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 index which is based on cost function and is applicable both for normally and nonnormally distributed processes. In order to estimate the process capability index 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 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 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.
1. Introduction
Vannman [1] constructed a superstructure PCI which was referred to as and is defined by where and are the process mean and standard deviation, is the target value and , and . Let USL and LSL be the upper and lower specification limits. The four basic indices, , , , and , are special cases of , by letting and . Later, in the year 1997, Chen and Pearn [2] generalized the work of Vannman [1] and proposed a new quantile-based PCI superstructure index for any underlying distribution which is defined as where is the th percentile and is the median of the distribution. It has been observed that for skewed distributions, process median is a more robust measure than the process mean . Considering the values of , we can obtain the following four indices for any underlying distribution, such as
Clearly, when the underlying distribution is normal, then and , and hence, the index reduces to .
In this paper, we propose a cost-effective PCI, say , using the tolerance cost function as suggested by Jeang et al. [3] which is obtained by replacing the denominator of in Equation (3) by sum of quality loss, , and tolerance cost, , i.e., which in turn we obtain a new index, defined as where , are the coefficients for the tolerance cost function and 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, , 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 using normal distribution. Therefore, based on the aforementioned ten traditional estimation techniques, we consider five bootstrap confidence intervals (BCIs), namely, the standard bootstrap , percentile bootstrap , Student’s bootstrap , bias-corrected percentile bootstrap , and bias-corrected accelerated bootstrap . 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 for when the processes follow LED. Thus far, we have not come across any report for calculating PCI, , 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 Section 2. In Section 3, we introduce the PCI for LED. In Section 4, we describe the considered methods of estimation for . In Section 5, five BCIs (, , , , and ) have been discussed for the PCI 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 using a distribution function for a specific PCI is defined as
When values are positive, the distribution is observed to become less resilient (or more sensitive) at upper specification limit compared to lower specification; conversely, when 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 values are low (in absolute sense). Table 1 has the values for the gamma, Weibull, and LE distributions. The mathematical NS values are expressed as defective per million values. Table 1 shows that the Weibull and gamma distributions are more sensitive (or less robust) than LED for and , respectively.
3. 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 where is the shape parameter and is scale parameter of the two parameter LED, respectively. This distribution is a generalization of exponential distribution, and it can be obtained by taking . This distribution is in the bathtub and upside-down bathtub classes for and , respectively. Then, the index (see Chen and Pearn [2]) and the proposed index for the two parameter LED are, respectively, given as where th is the quantile of the two parameter LED with parameters .
4. Estimation of
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 of .
4.1. Maximum Likelihood Estimators
Let be a random sample of size drawn from two parameter LED (9), and then, the likelihood function is given by
Taking logarithm on both the sides of Equation (13), we have
The MLEs of and , say and , respectively, can be obtained as an iterative solutions of the following two equations:
Equations (15) and (16) can be solved for and using any numerical iterative procedure. Since the MLEs ofandare 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, sayand, and obtaining the estimates ofandasand, respectively. Consequently, the MLE of can be obtained as where the MLE of the quantile function with parameters is
4.2. Least Squares Estimators
We minimize the following function with respect to and for obtaining the LSEs denoted by and . where is the CDF, given in Equation (10) and is the th order statistic of a random sample . Equivalently, they can be obtained by solving where
Substituting the LSEs, we can get the estimator of as
4.3. Weighted Least Squares Estimators
The WLSEs, and , can be obtained by minimizing the following function:
The estimators and of the parameters and can be obtained by solving the following nonlinear equations: where , , and are defined in Equations (21) and (22), respectively. Substituting the WLSEs, we can get the estimator of as
4.4. Percentile Estimators
The percentile estimates and of the parameters and can be obtained by minimizing the following function with respect to and :
Several estimators of can be used here; see, for example, Mann et al. [17]. In this paper, we have consider . Substituting the PCEs, we can get the estimator of as
4.5. Cramèr-von Mises Estimators
The Cramèr-von Mises estimators of and , say and , can be obtained by minimizing the following function with respect to and .
The estimators and of the parameters and can be obtained by solving the following nonlinear equations: where and are given by Equations (21) and (22), respectively. Substituting the CMEs, we can get the estimator of as
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 where , is the th order statistic of a random sample . Note that and . The MPSEs, and , of the parameters and can be obtained by maximizing the following geometric mean of the spacing function with respect to and or, equivalently, by maximizing the function
The estimators and of the parameters and can be obtained by solving the nonlinear equations: where and are given by Equations (21) and (22), respectively. Substituting the MPSEs, we can get the estimator of as
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 and can be obtained by minimizing the following function: with respect to and , respectively. The estimators and of the parameters and can be obtained by solving the following nonlinear equations: where , , and are defined in Equations (21) and (22), respectively. Substituting the MDEs, we can get the estimator of as
4.8. Minimum Spacing Absolute Log-Distance Estimators
Torabi [20] proposed the minimum spacing absolute-log distance estimators (MLDE). The MLDEs of the parameters and can be obtained by minimizing the function:
The estimators and of the parameters and can be obtained by solving the following nonlinear equations: where , , and are defined in Equations (21) and (22), respectively. Substituting the MLDEs, we can get the estimator of as
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 and of the parameters and of L-E distribution are obtained by minimizing the function:
These estimators can also be obtained by solving the following nonlinear equations: where and are defined in Equations (21) and (22), respectively. Substituting the ADEs, we can get the estimator of as
Similarly, the right-tail Anderson-Darling (RAD) estimators and of the parameters and are obtained by minimizing the following function:
These estimators can also be obtained by solving the nonlinear equations: where and are defined in Equations (21) and (22), respectively. Substituting the RADEs, we can get the estimator of as
5. Bootstrap Confidence Intervals
In this section, we use bootstrap technique to construct confidence intervals for the PCI using all considered methods of estimation. Here, we consider five methods of bootstrap CIs, namely, (i) , (ii) , (iii) , (iv) , and (v) . We discuss the algorithm for the bootstrap methods based on maximum likelihood only. (1)Let () be a random sample of size drawn from LED (). Compute MLEs () of (). A bootstrap sample () is obtained by multiplying as mass at each point from the original sample(2)Compute the MLEs () of () as well as of . The th bootstrap estimator of is computed as (3)There are total number of resamples. From these resamples, the entire collection of values of from smallest to largest would constitute an empirical bootstrap distribution as
5.1.
Let and be the sample mean and standard deviation of , i.e., respectively. A confidence interval of is given as where is obtained by using upper th point of the standard normal deviate.
5.2.
Let be the percentile of , i.e., is such that where is an indicator function. Then, a confidence interval of is given as
5.3.
Let be the percentile of , i.e., is such that where is defined above. A confidence interval of is given by
5.4.
At first, locate the observed in the order statistics . Next, we compute and to calculate and where respectively. Then, confidence interval of is
5.5.
Calculate where is called the acceleration factor and is the MLE of based on observations after excluding the th observation.
Then, a confidence interval of is given as where and , respectively.
6. Simulation and Discussion
Here, we conduct a simulation research to evaluate the behaviour of the different PCI 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, , and the lower and upper specification limitations as 0.50 and 9.50 as well as the 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 , , and . The values of , , and are then determined at . The following stages are used to present the method for obtaining the average estimations of the index and the accompanying MSEs: (1)Draw a random sample of size from (2)Estimate the parameters , using MLE(3)Estimate the PCI using the estimates of parameters and (4)Repeat steps 1-3; times(5)Calculate the average estimate of and MSEs based on repetitions
In a similar way, we can find the estimates of 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 (, , , , and ) is discussed in details in Section 4. For each design, bootstrap samples with each of size are drawn from the original sample using the estimates of the parameters and replicated times. The results of the 95% BCIs, viz., , , , , and , are constructed by each of the classical methods of estimation for 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 , are given as Appendix at last.
Simulated outcomes of considered estimators for the index 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 increases, the value of the index increases. Results of the estimated AW and CPs of BCIs of the index 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 provides smaller AW, while CI of 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: < < < < for all settings considered in this study. Therefore, we conclude that 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 its 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 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 the 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 . In case of L-E distribution, MLEs of the parameters and are and . 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:
1,10.4,8.8,9.7,7.8,9.9,11.7,8.0,9.3,9.0,8.2,8.9,10.1,9.4,9.2,7.9,9.5,10.9,7.8,8.3,9.1,8.4,9.6,11.1,7.9,8.5,8.7,7.8,10.5,8.5,11.5,8.0,7.9,8.3,8.7,10.0,9.4,9.0,9.2,10.7,9.3,9.7,8.7,8.2,8.9,8.6,9.5,9.4,8.8,8.3,8.4,9.1,10.1,7.8,8.1,8.8,8.0,9.2,8.4,7.8,7.9,8.5,9.2,8.7,10.2,7.9,9.8,8.3,9.0,9.6,9.9,10.6,8.6,9.4,8.8,8.2,10.5,9.7,9.1,8.0,8.7,9.8,8.5,8.9,9.1,8.4,8.1,9.5,8.7,9.3,8.1,10.1,9.6,8.3,8.0,9.8,9.0,8.9,8.1,9.7,8.5,8.2,9.0,10.2,9.5,8.3,8.9,9.1,10.3,8.4,8.6,9.2,8.5,9.6,9.0,10.7,8.6,10.0,8.8,8.6.
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 mil and mil ( in =0.0254 mm). Here, we have considered the target value mil. In case of L-E distribution, MLEs of the parameters and are and . 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:
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 Table 14, we report the point estimates of and and the width of BCIs based on different methods of estimation. For both the data sets, we observe that the width of 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.
8. Conclusions and Future Works
In this work, we evaluate five BCIs of the PCI 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 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]).
Appendix
The R Codes for Calculation of Point Estimate, Corresponding to MSEs and AWs and CPs of BCIs of , are given below:
g = function(L,T,U);
{
QU = (1/a)*log(1 + (0.99865/(1-0.99865))^(1/b)); QM = (1/a)*log(1 + (0.5/(1-0.5))^(1/b));
QL = (1/a)*log(1 + (0.00135/(1-0.00135))^(1/b)); QSe = (QU-QL)/6; Qe = sqrt(QSe^2 + (QM-T)^2);
c0 =1; c1 = 3; c2 = 2; t = .75; ;Cost=c0+c1*exp(-c2*t);Qec=sqrt(QSe^2+(QM-T)^2+Cost);
cnpm=(U-L)/(6*Qe);cnpm;cnpmc=(U-L)/(6*Qec);cnpmc
#Random number generation from LED($\varrho$;$\vartheta$)
# maximum likelihood estimate .... # Maximum product spacings estimate
}
reep=t(replicate(1000,g()))
# Obtain Average estimate and corresponding MSE
#Repeat function
fb=function(th)
{
a=th[1];b=th[2]
l=n*log(a)+n*log(b)+(b-1)*sum(log(exp(a*x)-1))-a*sum(x)
-2*sum(log(1+(exp(a*x)-1)^b))
return(-l)
}
z=nlm(fb,c(0.8,0.8));z
a_cap=z$estimate[1];a_cap
b_cap=z$estimate[2];b_cap
fd=function(th)
{a_cap=th[1];b_cap=th[2]
l2=n*log(a_cap)+n*log(b_cap)+(b_cap-1)*sum(log(exp(a_cap*x)-1))-a_cap*sum(x)
-2*sum(log(1+(exp(a_cap*x)-1)^b_cap))
return(-l2)}
a_cap_boot_mle=mean(a_cap_boot);
b_cap_boot_mle=mean(b_cap_boot);
cnpmc_boot=sort(cnpmc_boot);
cnpmc_boot_mle=mean(cnpmc_boot);
cnpmc_boot_sd=sd(cnpmc_boot);
# Standard bootstrap,..., bias-corrected accelerated bootstrap
Data Availability
The link of the dataset used in this study is included within the article.
Conflicts of Interest
The authors declare no conflict of interest.
Authors’ Contributions
All authors have contributed equally and agreed to the published version of the manuscript.
Acknowledgments
This research was funded by the Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R50), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.