(14)∑i=1nFzi:nα,β−in+1ζ1zi:nα,β=0,∑i=1nFzi:nα,β−in+1ζ2zi:nα,β=0,where(15)ζ1zi:nα,β=β2e−βzi:nαzi:nαlogzi:n1+zi:nαβ+1,(16)ζ2zi:nα,β=θzi:nαe−βzi:nα2+β+βzi:nα+zi:nαβ+12.Substituting the LSEs in equations (6) and (7), we can get the estimators of Spmk and Spmkc as(17)S^pmklse=Φ−11−p^lse/231+z¯−T/s2,S^pmkclse=Φ−11−p^lse/231+z¯−T/s2+CMt.The WLSEs,α^wlse and β^wlse, can be obtained by minimizing the following function with respect to a and β, respectively:(18)Wα,β=∑i=1nn+12n+2in−i+1Fzi:nα,β−in+12.Equivalently, WLSEs can be obtained by solving the following equations:(19)∑i=1nn+12n+2in−i+1Fzi:nα,β−in+1ζ1zi:nα,β=0,∑i=1nn+12n+2in−i+1Fzi:nα,β−in+1ζ2zi:nα,β=0,where ζ1α,β and ζ2α,β are defined in equations (15) and (16), respectively. Substituting the WLSEs in equations (6) and (7), we can get the estimators Spmk and Spmkc as(20)S^pmkwlse=Φ−11−p^wlse/231+z¯−T/s2,S^pmkcwlse=Φ−11−p^wlse/231+z¯−T/s2+CMt.


### 3.3. Cramèr–von Mises Estimators

The Cramèr–von Mises (see [34, 35]) estimators of α and β can be obtained by minimizing the function(21)Cα,β=112n+∑i=1nFzi:nα,β−2i−12n2,with respect to α and β. These estimators can also be obtained by solving the following nonlinear equations:(22)∑i=1nFzi:nα,β−2i−12nζ1zi:nα,β=0,∑i=1nFzi:nα,β−2i−12nζ2zi:nα,β=0,where ζ1α,β and ζ2α,β are defined in equations (15) and (16), respectively. Substituting the CMEs in equations (6) and (7), we can get the estimators of Spmk and Spmkc as(23)S^pmkcme=Φ−11−p^cme/231+z¯−T/s2,S^pmkccme=Φ−11−p^cme/231+z¯−T/s2+CMt.


### 3.4. Maximum Product Spacing Estimators

Cheng and Amin [36] studied this method as an alternative to the maximum likelihood method. The uniform spacings of a random sample of size n is defined as(24)Diα,β=Fzi:nα,β−Fzi−1:nα,β,i=1,2,…,n+1,where Fz0:nα,β=0 and Fzn+1:nα,β=1. Clearly, ∑i=1n+1Diα,β=1.The MPSEsα^mpse and β^mpse, of the parameters a and β are obtained by maximizing with respect to a and β, the geometric mean of the spacings:(25)Gα,β=∏i=1n+1Diα,β1/n+1,or, equivalently, by maximizing the function(26)Hα,β=1n+1∑i=1n+1logDiα,β.The estimatorsα^mpse and β^mpse of the parameters α and β can be obtained by solving the following nonlinear equations:(27)∂∂αHα,β=1n+1∑i=1n+11Diα,βζ1zi:nα,β−ζ1zi−1:nα,β=0,∂∂βHα,β=1n+1∑i=1n+11Diα,βζ2zi:nα,β−ζ2zi−1:nα,β=0,where ζ1α,β and ζ2α,β are defined in equations (15) and (16), respectively. Substituting the MPSEs in equations (6) and (7), we can get the estimator of Spmk and Spmkc as(28)S^pmkmpse=Φ−11−p^mpse/231+z¯−T/s2,S^pmkcmpse=Φ−11−p^mpse/231+z¯−T/s2+CMt.


### 3.5. Anderson–Darling and Right-Tail Anderson–Darling Estimators

The Anderson and Darling estimators (ADEs) (see [37]) α^ade and β^ade of the parameters α and β can be obtained by minimizing the following function with respect to α and β:(29)Aα,β=−n−1n∑i=1n2i−1logFzi:nα,β+logF¯zn+1−i:nα,β.These estimators can also be obtained by solving the following nonlinear equations:(30)∑i=1n2i−1ζ1zi:nα,βFzi:nα,β−ζ1zn+1−i:nα,βF¯zn+1−i:nα,β=0,∑i=1n2i−1ζ2zi:nα,βFzi:nα,β−ζ2zn+1−i:nα,βF¯zn+1−i:nα,β=0,where ζ1α,β and ζ2α,β are defined in equations (15) and (16), respectively. Substituting the ADEs in equations (6) and (7), we can get the estimators of Spmk and Spmkc as(31)S^pmkade=Φ−11−p^ade/231+z¯−T/s2,S^pmkcade=Φ−11−p^ade/231+z¯−T/s2+CMt.The RTADE estimatorsα^rtade and β^rtade of the parameters α and β are obtained by minimizing the following function with respect to α and β:(32)ℛα,β=n2−2∑i=1nFzi:nα,β−1n∑i=1n2i−1logF¯zn+1−i:nα,β.These estimators can also be obtained by solving the following nonlinear equations:(33)−2∑i=1nζ1xi:nα,β+1n∑i=1n2i−1ζ1zn+1−i:nα,βF¯zn+1−i:nα,β=0,−2∑i=1nζ2zi:nα,β+1n∑i=1n2i−1ζ2zn+1−i:nα,βSzn+1−i:nα,β=0.Substituting the RTADEs in equations (6) and (7), we can get the estimators of Spmk and Spmkc as(34)S^pmkrtade=Φ−11−p^rade/231+z¯−T/s2,S^pmkcrtade=Φ−11−p^rade/231+z¯−T/s2+CMt.


## 3.1. Maximum Likelihood Estimators

Letz1,z2,…,zn be random samples of size n taken from the PLD with parameter (α,β). The likelihood function is given by(9)ℒα,βz=αβ21+βne−β∑i=1nziα∏i=1n1+ziαziα−1.Taking logarithm on both the sides of equation (9), we get(10)logℒα,βz=nlogα+2nlogβ−nlog1+β−β∑i=1nziα+∑i=1nlog1+ziα+α−1∑i=1nlogzi.The resulting partial derivatives with respect toα and β of the log-likelihood function and equating to zero yields the solutions of α and β and are given by(11)nα−β∑i=1nziαlogzi+∑i=1nlogzi+∑i=1nziαlogzi1+ziα=0,n2+ββ1+β−∑i=1nziα=0.The MLEs ofα and β denoted by α^mle and β^mle can be obtained by solving the above equations simultaneously or by maximizing the log-likelihood function given in equation (10) with respect to α and β. Substituting the MLEs (α^mle,β^mle) of (α,β) and by using the invariance property of MLE, we can get the estimators of Spmk and Spmkc as(12)S^pmkmle=Φ−11−p^mle/231+z¯−T/s2,S^pmkcmle=Φ−11−p^mle/231+z¯−T/s2+CMt,where z¯, s, and p^ are the sample mean, sample standard deviation, and sample proportion of nonconformity, respectively.


## 3.2. Least Squares and Weighted Least Squares Estimators

Here, we compute least square and weighted least square estimators of the indicesSpmk and Spmkc. This method was initially studied in [33] for estimation problems related to beta distribution. However, this method can be applied to some other distributions as well. Let z1:n<z2:n<⋯<zn:n be the order observations obtained from a sample of size n from Fα,β. Therefore, the least square estimators of α and β, say, α^lse and β^lse, can be obtained by minimizing the following function:(13)Sα,β=∑i=1nFzi:nα,β−in+12,with respect to α and β, respectively. Equivalently, they can be obtained by solving the following equations:(14)∑i=1nFzi:nα,β−in+1ζ1zi:nα,β=0,∑i=1nFzi:nα,β−in+1ζ2zi:nα,β=0,where(15)ζ1zi:nα,β=β2e−βzi:nαzi:nαlogzi:n1+zi:nαβ+1,(16)ζ2zi:nα,β=θzi:nαe−βzi:nα2+β+βzi:nα+zi:nαβ+12.Substituting the LSEs in equations (6) and (7), we can get the estimators of Spmk and Spmkc as(17)S^pmklse=Φ−11−p^lse/231+z¯−T/s2,S^pmkclse=Φ−11−p^lse/231+z¯−T/s2+CMt.The WLSEs,α^wlse and β^wlse, can be obtained by minimizing the following function with respect to a and β, respectively:(18)Wα,β=∑i=1nn+12n+2in−i+1Fzi:nα,β−in+12.Equivalently, WLSEs can be obtained by solving the following equations:(19)∑i=1nn+12n+2in−i+1Fzi:nα,β−in+1ζ1zi:nα,β=0,∑i=1nn+12n+2in−i+1Fzi:nα,β−in+1ζ2zi:nα,β=0,where ζ1α,β and ζ2α,β are defined in equations (15) and (16), respectively. Substituting the WLSEs in equations (6) and (7), we can get the estimators Spmk and Spmkc as(20)S^pmkwlse=Φ−11−p^wlse/231+z¯−T/s2,S^pmkcwlse=Φ−11−p^wlse/231+z¯−T/s2+CMt.


## 3.3. Cramèr–von Mises Estimators

The Cramèr–von Mises (see [34, 35]) estimators of α and β can be obtained by minimizing the function(21)Cα,β=112n+∑i=1nFzi:nα,β−2i−12n2,with respect to α and β. These estimators can also be obtained by solving the following nonlinear equations:(22)∑i=1nFzi:nα,β−2i−12nζ1zi:nα,β=0,∑i=1nFzi:nα,β−2i−12nζ2zi:nα,β=0,where ζ1α,β and ζ2α,β are defined in equations (15) and (16), respectively. Substituting the CMEs in equations (6) and (7), we can get the estimators of Spmk and Spmkc as(23)S^pmkcme=Φ−11−p^cme/231+z¯−T/s2,S^pmkccme=Φ−11−p^cme/231+z¯−T/s2+CMt.


## 3.4. Maximum Product Spacing Estimators

Cheng and Amin [36] studied this method as an alternative to the maximum likelihood method. The uniform spacings of a random sample of size n is defined as(24)Diα,β=Fzi:nα,β−Fzi−1:nα,β,i=1,2,…,n+1,where Fz0:nα,β=0 and Fzn+1:nα,β=1. Clearly, ∑i=1n+1Diα,β=1.The MPSEsα^mpse and β^mpse, of the parameters a and β are obtained by maximizing with respect to a and β, the geometric mean of the spacings:(25)Gα,β=∏i=1n+1Diα,β1/n+1,or, equivalently, by maximizing the function(26)Hα,β=1n+1∑i=1n+1logDiα,β.The estimatorsα^mpse and β^mpse of the parameters α and β can be obtained by solving the following nonlinear equations:(27)∂∂αHα,β=1n+1∑i=1n+11Diα,βζ1zi:nα,β−ζ1zi−1:nα,β=0,∂∂βHα,β=1n+1∑i=1n+11Diα,βζ2zi:nα,β−ζ2zi−1:nα,β=0,where ζ1α,β and ζ2α,β are defined in equations (15) and (16), respectively. Substituting the MPSEs in equations (6) and (7), we can get the estimator of Spmk and Spmkc as(28)S^pmkmpse=Φ−11−p^mpse/231+z¯−T/s2,S^pmkcmpse=Φ−11−p^mpse/231+z¯−T/s2+CMt.


## 3.5. Anderson–Darling and Right-Tail Anderson–Darling Estimators

The Anderson and Darling estimators (ADEs) (see [37]) α^ade and β^ade of the parameters α and β can be obtained by minimizing the following function with respect to α and β:(29)Aα,β=−n−1n∑i=1n2i−1logFzi:nα,β+logF¯zn+1−i:nα,β.These estimators can also be obtained by solving the following nonlinear equations:(30)∑i=1n2i−1ζ1zi:nα,βFzi:nα,β−ζ1zn+1−i:nα,βF¯zn+1−i:nα,β=0,∑i=1n2i−1ζ2zi:nα,βFzi:nα,β−ζ2zn+1−i:nα,βF¯zn+1−i:nα,β=0,where ζ1α,β and ζ2α,β are defined in equations (15) and (16), respectively. Substituting the ADEs in equations (6) and (7), we can get the estimators of Spmk and Spmkc as(31)S^pmkade=Φ−11−p^ade/231+z¯−T/s2,S^pmkcade=Φ−11−p^ade/231+z¯−T/s2+CMt.The RTADE estimatorsα^rtade and β^rtade of the parameters α and β are obtained by minimizing the following function with respect to α and β:(32)ℛα,β=n2−2∑i=1nFzi:nα,β−1n∑i=1n2i−1logF¯zn+1−i:nα,β.These estimators can also be obtained by solving the following nonlinear equations:(33)−2∑i=1nζ1xi:nα,β+1n∑i=1n2i−1ζ1zn+1−i:nα,βF¯zn+1−i:nα,β=0,−2∑i=1nζ2zi:nα,β+1n∑i=1n2i−1ζ2zn+1−i:nα,βSzn+1−i:nα,β=0.Substituting the RTADEs in equations (6) and (7), we can get the estimators of Spmk and Spmkc as(34)S^pmkrtade=Φ−11−p^rade/231+z¯−T/s2,S^pmkcrtade=Φ−11−p^rade/231+z¯−T/s2+CMt.


## 4. Bootstrap Confidence Intervals

In this section, three parametric bootstrap confidence intervals ofSpmk and Spmkc are obtained. Particularly, S-boot, P-boot, and BCp-boot CIs are constructed for Spmk and Spmkc (see [38]):(1)
Based on the original sampleZ=z1,z2…,zm, obtain Θ^=α^,β^, the MLE of Θ.(2)
Simulate the firstm-order statistics from a sample of size n from power Lindley distribution with parameter Θ^: z1=z1:n1,z2:n1,…,zm:n1.(3)
Compute the MLE ofΘ as well as Spmk based on z1, say, Θ^1∗ and S^pmk1∗.(4)
Repeat steps 2-3 B times and obtainS^pmk1∗,S^pmk2∗,…,S^pmkB∗(5)
ArrangeS^pmk1∗,S^pmk2∗,…,S^pmkB∗ in the ascending order and obtain S^pmkj∗;j=11B.


### 4.1.S-Boot

LetS^¯pmk∗ be the sample mean and s∗ be the sample standard deviation of S^pmkj∗;j=11B, i.e.,(35)S^¯pmk∗=1B∑j=1BS^pmkj∗,s∗=1B−1∑j=1BS^pmkj∗−S^¯pmk∗2.A1001−γ%S-boot CI of Spmk is given by(36)S^¯pmk∗−Zγ/2⋅s∗,S^¯pmk∗+Zγ/2⋅s∗.Here,Zγ/2 is the upper γ/2-th point of the standard normal variate.


### 4.2.P-Boot

LetS^pmk∗τ be the τ percentile of S^pmkj∗,j=11B, i.e., S^pmk∗τ is such that(37)1B∑j=1BInS^pmkj∗≤S^pmk∗τ=τ;0<τ<1,where In. is an indicator function. A 1001−γ%P-boot CI of Spmk is given by(38)S^pmk∗B.γ/2,S^pmk∗B.1−γ/2.


### 4.3.BCp-Boot

In this method, we first locate the observedS^pmk in the ordered distributions S^pmk1∗≤S^pmk2∗≤⋯≤S^pmkB∗. Next, we compute the probability(39)Ψ0=1B∑j=1BInS^pmkj∗≤S^pmk.Then, calculateΠ0=Φ−1Ψ0, where Φ. is the CDF of the standard normal variate and ψl and ψu are given by(40)ψl=Φ2Π0−Z1−γ/2,ψu=Φ2Π0+Z1−γ/2.Then,1001−γ%BCp-boot CI of Spmk is(41)S^pmk∗B.ψl,S^pmk∗B.ψu.In a similar way, we can also find out the empirical bootstrap distribution ofSpmkc, given as S^pmkc∗j;j=11B, i.e., S^pmkc∗1≤S^pmkc∗2≤⋯≤S^pmkc∗B.To study the different bootstrap CIs, we consider their estimated AWs, CPs, and RCs. The AWs, CPs, and RCs of the BCIs are calculated based onK=5,000 different trials and are defined as(42)AW=∑i=1KUi−LiK,CP=LW≤Spmk≤UpK,RC=CPAW,where LW and UP denote the 1001−γ% CIs based on K repetitions.


## 4.1.S-Boot

LetS^¯pmk∗ be the sample mean and s∗ be the sample standard deviation of S^pmkj∗;j=11B, i.e.,(35)S^¯pmk∗=1B∑j=1BS^pmkj∗,s∗=1B−1∑j=1BS^pmkj∗−S^¯pmk∗2.A1001−γ%S-boot CI of Spmk is given by(36)S^¯pmk∗−Zγ/2⋅s∗,S^¯pmk∗+Zγ/2⋅s∗.Here,Zγ/2 is the upper γ/2-th point of the standard normal variate.


## 4.2.P-Boot

LetS^pmk∗τ be the τ percentile of S^pmkj∗,j=11B, i.e., S^pmk∗τ is such that(37)1B∑j=1BInS^pmkj∗≤S^pmk∗τ=τ;0<τ<1,where In. is an indicator function. A 1001−γ%P-boot CI of Spmk is given by(38)S^pmk∗B.γ/2,S^pmk∗B.1−γ/2.


## 4.3.BCp-Boot

In this method, we first locate the observedS^pmk in the ordered distributions S^pmk1∗≤S^pmk2∗≤⋯≤S^pmkB∗. Next, we compute the probability(39)Ψ0=1B∑j=1BInS^pmkj∗≤S^pmk.Then, calculateΠ0=Φ−1Ψ0, where Φ. is the CDF of the standard normal variate and ψl and ψu are given by(40)ψl=Φ2Π0−Z1−γ/2,ψu=Φ2Π0+Z1−γ/2.Then,1001−γ%BCp-boot CI of Spmk is(41)S^pmk∗B.ψl,S^pmk∗B.ψu.In a similar way, we can also find out the empirical bootstrap distribution ofSpmkc, given as S^pmkc∗j;j=11B, i.e., S^pmkc∗1≤S^pmkc∗2≤⋯≤S^pmkc∗B.To study the different bootstrap CIs, we consider their estimated AWs, CPs, and RCs. The AWs, CPs, and RCs of the BCIs are calculated based onK=5,000 different trials and are defined as(42)AW=∑i=1KUi−LiK,CP=LW≤Spmk≤UpK,RC=CPAW,where LW and UP denote the 1001−γ% CIs based on K repetitions.


## 5. Simulation Study

In order to examine the behaviour of the proposed BCIs (S-boot, P-boot, and BCp-boot) of the indices Spmk and Spmkc under different methods of estimation, a Monte Carlo simulation study is performed. The performance of point estimators of the indices Spmk and Spmkc is compared in terms of MSEs. We have considered n=10,20,30,50 in the simulation study. We set lower, target, and upper specification limits as L=0.5, T=2.0, and U=5.5 and α,β=1.50,0.50,2.00,0.50,2.50,0.75,and7.50,0.25, respectively. The coefficients in the tolerance cost function are considered as C0=15, C1=30, and C2=5. At t=0.5, the values of Spmkc are also calculated. The number of bootstrap samples used in the construction of the respective confidence interval was set to B=1,000, and results are based on K=5,000 replications. Next, using 1000 bootstrap samples, 95% BCIs are then constructed for each of the seven methods of estimation.Point estimates of the parameters under different methods of estimation are reported in Table1, while point estimates of Spmk and Spmkc along with their MSEs are given in Table 2. The values of BCIs are reported in Tables 3–9. From Table 1, we observe that the estimated value of α generally increases with increasing α for fixed β for all classical methods of estimation. It is quite clear from Table 2 that as sample size increases, the MSEs decrease for all methods of estimation. It verifies the consistency of all methods of estimation. In terms of performance of the methods of estimation, it is observed that among all the considered classical methods of estimation, MLE has the minimum MSE for most of the configurations considered in our studies and performs better than their counterparts for all cases and the order of the best method of estimation is MLE<WLSE<MPSE<CME<LSE<RTADE<ADE in terms of MSEs. To assess the performance of BCIs, we report the estimated average widths, coverage probabilities, and relative coverages of BCIs of the indices Spmk and Spmkc for PLD using MLE, LSE, WLSE, CME, MPSE, ADE, and RTADE, respectively, in Tables 3–9. The comparisons are based on lower AWs, higher CPs, and higher RCs. The calculated coverage probabilities are compared with the nominal value of 95%. The simulation results show that the BCp-boot confidence interval provide smaller AW and higher RC than P-boot and S-boot for almost all sample sizes and for all the considered methods of estimation for both the indices Spmk and Spmkc, and the order of the best method of BCI in terms of RC is BCp<P<S. Therefore, we can conclude that the performance of the BCp-boot method is better as compared to the other two methods. Besides, it is observed that when α=7.5 and β=0.25, Spmk=1.00321 which indicates that the process is capable, whereas when the value of α=7.5 and β=0.50, the corresponding value of Spmk=0.82754 which indicates that the process is incapable (not shown in the simulation study). Furthermore, AWs and CPs of BCIs of Spmkc are smaller than Spmk for all the configurations considered in our simulation study.Table 1
Classical estimates of the parametersα and β using different methods.
Parameters
n
Estimates
MLEs
LSEs
WLSEs
CMEs
MPSEs
ADEs
RTADEs
10
α
^
1.76734
1.65161
1.63494
1.68981
1.62149
1.69094
1.70079
β
^
0.65521
0.63513
0.65418
0.67247
0.62064
0.66606
0.65932
α = 1.50
20
α
^
1.72966
1.63456
1.59965
1.64055
1.56868
1.63674
1.63584
β
^
0.63231
0.57784
0.58071
0.62074
0.58616
0.61944
0.61191
β = 0.50
30
α
^
1.64579
1.56177
1.54958
1.57218
1.53307
1.57661
1.59127
β
^
0.57224
0.54852
0.54651
0.56965
0.54727
0.56964
0.56049
50
α
^
1.53862
1.52989
1.51685
1.52746
1.21246
1.53517
1.54239
β
^
0.52083
0.51634
0.52589
0.54450
0.51544
0.53142
0.54413
10
α
^
2.51384
2.48984
2.43719
2.77994
2.42864
2.42864
2.67545
β
^
0.62635
0.68777
0.71527
0.61097
0.72919
0.65469
0.69694
α = 2.00
20
α
^
2.44839
2.36855
2.31785
2.57792
2.37323
2.29830
2.55131
β
^
0.56407
0.56846
0.66575
0.53423
0.67279
0.56507
0.60958
β = 0.50
30
α
^
2.35788
2.29573
2.26426
2.47782
2.30075
2.23093
2.42243
β
^
0.53185
0.54211
0.55079
0.52473
0.57378
0.53497
0.53931
50
α
^
2.18335
2.22406
2.15245
2.27591
2.17762
2.19035
2.24006
β
^
0.52274
0.51361
0.51088
0.50192
0.52799
0.50251
0.52560
10
α
^
2.71771
2.67307
2.66022
2.79007
2.82722
2.82416
2.83159
β
^
0.92929
0.83002
0.82454
0.87826
0.81764
0.87955
0.84966
α = 2.50
20
α
^
2.64297
2.59323
2.64268
2.74312
2.74263
2.74329
2.74326
β
^
0.89968
0.78031
0.79258
0.79511
0.78386
0.81821
0.82088
β = 0.75
30
α
^
2.60260
2.56903
2.61764
2.68423
2.68474
2.67252
2.61235
β
^
0.82096
0.76587
0.76992
0.76217
0.77752
0.76915
0.76824
50
α
^
2.53792
2.49415
2.54408
2.54941
2.56324
2.51164
2.55552
β
^
0.78838
0.74181
0.74455
0.73030
0.74173
0.73849
0.73005
10
α
^
7.75678
7.81757
7.86120
7.74842
7.69410
7.77461
7.81194
β
^
0.31787
0.35308
0.38158
0.28888
0.32420
0.32665
0.31735
α = 7.50
20
α
^
7.66016
7.73530
7.73144
7.69019
7.64883
7.64127
7.69165
β
^
0.28845
0.30230
0.33050
0.27387
0.27150
0.27394
0.27054
β = 0.25
30
α
^
7.53170
7.61765
7.69628
7.65637
7.54035
7.54510
7.66296
β
^
0.27608
0.32751
0.30899
0.26808
0.26692
0.26817
0.26396
50
α
^
7.52985
7.56082
7.51582
7.54020
7.47283
7.53227
7.52185
β
^
0.26037
0.26529
0.27228
0.26341
0.24049
0.25071
0.24701Table 2
Classical estimates and corresponding MSEs of PCIsSpmk and Spmkc.
True PCIs
n
MLE
LSE
WLSE
CME
MPSE
ADE
RTADE
Parameters
(MSE)
(MSE)
(MSE)
(MSE)
(MSE)
(MSE)
(MSE)
S
pmk = 0.60004
10
0.63664
0.63661
0.63554
0.63431
0.63341
0.63667
0.63668
(0.02140)
(0.02631)
(0.02442)
(0.02533)
(0.02446)
(0.02644)
(0.02643)
α = 1.50
20
0.61375
0.62664
0.62665
0.62639
0.62589
0.62664
0.62666
(0.00837)
(0.00998)
(0.00936)
(0.00989)
(0.00942)
(0.01012)
(0.00999)
β = 0.50
30
0.60966
0.62389
0.62113
0.62312
0.62087
0.62386
0.62389
(0.00513)
(0.00796)
(0.00667)
(0.00786)
(0.00669)
(0.00808)
(0.00799)
50
0.60364
0.61299
0.61106
0.61287
0.61087
0.61296
0.61302
(0.00274)
(0.00348)
(0.00332)
(0.00343)
(0.00338)
(0.00355)
(0.00354)
S
pmkc = 0.13973
10
0.14756
0.14760
0.14666
0.14758
0.14654
0.14771
0.14763
(0.01353)
(0.01369)
(0.01359)
(0.01367)
(0.01364)
(0.01373)
(0.01371)
α = 1.50
20
0.14166
0.14199
0.14097
0.14169
0.14096
0.14289
0.14267
(0.01003)
(0.01021)
(0.01009)
(0.01016)
(0.01013)
(0.01014)
(0.01008)
β = 0.50
30
0.13999
0.14108
0.13989
0.14022
0.13984
0.14256
0.14232
(0.00857)
(0.00901)
(0.00863)
(0.00871)
(0.00867)
(0.00892)
(0.00887)
50
0.13974
0.13999
0.13973
0.13989
0.13969
0.14171
0.14154
(0.00553)
(0.00591)
(0.00559)
(0.00564)
(0.00562)
(0.00617)
(0.00605)
S
pmk = 0.65709
10
0.69575
0.69611
0.69274
0.69589
0.69272
0.69656
0.69651
(0.02478)
(0.02487)
(0.02481)
(0.02483)
(0.02483)
(0.02493)
(0.02488)
α = 2.00
20
0.67389
0.67543
0.67329
0.67511
0.67323
0.67589
0.67583
(0.01048)
(0.01089)
(0.01069)
(0.01071)
(0.01069)
(0.01113)
(0.01097)
β = 0.50
30
0.66514
0.66638
0.66501
0.66611
0.66501
0.66654
0.66644
(0.00656)
(0.00777)
(0.00762)
(0.00767)
(0.00762)
(0.00789)
(0.00784)
50
0.66307
0.66454
0.65996
0.66437
0.65998
0.66489
0.66478
(0.00364)
(0.00429)
(0.00421)
(0.00424)
(0.00422)
(0.00438)
(0.00433)
S
pmkc = 0.15609
10
0.15729
0.15736
0.15723
0.15733
0.15717
0.15739
0.15737
(0.01449)
(0.01467)
(0.01452)
(0.01462)
(0.01461)
(0.01471)
(0.01471)
α = 2.00
20
0.15704
0.15724
0.15711
0.15719
0.15713
0.15723
0.15721
(0.01237)
(0.01261)
(0.01244)
(0.01255)
(0.01253)
(0.01264)
(0.01262)
β = 0.50
30
0.15653
0.15672
0.15669
0.15668
0.15668
0.15674
0.15671
(0.00988)
0.01022
(0.00993)
(0.01017)
(0.01007)
(0.0128)
(0.01025)
50
0.15615
0.15629
0.15626
0.15626
0.15621
0.15629
0.15628
(0.00667)
(0.00861)
(0.00669)
(0.0852)
(0.00848)
(0.00877)
(0.00876)
S
pmk = 0.52232
10
0.55430
0.55444
0.55429
0.55439
0.55423
0.55432
0.55431
(0.02202)
(0.02213)
(0.02204)
(0.02211)
(0.02209)
(0.02214)
(0.02214)
α = 2.50
20
0.54297
0.54323
0.54268
0.54312
0.54263
0.54329
0.54326
(0.01403)
(0.01422)
(0.01414)
(0.01419)
(0.01417)
(0.01409)
(0.01407)
β = 0.75
30
0.54129
0.54203
0.54101
0.54137
0.54097
0.54222
0.54211
(0.00861)
(0.00881)
(0.00867)
(0.00872)
(0.00869)
(0.00871)
(0.00867)
50
0.53866
0.54012
0.53856
0.53897
0.53888
0.54037
0.54025
(0.00467)
(0.00484)
(0.00473)
(0.00479)
(0.00477)
(0.00488)
(0.00485)
S
pmkc = 0.14535
10
0.14798
0.14816
0.14781
0.14813
0.14776
0.14822
0.14821
(0.01379)
(0.01434)
(0.01382)
(0.01388)
(0.01384)
(0.01489)
(0.01489)
α = 2.50
20
0.14622
0.14642
0.14617
0.14633
0.14614
0.14623
0.14623
(0.01273)
(0.01322)
(0.01277)
(0.01286)
(0.01277)
(0.01393)
(0.01392)
β = 0.75
30
0.14579
0.14589
0.14571
0.14583
0.14568
0.14593
0.14590
(0.01006)
(0.01106)
(0.01013)
(0.01023)
(0.01014)
(0.01123)
(0.01119)
50
0.14538
0.14551
0.14536
0.14541
0.14527
0.14541
0.14538
(0.00897)
(0.01013)
(0.00907)
(0.00923)
(0.00909)
(0.01024)
(0.01023)
S
pmk = 1.00322
10
1.01354
1.01362
1.01352
1.01358
1.01348
1.01355
1.01354
(0.03678)
(0.03728)
(0.03684)
(0.03688)
(0.03685)
(0.03754)
(0.03756)
α = 7.50
20
1.01322
1.01333
1.01329
1.01336
1.01332
1.01334
1.01337
(0.02424)
(0.02488)
(0.02430)
(0.02434)
(0.02432)
(0.02493)
(0.02493)
β = 0.25
30
1.00778
1.00781
1.00780
1.00792
1.00784
1.00785
1.00785
(0.00864)
(0.00883)
(0.00866)
(0.00871)
(0.00866)
(0.00888)
(0.00889)
50
1.00522
1.00527
1.00524
1.00526
1.00525
1.00528
1.00529
(0.00524)
(0.00554)
(0.00527)
(0.00528)
(0.00528)
(0.00558)
(0.00559)
S
pmkc = 0.27863
10
0.28355
0.28411
0.28377
0.28409
0.28376
0.28416
0.28417
(0.02446)
(0.02502)
(0.02449)
(0.02451)
(0.02449)
(0.02508)
(0.02509)
α = 7.50
20
0.28322
0.28404
0.28373
0.28402
0.28372
0.28411
0.28413
(0.02211)
(0.02221)
(0.02213)
(0.02214)
(0.02214)
(0.02226)
(0.02228)
β = 0.25
30
0.28114
0.28136
0.28121
0.28124
0.28117
0.28124
0.28128
(0.01878)
(0.01893)
(0.01879)
(0.01883)
(0.01882)
(0.01896)
(0.01896)
50
0.27889
0.27922
0.28113
0.27930
0.27892
0.28114
0.28115
(0.01322)
(0.01338)
(0.01324)
(0.01327)
(0.01324)
(0.01339)
(0.01339)Table 3
BCIs (AWs, CPs, and RCs) of PCIsSpmk and Spmkc using MLEs of α and β.
PCIs
n
S-boot
P-boot
B
C
p-boot
Parameters
AWs
CPs
RCs
AWs
CPs
RCs
AWs
CPs
RCs
S
pmk = 0.60004
10
0.41186
0.837
2.03224
0.39172
0.837
2.13417
0.39169
0.836
2.13434
α = 1.50
20
0.38221
0.889
2.32594
0.35226
0.888
2.52086
0.35221
0.889
2.52406
β = 0.50
30
0.35267
0.911
2.58315
0.33249
0.910
2.73692
0.33238
0.911
2.74083
50
0.33355
0.926
2.77619
0.32313
0.926
2.86571
0.31394
0.927
2.95279
S
pmkc = 0.13973
10
0.14273
0.842
5.89925
0.12217
0.841
6.88385
0.12211
0.841
6.88723
α = 1.50
20
0.12399
0.891
7.18606
0.10316
0.890
8.62737
0.10311
0.891
8.64125
β = 0.50
30
0.11256
0.912
8.10234
0.09244
0.911
9.85504
0.09223
0.912
9.88832
50
0.09383
0.928
9.89022
0.07328
0.927
12.65010
0.07308
0.926
12.67104
S
pmk = 0.65709
10
0.42786
0.837
1.95624
0.40992
0.837
2.04186
0.40893
0.837
2.04680
α = 2.00
20
0.38836
0.889
2.28911
0.35689
0.888
2.48816
0.35672
0.889
2.49215
β = 0.50
30
0.36112
0.911
2.52270
0.32245
0.910
2.82214
0.32213
0.911
2.82805
50
0.32778
0.926
2.82506
0.30898
0.926
2.99695
0.30893
0.927
3.00067
S
pmkc = 0.15609
10
0.14522
0.842
5.79809
0.12681
0.841
6.63196
0.12674
0.841
6.63563
α = 2.00
20
0.12468
0.891
7.14629
0.10472
0.890
8.49885
0.10461
0.891
8.51735
β = 0.50
30
0.11883
0.912
7.67482
0.09622
0.911
9.46788
0.09614
0.912
9.48616
50
0.09947
0.928
9.32944
0.07570
0.927
12.24570
0.07558
0.926
12.25191
S
pmk = 0.52232
10
0.38283
0.837
2.18634
0.36780
0.837
2.27569
0.36773
0.837
2.27612
α = 2.50
20
0.35362
0.889
2.51399
0.33237
0.888
2.67172
0.33223
0.889
2.67585
β = 0.75
30
0.32316
0.911
2.81903
0.30873
0.910
2.94755
0.30864
0.911
2.95165
50
0.30891
0.926
2.99763
0.28573
0.926
3.24082
0.28568
0.927
3.24488
S
pmkc = 0.14535
10
0.12332
0.842
6.82776
0.10659
0.841
7.89004
0.10652
0.841
7.89523
α = 2.50
20
0.10897
0.891
8.17656
0.09230
0.890
9.64247
0.09223
0.891
9.66063
β = 0.75
30
0.08974
0.912
10.16269
0.07461
0.911
12.21015
0.07452
0.912
12.23832
50
0.06874
0.927
13.48559
0.05581
0.927
16.60992
0.05572
0.926
16.61880
S
pmk = 1.00322
10
0.66274
0.838
1.26445
0.64568
0.837
1.29631
0.64562
0.837
1.29642
α = 7.50
20
0.61472
0.890
1.44781
0.58994
0.888
1.50523
0.58988
0.889
1.50708
β = 0.25
30
0.56332
0.912
1.61897
0.54297
0.911
1.67780
0.54291
0.911
1.68078
50
0.52221
0.926
1.77323
0.50442
0.925
1.83378
0.50437
0.926
1.83595
S
pmkc = 0.27863
10
0.21782
0.841
3.86098
0.19436
0.840
4.32187
0.19433
0.841
4.32769
α = 7.50
20
0.17894
0.892
4.98491
0.15447
0.891
5.76811
0.15442
0.891
5.76997
β = 0.25
30
0.14337
0.911
6.35418
0.12668
0.910
7.18345
0.12664
0.910
7.18572
50
0.11645
0.926
7.95191
0.08896
0.925
10.39793
0.08895
0.926
10.41034Table 4
BCIs (AWs, CPs, and RCs) of PCIsSpmk and Spmkc using LSEs of α and β.
PCIs
n
S-boot
P-boot
B
C
p-boot
Parameters
AWs
CPs
RCs
AWs
CPs
RCs
AWs
CPs
RCs
S
pmk = 0.60004
10
0.41211
0.834
2.02373
0.39177
0.836
2.13390
0.39173
0.835
2.13157
α = 1.50
20
0.38322
0.887
2.31459
0.35232
0.888
2.52043
0.35224
0.888
2.52100
β = 0.50
30
0.35312
0.909
2.57419
0.33255
0.911
2.73943
0.33243
0.910
2.73741
50
0.33407
0.925
2.76888
0.32317
0.926
2.86536
0.31398
0.926
2.94923
S
pmkc = 0.13973
10
0.14321
0.840
5.86551
0.12221
0.841
6.88159
0.12214
0.840
6.87735
α = 1.50
20
0.12467
0.890
7.13884
0.10322
0.890
8.62236
0.10316
0.890
8.62737
β = 0.50
30
0.11311
0.912
8.06294
0.09246
0.912
9.86372
0.09227
0.911
9.87319
50
0.09409
0.925
9.83101
0.07333
0.926
12.62784
0.07315
0.926
12.65892
S
pmk = 0.65709
10
0.42813
0.835
1.95034
0.40999
0.837
2.04151
0.40896
0.836
2.04420
α = 2.00
20
0.38922
0.889
2.28405
0.35692
0.890
2.49355
0.35677
0.889
2.49180
β = 0.50
30
0.36248
0.910
2.51048
0.32248
0.910
2.82188
0.32217
0.910
2.82459
50
0.32825
0.926
2.82102
0.30904
0.927
2.99961
0.30895
0.928
3.00372
S
pmkc = 0.15609
10
0.14589
0.840
5.75776
0.12684
0.841
6.63040
0.12677
0.840
6.62617
α = 2.00
20
0.12498
0.889
7.11313
0.10478
0.890
8.49398
0.10463
0.891
8.51572
β = 0.50
30
0.11941
0.910
7.62080
0.09626
0.911
9.46395
0.09618
0.911
9.47182
50
0.10127
0.926
9.14387
0.07576
0.927
12.23600
0.07563
0.927
12.25704
S
pmk = 0.52232
10
0.38334
0.835
2.17822
0.36783
0.837
2.27550
0.36777
0.836
2.27315
α = 2.50
20
0.35387
0.889
2.51222
0.33244
0.888
2.67115
0.33228
0.889
2.67545
β = 0.75
30
0.32379
0.909
2.80737
0.30879
0.910
2.94698
0.30866
0.910
2.94822
50
0.30957
0.925
2.98801
0.28579
0.926
3.24014
0.28574
0.927
3.24420
S
pmkc = 0.14535
10
0.12389
0.839
6.77213
0.10664
0.841
7.88634
0.10657
0.840
7.88214
α = 2.50
20
0.10944
0.890
8.13230
0.09236
0.890
9.63620
0.09228
0.890
9.64456
β = 0.75
30
0.09144
0.910
9.95188
0.07467
0.912
12.21374
0.07458
0.911
12.21507
50
0.06939
0.926
13.3448
0.05585
0.927
16.59803
0.05574
0.928
16.64872
S
pmk = 1.00322
10
0.66283
0.840
1.26729
0.64575
0.838
1.29771
0.64571
0.839
1.29934
α = 7.50
20
0.61477
0.891
1.44932
0.58999
0.890
1.50850
0.58994
0.890
1.50862
β = 0.25
30
0.56341
0.912
1.61871
0.54306
0.911
1.67531
0.54304
0.911
1.67759
50
0.52228
0.929
1.77873
0.50448
0.927
1.83753
0.50444
0.927
1.83768
S
pmkc = 0.27863
10
0.21791
0.842
3.86398
0.19444
0.841
4.32524
0.19439
0.842
4.33149
α = 7.50
20
0.17899
0.893
4.98910
0.15453
0.892
5.77234
0.15450
0.892
5.77436
β = 0.25
30
0.14340
0.912
6.35983
0.12674
0.911
7.18794
0.12671
0.911
7.18964
50
0.11653
0.927
7.95503
0.08908
0.926
10.39515
0.08906
0.927
10.40871Table 5
BCIs (AWs, CPs, and RCs) of PCIsSpmk and Spmkc using WLSEs of α and β.
PCIs
n
S-boot
P-boot
B
C
p-boot
Parameters
AWs
CPs
RCs
AWs
CPs
RCs
AWs
CPs
RCs
S
pmk = 0.60004
10
0.41784
0.836
2.00076
0.39610
0.837
2.11310
0.39599
0.837
2.11368
α = 1.50
20
0.38759
0.889
2.29366
0.35457
0.889
2.50726
0.35446
0.889
2.50804
β = 0.50
30
0.35688
0.912
2.55548
0.33454
0.912
2.72613
0.33445
0.913
2.72985
50
0.33884
0.928
2.73875
0.32499
0.930
2.86162
0.31838
0.929
2.91789
S
pmkc = 0.13973
10
0.14824
0.842
5.67997
0.12779
0.843
6.59676
0.12774
0.842
6.59151
α = 1.50
20
0.12722
0.891
7.00361
0.10689
0.892
8.34502
0.10683
0.893
8.35907
β = 0.50
30
0.11847
0.914
7.71503
0.09589
0.914
9.53175
0.09578
0.914
9.54270
50
0.09455
0.928
9.81491
0.07898
0.929
11.76247
0.07884
0.929
11.78335
S
pmk = 0.65709
10
0.42884
0.837
1.95177
0.40910
0.837
2.04595
0.40898
0.838
2.04899
α = 2.00
20
0.38991
0.888
2.27744
0.35692
0.888
2.48795
0.35679
0.889
2.49166
β = 0.50
30
0.36889
0.911
2.46957
0.32474
0.911
2.80532
0.32467
0.912
2.80900
50
0.32888
0.927
2.81865
0.30956
0.927
2.99457
0.30924
0.928
3.00090
S
pmkc = 0.15609
10
0.14667
0.842
5.74077
0.12692
0.844
6.64985
0.12687
0.843
6.64459
α = 2.00
20
0.12534
0.890
7.10068
0.10474
0.890
8.49723
0.10468
0.891
8.51165
β = 0.50
30
0.11993
0.913
7.61277
0.09635
0.915
9.49662
0.09627
0.914
9.49413
50
0.10462
0.928
8.87019
0.07584
0.929
12.24947
0.07577
0.930
12.27398
S
pmk = 0.52232
10
0.38379
0.838
2.18348
0.36788
0.839
2.28063
0.36781
0.838
2.27835
α = 2.50
20
0.35451
0.889
2.50768
0.33246
0.891
2.68002
0.33237
0.891
2.68074
β = 0.75
30
0.32511
0.911
2.80218
0.30887
0.913
2.95593
0.30872
0.912
2.95413
50
0.30982
0.927
2.99205
0.28590
0.927
3.24239
0.28582
0.928
3.24679
S
pmkc = 0.14535
10
0.12478
0.841
6.73986
0.10669
0.841
7.88265
0.10658
0.842
7.90016
α = 2.50
20
0.10993
0.890
8.09606
0.09244
0.891
9.63868
0.09237
0.891
9.64598
β = 0.75
30
0.09251
0.913
9.86920
0.07464
0.915
12.25884
0.07455
0.914
12.26022
50
0.06911
0.928
13.42786
0.05589
0.929
16.62193
0.05577
0.929
16.65770
S
pmk = 1.00322
10
0.66281
0.840
1.26733
0.64572
0.838
1.29777
0.64569
0.838
1.29729
α = 7.50
20
0.61475
0.891
1.44936
0.58996
0.889
1.50688
0.58992
0.890
1.50867
β = 0.25
30
0.56337
0.912
1.61882
0.54301
0.910
1.67584
0.54299
0.910
1.67590
50
0.52224
0.928
1.77696
0.50444
0.926
1.83569
0.50439
0.927
1.83786
S
pmkc = 0.27863
10
0.21787
0.842
3.86468
0.19439
0.841
4.32635
0.19436
0.841
4.32702
α = 7.50
20
0.17898
0.893
4.98938
0.15449
0.891
5.76736
0.15447
0.892
5.77458
β = 0.25
30
0.14336
0.912
6.36160
0.12670
0.910
7.18232
0.12669
0.910
7.18288
50
0.11648
0.926
7.94986
0.08901
0.925
10.39209
0.08899
0.926
10.40566Table 6
BCIs (AWs, CPs, and RCs) of PCIsSpmk and Spmkc using CMEs of α and β.
PCIs
n
S-boot
P-boot
B
C
p-boot
Parameters
AWs
CPs
RCs
AWs
CPs
RCs
AWs
CPs
RCs
S
pmk = 0.60004
10
0.41832
0.834
1.99368
0.39613
0.837
2.11294
0.39602
0.836
2.11100
α = 1.50
20
0.38803
0.889
2.29105
0.35460
0.889
2.50705
0.35448
0.889
2.50789
β = 0.50
30
0.35733
0.911
2.54946
0.33458
0.913
2.72879
0.33447
0.914
2.73268
50
0.33929
0.927
2.73217
0.32504
0.928
2.85503
0.31841
0.929
2.91762
S
pmkc = 0.13973
10
0.14876
0.840
5.64667
0.12784
0.841
6.57853
0.12777
0.841
6.58213
α = 1.50
20
0.12769
0.890
6.97000
0.10692
0.892
8.34268
0.10685
0.892
8.34815
β = 0.50
30
0.11903
0.910
7.64513
0.09592
0.911
9.49749
0.09581
0.913
9.52927
50
0.09498
0.928
9.77047
0.07902
0.928
11.74386
0.07887
0.929
11.77887
S
pmk = 0.65709
10
0.42930
0.837
1.94968
0.40912
0.837
2.04585
0.40901
0.838
2.04884
α = 2.00
20
0.39034
0.888
2.27493
0.35694
0.888
2.48781
0.35681
0.889
2.49152
β = 0.50
30
0.36924
0.911
2.46722
0.32477
0.911
2.80506
0.32469
0.912
2.80883
50
0.32941
0.927
2.81412
0.30959
0.927
2.99428
0.30927
0.928
3.00061
S
pmkc = 0.15609
10
0.14712
0.841
5.71642
0.12695
0.842
6.63253
0.12689
0.842
6.63566
α = 2.00
20
0.12583
0.888
7.05714
0.10475
0.890
8.49642
0.10469
0.890
8.50128
β = 0.50
30
0.12133
0.911
7.50844
0.09639
0.913
9.47193
0.09629
0.913
9.48177
50
0.10509
0.927
8.82101
0.07585
0.928
12.23467
0.07579
0.929
12.25755
S
pmk = 0.52232
10
0.38471
0.836
2.17306
0.36793
0.837
2.27488
0.36786
0.837
2.27532
α = 2.50
20
0.35541
0.888
2.49852
0.33251
0.890
2.67661
0.33242
0.890
2.67733
β = 0.75
30
0.32611
0.912
2.79660
0.30893
0.914
2.95859
0.30877
0.913
2.95689
50
0.31162
0.927
2.97477
0.28595
0.929
3.24881
0.28587
0.929
3.24972
S
pmkc = 0.14535
10
0.12522
0.840
6.70819
0.10673
0.840
7.87032
0.10661
0.841
7.88856
α = 2.50
20
0.11134
0.889
7.98455
0.09248
0.891
9.63451
0.09239
0.891
9.64390
β = 0.75
30
0.09306
0.912
9.80012
0.07467
0.914
12.24052
0.07458
0.913
12.24188
50
0.06964
0.928
13.32567
0.05592
0.929
16.61659
0.05579
0.929
16.65172
S
pmk = 1.00322
10
0.66290
0.839
1.26565
0.64569
0.838
1.29783
0.64564
0.838
1.29793
α = 7.50
20
0.61471
0.891
1.44946
0.58993
0.889
1.50695
0.58989
0.890
1.50875
β = 0.25
30
0.56332
0.912
1.61897
0.54298
0.911
1.67777
0.54291
0.911
1.67799
50
0.52223
0.927
1.77507
0.50445
0.926
1.83566
0.50443
0.926
1.83573
S
pmkc = 0.27863
10
0.21787
0.842
3.86468
0.19439
0.841
4.32635
0.19438
0.841
4.32657
α = 7.50
20
0.17893
0.892
4.98518
0.15452
0.891
5.76624
0.15447
0.892
5.77458
β = 0.25
30
0.14339
0.912
6.36027
0.12671
0.911
7.18964
0.12668
0.911
7.19134
50
0.11649
0.927
7.95776
0.08899
0.925
10.39442
0.08896
0.926
10.40917Table 7
BCIs (AWs, CPs, and RCs) of PCIsSpmk and Spmkc using MPSEs of α and β.
PCIs
n
S-boot
P-boot
B
C
p-boot
Parameters
AWs
CPs
RCs
AWs
CPs
RCs
AWs
CPs
RCs
S
pmk = 0.60004
10
0.41788
0.837
2.00296
0.39612
0.839
2.11804
0.39602
0.838
2.11605
α = 1.50
20
0.38771
0.890
2.29553
0.35462
0.890
2.50972
0.35450
0.890
2.51057
β = 0.50
30
0.35693
0.914
2.56072
0.33459
0.915
2.73469
0.33454
0.916
2.73808
50
0.33895
0.929
2.74081
0.32502
0.930
2.86136
0.31844
0.931
2.92362
S
pmkc = 0.13973
10
0.14833
0.840
5.66304
0.12784
0.840
6.57071
0.12777
0.841
6.58213
α = 1.50
20
0.12730
0.890
6.99135
0.10693
0.892
8.34190
0.10686
0.892
8.34737
β = 0.50
30
0.11855
0.913
7.70139
0.09594
0.913
9.51636
0.09582
0.914
9.53871
50
0.09458
0.927
9.80122
0.07899
0.928
11.74832
0.07885
0.928
11.76918
S
pmk = 0.65709
10
0.42888
0.837
1.95159
0.40913
0.838
2.04824
0.40899
0.839
2.05139
α = 2.00
20
0.38994
0.888
2.27727
0.35693
0.890
2.49348
0.35682
0.889
2.49145
β = 0.50
30
0.36893
0.912
2.47201
0.32477
0.915
2.81737
0.32469
0.914
2.81499
50
0.32892
0.928
2.82135
0.30959
0.930
3.00397
0.30927
0.930
3.00708
S
pmkc = 0.15609
10
0.14672
0.842
5.73882
0.12696
0.844
6.64776
0.12689
0.843
6.64354
α = 2.00
20
0.12538
0.890
7.09842
0.10479
0.893
8.52180
0.10472
0.892
8.51795
β = 0.50
30
0.11998
0.913
7.60960
0.09638
0.915
9.49367
0.09632
0.915
9.49958
50
0.10468
0.928
8.86511
0.07586
0.930
12.25942
0.07581
0.931
12.28070
S
pmk = 0.52232
10
0.38385
0.838
2.18314
0.36793
0.839
2.28032
0.36784
0.838
2.27816
α = 2.50
20
0.35462
0.890
2.50972
0.33253
0.891
2.67945
0.33241
0.891
2.68042
β = 0.75
30
0.32524
0.912
2.80483
0.30893
0.913
2.95536
0.30883
0.913
2.95631
50
0.30988
0.927
2.99148
0.28599
0.929
3.24836
0.28587
0.929
3.24972
S
pmkc = 0.14535
10
0.12483
0.842
6.74517
0.10672
0.842
7.88980
0.10661
0.843
7.90732
α = 2.50
20
0.10997
0.890
8.09311
0.09249
0.891
9.63347
0.09240
0.891
9.64285
β = 0.75
30
0.09256
0.912
9.85306
0.07468
0.914
12.23885
0.07458
0.913
12.24188
50
0.06917
0.928
13.41622
0.05593
0.929
16.61004
0.05582
0.930
16.66069
S
pmk = 1.00322
10
0.66293
0.839
1.26559
0.64572
0.837
1.29622
0.64564
0.838
1.29736
α = 7.50
20
0.61477
0.891
1.44932
0.58996
0.889
1.50688
0.58992
0.890
1.50867
β = 0.25
30
0.56337
0.912
1.61882
0.54302
0.911
1.67765
0.54297
0.912
1.67965
50
0.52224
0.926
1.77313
0.50444
0.925
1.83371
0.50440
0.925
1.83386
S
pmkc = 0.27863
10
0.21785
0.842
3.86504
0.19438
0.841
4.32657
0.19437
0.841
4.32679
α = 7.50
20
0.17898
0.892
4.98379
0.15452
0.891
5.76624
0.15449
0.892
5.77383
β = 0.25
30
0.14344
0.912
6.35805
0.12671
0.911
7.18964
0.12668
0.911
7.19134
50
0.11651
0.926
7.94781
0.08899
0.925
10.39442
0.08896
0.926
10.40917Table 8
BCIs (AWs, CPs, and RCs) of PCIsSpmk and Spmkc using ADEs of α and β.
PCIs
n
S-boot
P-boot
B
C
p-boot
Parameters
AWs
CPs
RCs
AWs
CPs
RCs
AWs
CPs
RCs
S
pmk = 0.60004
10
0.41851
0.837
1.99995
0.39615
0.839
2.11788
0.39606
0.839
2.11836
α = 1.50
20
0.38813
0.889
2.29046
0.35466
0.890
2.50944
0.35454
0.890
2.51029
β = 0.50
30
0.35744
0.914
2.55707
0.33463
0.914
2.73137
0.33457
0.915
2.73485
50
0.33941
0.928
2.73415
0.32507
0.929
2.85784
0.31848
0.930
2.92012
S
pmkc = 0.13973
10
0.14884
0.838
5.63020
0.12787
0.839
6.56135
0.12782
0.840
6.57174
α = 1.50
20
0.12779
0.888
6.94890
0.10698
0.889
8.30996
0.10689
0.890
8.32631
β = 0.50
30
0.11908
0.911
7.65031
0.09599
0.912
9.50098
0.09584
0.913
9.52629
50
0.09504
0.927
9.75378
0.07904
0.927
11.72823
0.07888
0.928
11.76470
S
pmk = 0.65709
10
0.42936
0.836
1.94708
0.40917
0.839
2.05049
0.40903
0.840
2.05363
α = 2.00
20
0.39134
0.888
2.26912
0.35698
0.890
2.49313
0.35685
0.890
2.49404
β = 0.50
30
0.37138
0.911
2.45301
0.32483
0.913
2.81070
0.32471
0.913
2.81173
50
0.33145
0.927
2.79680
0.30964
0.928
2.99702
0.30933
0.929
3.00326
S
pmkc = 0.15609
10
0.14722
0.840
5.70574
0.12699
0.842
6.63044
0.12691
0.842
6.63462
α = 2.00
20
0.12589
0.889
7.06172
0.10484
0.891
8.49866
0.10475
0.891
8.50596
β = 0.50
30
0.12136
0.911
7.50659
0.09641
0.913
9.46997
0.09634
0.914
9.48723
50
0.10511
0.928
8.82884
0.07589
0.929
12.24140
0.07584
0.930
12.26265
S
pmk = 0.52232
10
0.38536
0.837
2.17199
0.36797
0.839
2.28007
0.36787
0.839
2.28069
α = 2.50
20
0.35521
0.888
2.49992
0.33258
0.890
2.67604
0.33244
0.890
2.67717
β = 0.75
30
0.32577
0.912
2.79952
0.30897
0.914
2.95821
0.30886
0.914
2.95926
50
0.31133
0.928
2.98075
0.28604
0.929
3.24779
0.28589
0.930
3.25299
S
pmkc = 0.14535
10
0.12533
0.841
6.71028
0.10676
0.842
7.88684
0.10664
0.842
7.89572
α = 2.50
20
0.11146
0.889
7.97595
0.09254
0.891
9.62826
0.09244
0.891
9.63868
β = 0.75
30
0.09308
0.912
9.79802
0.07473
0.915
12.24407
0.07462
0.915
12.26212
50
0.06961
0.928
13.33141
0.05596
0.929
16.60114
0.05584
0.930
16.65472
S
pmk = 1.00322
10
0.66291
0.838
1.26412
0.64569
0.837
1.29628
0.64566
0.837
1.29634
α = 7.50
20
0.61474
0.892
1.45101
0.58994
0.890
1.50862
0.58992
0.890
1.50867
β = 0.25
30
0.56336
0.912
1.61885
0.54300
0.911
1.67771
0.54298
0.911
1.67777
50
0.52222
0.926
1.77319
0.50441
0.925
1.83382
0.50440
0.926
1.83584
S
pmkc = 0.27863
10
0.21783
0.842
3.86539
0.19436
0.841
4.32702
0.19434
0.841
4.32746
α = 7.50
20
0.17896
0.892
4.98435
0.15451
0.891
5.76661
0.15450
0.891
5.76699
β = 0.25
30
0.14342
0.912
6.35894
0.12669
0.911
7.19078
0.12666
0.911
7.19248
50
0.11651
0.927
7.95639
0.08898
0.925
10.39559
0.08897
0.925
10.39910Table 9
BCIs (AWs, CPs, and RCs) of PCIsSpmk and Spmkc using RTADEs of α and β.
PCIs
n
S-boot
P-boot
B
C
p-boot
Parameters
AWs
CPs
RCs
AWs
CPs
RCs
AWs
CPs
RCs
S
pmk = 0.60004
10
0.41882
0.837
1.99847
0.39619
0.840
2.12019
0.39609
0.840
2.12073
α = 1.50
20
0.38844
0.889
2.28864
0.35469
0.890
2.50923
0.35456
0.890
2.51015
β = 0.50
30
0.35776
0.912
2.54919
0.33466
0.915
2.73411
0.33460
0.914
2.73161
50
0.33973
0.928
2.73158
0.32511
0.929
2.85749
0.31850
0.930
2.91993
S
pmkc = 0.13973
10
0.14912
0.838
5.61963
0.12789
0.840
6.56814
0.12784
0.841
6.57853
α = 1.50
20
0.12811
0.888
6.93154
0.10702
0.889
8.30685
0.10692
0.890
8.32398
β = 0.50
30
0.11935
0.912
7.64139
0.09603
0.915
9.52827
0.09587
0.914
9.53374
50
0.09537
0.928
9.73052
0.07908
0.929
11.74759
0.07891
0.930
11.78557
S
pmk = 0.65709
10
0.42964
0.837
1.94814
0.40919
0.840
2.05283
0.40907
0.841
2.05588
α = 2.00
20
0.39167
0.889
2.26976
0.35702
0.890
2.49285
0.35688
0.890
2.49383
β = 0.50
30
0.37171
0.913
2.45621
0.32485
0.915
2.81668
0.32474
0.914
2.81455
50
0.33177
0.928
2.79711
0.30969
0.929
2.99977
0.30937
0.930
3.00610
S
pmkc = 0.15609
10
0.14755
0.841
5.69976
0.12704
0.842
6.62783
0.12693
0.842
6.63357
α = 2.00
20
0.12624
0.889
7.04214
0.10488
0.891
8.49542
0.10477
0.891
8.50434
β = 0.50
30
0.12173
0.913
7.50020
0.09644
0.915
9.48776
0.09636
0.915
9.49564
50
0.10537
0.928
8.80706
0.07590
0.930
12.25296
0.07586
0.930
12.25942
S
pmk = 0.52232
10
0.38561
0.838
2.17318
0.36799
0.840
2.28267
0.36789
0.840
2.28329
α = 2.50
20
0.35553
0.888
2.49767
0.33260
0.890
2.67588
0.33246
0.891
2.68002
β = 0.75
30
0.32593
0.914
2.80428
0.30901
0.915
2.96106
0.30889
0.916
2.96545
50
0.31150
0.929
2.98234
0.28608
0.929
3.24734
0.28592
0.930
3.25265
S
pmkc = 0.14535
10
0.12549
0.841
7.53111
0.10679
0.842
7.88463
0.10667
0.842
7.89350
α = 2.50
20
0.11167
0.889
7.96095
0.09257
0.891
9.62514
0.09246
0.891
9.63659
β = 0.75
30
0.09333
0.913
9.78249
0.07477
0.914
12.22415
0.07465
0.915
12.25720
50
0.06979
0.928
13.29703
0.05599
0.929
16.59224
0.05587
0.930
16.64578
S
pmk = 1.00322
10
0.66294
0.838
1.26406
0.64573
0.837
1.29620
0.64569
0.837
1.29628
α = 7.50
20
0.61478
0.892
1.45092
0.58997
0.890
1.50855
0.58994
0.890
1.50862
β = 0.25
30
0.56339
0.912
1.61877
0.54303
0.911
1.67762
0.54299
0.911
1.67774
50
0.52225
0.926
1.77309
0.50446
0.925
1.83364
0.50443
0.925
1.83375
S
pmkc = 0.27863
10
0.21786
0.842
3.86486
0.19439
0.841
4.32635
0.19438
0.841
4.32657
α = 7.50
20
0.17899
0.892
4.98351
0.15454
0.891
5.76549
0.15451
0.891
5.76661
β = 0.25
30
0.14347
0.912
6.35672
0.12674
0.911
7.18794
0.12671
0.911
7.18964
50
0.11653
0.927
7.95503
0.08898
0.925
10.39559
0.08896
0.925
10.39793


## 6. Applications

Here, we consider two different examples taken from the literature in order to assess the performance of the BCIs of the indicesSpmk and Spmkc:(i)
Data set I: the data set represents the ball size (in mm), and the process was monitored for this quality characteristic withLSL=0.50mm and USL=8.00mm, respectively, and are taken from [9]. Here, we assume that the target value T=4.0mm for the analysis purpose. The data are(43)2.891,4.035,4.495,2.890,2.312,3.158,5.228,3.334,5.896,5.639,3.842,1.590,1.842,0.680,2.752,1.301,2.260,0.889,2.381,0.619,2.788,1.050,3.750,3.508,6.549,5.954,2.207,4.417,4.805,1.516,2.227,2.797,1.636,1.066,0.940,4.101,1.295,1.770,3.492,5.706,3.722,6.644,2.472,1.383,4.494,1.694,2.892,2.111,2.093,3.222,2.891,2.582,0.665,3.234,1.102,1.083,1.508,1.811,2.803,6.659,6.229,3.177,2.333,1.311,4.419,2.495,0.921,4.061,9.725,1.600,4.281,3.360,1.618,4.489,3.696,1.982,2.413,5.480,1.992,2.573,1.845,4.620,6.221,1.694,1.380,3.982,2.260,2.366,2.899,3.782,2.336,1.175,3.055,1.954,6.123,4.542,4.882,1.131,0.923,3.591.(ii)
Data set II: the data set II represents the protein amount (in g) in the restricted diet for adult patients in Hospital Carlos Van Buren of the city of Valparaiso, Chile, and was already considered in [9]. In this data set, specification limits are given as LSL=30g and USL=96g, respectively. Here, we assume that the target value T=45.0g for the analysis purpose. The data are(44)66.09,146.48,97.99,56.77,74.23,76.20,108.90,89.51,62.40,141.15,68.22,80.44,89.81,201.05,210.32,64.25,39.43,43.87,59.56,49.97,34.35,51.28,100.43,42.37,164.05,59.63,49.21,51.41,93.32,102.91,124.91,56.16,70.94,77.26,68.62,41.69,59.01,59.78,37.67,88.27,35.60,42.54,169.82,105.08,102.16,67.55,17.76,83.73,44.41,25.55,41.12,163.50,123.85,100.43,88.14,54.69,85.32,34.07,139.39,61.95,60.03.Descriptive statistics, viz., minimum, first quartile (Q1), second quartile (Q2), third quartile (Q3), mean, standard deviation (SD), maximum, coefficient of skewness (CS), and coefficient of kurtosis (CK) of the considered data sets are displayed in Table 9. For each data set, parameters are estimated using the MLE technique. The Kolmogorov–Smirnov (K-S) goodness-of-fit test and the corresponding p value, negative log-likelihood values calculated at MLEs, Akaike information criterion (AIC), and Bayesian information criterion (BIC) for each data set are displayed in Table 10. From Table 10, we can conclude that the PLD is a proper model to fit the two data sets. Also, theoretical and empirical CDFs of the considered data sets I and II are displayed in Figure 1. Next, we estimate the indices Spmk and Spmkc for the two data sets using aforementioned different methods of estimation and are reported in Table 11. It is worth mentioning that, among the considered methods of estimation, MLE gives the best results for data set I, while the ADE is the best for data set II. We also observe from Table 11 that among all BCIs, width of the confidence interval of the BCp-boot method is smaller than their counterparts for both PCIs. Therefore, it can be concluded that the performance of the BCp-boot method is better as compared to other two bootstrap methods (Table 12).Table 10
Descriptive statistics of the considered data sets I and II.
Data
n
Minimum
Q
1
Q
2
Q
3
Maximum
Mean
SD
CS
CK
I
100
0.619
1.694
2.770
4.042
9.725
3.036
1.715
0.998
4.130
II
61
17.760
51.280
68.220
100.430
210.320
80.440
42.289
1.198
4.092Figure 1
CDF and ECDF of the considered data sets (a) I and (b) II.
(a)
(b)Table 11
Model fitting summary of the considered data sets I and II.
Data
MLE
Negative L-L
AIC
BIC
K-S statistic
p value
I
α
^ = 1.47757, β^ = 0.30883
185.73750
375.47500
380.68530
0.06241
0.83080
II
α
^ = 1.42993, β^ = 0.00349
306.77020
617.5404
621.76210
0.08955
0.71220Table 12
Estimates and widths of BCIs of the indicesSpmk and Spmkc [C0=15, C1=30, C2=5, and t=0.5].
Data set
Methods of estimation
Estimates ofSpmk and Spmkc
Widths of BCIs
S-boot
P-boot
B
C
p-boot
I
MLE
S
^
pmk
mle = 0.65844
0.41386
0.39799
0.39767
S
^
pmkc
mle = 0.16141
0.13449
0.11676
0.11673
OLSE
S
^
pmk
olse = 0.63071
0.41402
0.40805
0.39788
S
^
pmkc
olse = 0.15569
0.13452
0.11678
0.11675
WLSE
S
^
pmk
wlse = 0.65470
0.41392
0.39777
0.39772
S
^
pmkc
wlse = 0.16117
0.13451
0.11677
0.11673
CME
S
^
pmk
cme = 0.63736
0.41391
0.39778
0.39773
S
^
pmkc
cme = 0.15755
0.13450
0.11678
0.11674
MPSE
S
^
pmk
mpse = 0.65378
0.41390
0.39776
0.39773
S
^
pmkc
mpse = 0.16111
0.13452
0.11677
0.11674
ADE
S
^
pmk
ade = 0.64574
0.41393
0.39781
0.39777
S
^
pmkc
ade = 0.15902
0.13453
0.11679
0.11677
RADE
S
^
pmk
rade = 0.62084
0.41392
0.39781
0.39778
S
^
pmkc
rade = 0.15301
0.13454
0.11680
0.11677
II
MLE
S
^
pmk
mle = 0.42365
0.27087
0.24350
0.24347
S
^
pmkc
mle = 0.12387
0.09233
0.08653
0.08651
OLSE
S
^
pmk
olse = 0.42765
0.27097
0.24351
0.24346
S
^
pmkc
olse = 0.12422
0.09233
0.08655
0.08652
WLSE
S
^
pmk
wlse = 0.42856
0.27093
0.24351
0.24347
S
^
pmkc
wlse = 0.12439
0.09234
0.08655
0.08653
CME
S
^
pmk
cme = 0.42298
0.27095
0.24350
0.24349
S
^
pmkc
cme = 0.12356
0.09235
0.08654
0.08653
MPSE
S
^
pmk
mpse = 0.42278
0.27095
0.24351
0.24348
S
^
pmkc
mpse = 0.12343
0.09235
0.08656
0.08653
ADE
S
^
pmk
ade = 0.42861
0.27103
0.24353
0.24350
S
^
pmkc
ade = 0.12444
0.09239
0.08658
0.08656
RADE
S
^
pmk
rade = 0.42859
0.27103
0.24354
0.24350
S
^
pmkc
rade = 0.12442
0.09240
0.08659
0.08656


## 7. Conclusions

In this article, seven classical methods of point estimation and their BCIs of the indicesSpmk and Spmkc have been considered. The proposed methods are illustrated using two practical examples. As it is not feasible to compare these methods theoretically, an extensive simulation study has been carried out to compare the performance of these methods with different sample sizes and combinations of the unknown parameters. True values and the estimates based on considered methods of estimation along with their MSEs of the indices Spmk and Spmkc have also been obtained.Further, the performance of BCIs is compared in terms of AWs, CPs, and RCs through the simulation study. From the simulation study, one can conclude that among BCIs, theBCp-boot method performs the best in terms of AW and RC. Real data analysis shows the same trend of inference as seen in the simulation study. Besides, it is observed that when α=7.5 and β=0.25, Spmk=1.00321 which indicates that the process is capable. However, a smaller value of Spmk implies higher expected loss, lower process yield, and poor process capability. It is to be noted that when the value of α=7.5 and β=0.50, the corresponding value of Spmk=0.82754 which indicates that the process is incapable. It is hoped that the results and proposed estimation methods will be of benefit to manufacturing industries in selecting appropriate methods of estimation and BCIs.



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*Source: 8974349-2020-02-25.xml*