Classical Estimation of the Index Spmk and Its Confidence Intervals for Power Lindley Distributed Quality Characteristics

'e 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 PCI Spmk 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 PCISpmk, 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 likelihoodmethod of estimation performs better than their counterparts in terms of mean squared error, while biascorrected 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.


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. e 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 C p , C pk , C pm , and C pmk 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 S pmk as follows: where p � 1 − F(USL) − F(LSL) { } 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 S pmk � C Npmk � C pmk � C pm , 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). erefore, it is important to consider the cost tolerance in evaluating PCIs. In this paper, we adopt the tolerance cost function as C M (t) � C 0 + C 1 exp − C 2 t (see [4]), where C 0 , C 1 , and C 2 are the coefficients for the tolerance cost function and t is the process tolerance. Replacing the denominator of S pmk in equation (1)  . (2) 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][9][10][11][12][13][14][15][16][17][18][19] and the references cited therein. e objective of this paper is three fold: First, the estimates of S pmk using seven different classical methods of estimation are obtained. e 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 wellknown distributions and in varied contexts, several authors have used different classical methods of estimation to estimate the parameters of the model (see [20][21][22][23][24][25] and many others). e performance of different estimation methods is demonstrated in terms of their mean squared errors (MSEs) using a simulation study. Second, three BCIs of S pmk based on the above-cited classical methods of estimation are constructed. e performance of the BCIs is demonstrated in terms of their estimated average widths (AWs), coverage probabilities (CPs), and relative coverages (RCs). ird, a new cost-effective PCI S pmkc incorporating a tolerance cost function in the index S pmk is developed. ese 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 S pmk and S pmkc which would be of significance to quality control engineers, where the item/subgroup quality characteristic follows power Lindley distribution (PLD). e rest of the paper is organized as follows: In Section 2, description of the PCIs S pmk and S pmkc for PLD is provided. In Section 3, different frequentist methods of estimation (MLE, LSE, WLSE, CME, MPSE, ADE, and RTADE) for the indices S pmk and S pmkc are described. In Section 4, three BCIs, namely, standard bootstrap (S-boot), percentile bootstrap (P-boot), and bias-corrected percentile bootstrap (BC p -boot) based on aforementioned methods of estimation of the PCIs S pmk and S pmkc 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 S pmk and S pmkc 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. e article ends with a brief conclusion in Section 7.
2. Indices S pmk and S pmkc for the PLD e power Lindley distribution (PLD) was recently proposed in [26]. e PDF and CDF of the PLD are given by 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][28][29][30][31][32] and many others. e r-th order raw moment (about origin) of the PLD is given by (see [26]) erefore, mean (μ) and variance (σ 2 ) of the PLD, respectively, are Now, the PCIs S pmk and S pmkc , where the quality characteristic T which follows the PLD, are, respectively, given by where is the exact proportion of nonconformity and μ and σ are the process mean and process standard deviation, respectively.

Estimation of the Indices S pmk and S pmkc
Here, we describe different classical estimators for the indices S pmk and S pmkc , respectively.

Maximum Likelihood Estimators.
Let z 1 , z 2 , . . . , z n be random samples of size n taken from the PLD with parameter (α, β). e likelihood function is given by Taking logarithm on both the sides of equation (9), we get e resulting partial derivatives with respect to α and β of the log-likelihood function and equating to zero yields the solutions of α and β and are given by e MLEs of α 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 S pmk and S pmkc as where z, s, and p are the sample mean, sample standard deviation, and sample proportion of nonconformity, respectively.

Least Squares and Weighted Least Squares Estimators.
Here, we compute least square and weighted least square estimators of the indices S pmk and S pmkc . is 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 z (1: n) < z (2: n) < · · · < z (n: n) be the order observations obtained from a sample of size n from F(α, β). erefore, the least square estimators of α and β, say, α lse and β lse , can be obtained by minimizing the following function: with respect to α and β, respectively. Equivalently, they can be obtained by solving the following equations: where Substituting the LSEs in equations (6) and (7), we can get the estimators of S pmk and S pmkc as Mathematical Problems in Engineering . (17) e WLSEs, α wlse and β wlse , can be obtained by minimizing the following function with respect to a and β, respectively: Equivalently, WLSEs can be obtained by solving the following equations: i�1 n (n + 1) 2 (n + 2) where ζ 1 (α, β) and ζ 2 (α, β) are defined in equations (15) and (16), respectively. Substituting the WLSEs in equations (6) and (7), we can get the estimators S pmk and S pmkc as

Cramèr-von Mises Estimators.
e Cramèr-von Mises (see [34,35]) estimators of α and β can be obtained by minimizing the function with respect to α and β. ese estimators can also be obtained by solving the following nonlinear equations: where ζ 1 (α, β) and ζ 2 (α, β) are defined in equations (15) and (16), respectively. Substituting the CMEs in equations (6) and (7), we can get the estimators of S pmk and S pmkc as

Maximum Product Spacing Estimators. Cheng and
Amin [36] studied this method as an alternative to the maximum likelihood method. e uniform spacings of a random sample of size n is defined as e MPSEs α 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: or, equivalently, by maximizing the function e estimators α mpse and β mpse of the parameters α and β can be obtained by solving the following nonlinear equations: where ζ 1 (α, β) and ζ 2 (α, β) are defined in equations (15) and (16), respectively. Substituting the MPSEs in equations (6) and (7), we can get the estimator of S pmk and S pmkc as

Anderson-Darling and Right-Tail Anderson-Darling
Estimators. e 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: ese estimators can also be obtained by solving the following nonlinear equations: where ζ 1 (α, β) and ζ 2 (α, β) are defined in equations (15) and (16), respectively. Substituting the ADEs in equations (6) and (7), we can get the estimators of S pmk and S pmkc as e RTADE estimators α rtade and β rtade of the parameters α and β are obtained by minimizing the following function with respect to α and β: ese estimators can also be obtained by solving the following nonlinear equations: Substituting the RTADEs in equations (6) and (7), we can get the estimators of S pmk and S pmkc as . (34)

Bootstrap Confidence Intervals
In this section, three parametric bootstrap confidence intervals of S pmk and S pmkc are obtained. Particularly, S-boot, P-boot, and BC p -boot CIs are constructed for S pmk and S pmkc (see [38]): Simulate the first m-order statistics from a sample of size n from power Lindley distribution with parameter Θ:
pmk is such that In S where In(.) is an indicator function. A 100(1 − c)% P-boot CI of S pmk is given by

BC p -Boot.
In this method, we first locate the observed S pmk in the ordered distributions S pmk . Next, we compute the probability is the CDF of the standard normal variate and ψ l and ψ u are given by (41) In a similar way, we can also find out the empirical bootstrap distribution of S pmkc , given as pmkc . To study the different bootstrap CIs, we consider their estimated AWs, CPs, and RCs. e AWs, CPs, and RCs of the BCIs are calculated based on K � 5, 000 different trials and are defined as where L W and U P denote the 100(1 − c)% CIs based on K repetitions.

Simulation Study
In order to examine the behaviour of the proposed BCIs (S-boot, P-boot, and BC p -boot) of the indices S pmk and S pmkc under different methods of estimation, a Monte Carlo simulation study is performed. e performance of point estimators of the indices S pmk and S pmkc 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), and (7.50, 0.25), respectively. e coefficients in the tolerance cost function are considered as C 0 � 15, C 1 � 30, and C 2 � 5. At t � 0.5, the values of S pmkc are also calculated. e 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 Table 1, while point estimates of S pmk and S pmkc along with their MSEs are given in Table 2.
e 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 S pmk and S pmkc for PLD using MLE, LSE, WLSE, CME, MPSE, ADE, and RTADE, respectively, in Tables 3-9.
e comparisons are based on lower AWs, higher CPs, and higher RCs. e calculated coverage probabilities are compared with the nominal value of 95%.
e simulation results show that the BC p -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 S pmk and S pmkc , and the order of the best method of BCI in terms of RC is BC p < P < S. erefore, we can conclude that the performance of the BC p -boot method is better as compared to the other two methods. Besides, it is observed that when α � 7.5 and β � 0.25, S pmk � 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 S pmk � 0.82754 which indicates that the process is incapable (not shown in the simulation study). Furthermore, AWs and CPs of BCIs of S pmkc are smaller than S pmk for all the configurations considered in our simulation study.

Applications
Here, we consider two different examples taken from the literature in order to assess the performance of the BCIs of the indices S pmk and S pmkc : (i) Data set I: the data set represents the ball size (in mm), and the process was monitored for this quality characteristic with LSL � 0.50 mm and USL � 8.00 mm, respectively, and are taken from [9]. Here, we assume that the target value T � 4.0 mm for the analysis purpose. e data are (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 Table 9: BCIs (AWs, CPs, and RCs) of PCIs S pmk and S pmkc using RTADEs of α and β.
Descriptive statistics, viz., minimum, first quartile (Q 1 ), second quartile (Q 2 ), third quartile (Q 3 ), 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.
e 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 S pmk and S pmkc 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 BC p -boot method is smaller than their counterparts for both PCIs. erefore, it can be concluded that the performance of the BC p -boot method is better as compared to other two bootstrap methods (Table 12).

Conclusions
In this article, seven classical methods of point estimation and their BCIs of the indices S pmk and S pmkc have been considered. e 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 S pmk and S pmkc 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, the BC p -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, S pmk � 1.00321 which indicates that the process is capable. However, a smaller value of S pmk 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 S pmk � 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.
Data Availability e authors declare that the data in the paper are original and available. e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.