Front. Appl. Math. Stat. Frontiers in Applied Mathematics and Statistics Front. Appl. Math. Stat. 2297-4687 Frontiers Media S.A. 10.3389/fams.2025.1668809 Applied Mathematics and Statistics Original Research Bootstrap confidence intervals of process capability indices Cpy and CNpmk using different methods of estimation for Frechet distribution Kanwal Tahira 1 Abbas Kamran 1 Zaman Mehwish 2 * Hussain Zamir 3 1Department of Statistics, King Abdullah Campus Chatter Kalas, The University of Azad Jammu and Kashmir, Muzaffarabad, Pakistan 2Via Cesare Battisti, University of Padova, Padova, PD, Italy 3School of Interdisciplinary Engineering and Sciences (SINES), National University of Sciences and Technology (NUST), Islamabad, Pakistan

Edited by: Stelios Psarakis, Athens University of Economics and Business, Greece

Reviewed by: Zakariya Yahya Algamal, University of Mosul, Iraq

Mahendra Saha, University of Delhi, India

 *Correspondence: Mehwish Zaman mehwish.zaman@studenti.unipd.it 14 10 2025 2025 11 1668809 18 07 2025 22 09 2025 Copyright © 2025 Kanwal, Abbas, Zaman and Hussain. 2025 Kanwal, Abbas, Zaman and Hussain

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Process capability analysis is the statistical evaluation of process capability to examine how well it meets or exceeds the customers satisfaction. In present work, we are intended to evaluate two critical metrics for process quality assessment, Cpy and CNpmk for asymmetric Frechet distribution using different classical estimation methods namely: maximum likelihood, least squares, weighted least squares, Cramer-von-Mises, Anderson-Darling, right-tail Anderson-Darling, along Bayesian estimation using reference and Jeffreys priors. Furthermore, Monte Carlo simulations are conducted for comparative analysis of aforementioned estimation methods, using mean squared error and width of bootstrap confidence intervals. Comparative analysis demonstrates that Bayesian estimation using reference prior consistently producing smaller coefficient of mean squared errors and reduced width of bootstrap intervals for small to moderately large sample sizes. Moreover, the real data analysis is also validating the advantages of Bayesian estimations for process capability analysis.

 process capability indices Bayesian estimators classical estimators bootstrap confidence intervals mean squared errors section-at-acceptance Statistics and Probability 1 Introduction

Statistical process control (SPC) is crucial for quality control in the highly competitive manufacturing industry. Process capability analysis is a vital component of SPC quality improvement programs. An essential part of SPC quality improvement initiatives is process capability analysis. Manufacturing process capabilities can be evaluated quantitatively and intuitively with the help of process capability indices (PCIs). These indices help identify areas for improvement and show how well a process meets customer needs. Over capability analysis, a process is evaluated using statistical methods to see whether it fulfills a set of customers preset specifications or not. Suppliers and manufacturers employ such measures to ensure that their finished products are of excellent quality and have the least amount of discrepancies. Capability indices are the most commonly used techniques for capability analysis. A PCI quantifies how well the process meets customer expectations.

Several PCIs have been suggested to quantify process performance. The readers may refer to Kane [1], addressed two often used indices: Cp and Cpk, Chan et al. [2] and Pearn et al. [3] devised two more advanced indices: Cpm and Cpmk. Choi and Owen [4] provided thorough comparisons of Cp, Cpk, Cpm, and Cpmk. Boyles [5] introduced PCI with asymmetric tolerance for Cpk, Cpm, and Cpmk. Many statisticians and quality researchers, including Kane [1], Chan et al. [2], Chou et al. [6], Pearn et al. [3], and Pearn and Chen [7], have discussed and analyzed point estimation and interval estimation of various indices.

The efficacy of these indicators is dependent on the process following a normal distribution [8–10]. However, the presence of various noise elements might occasionally result in abnormal manufacturing processes [11]. If the assumption is not met, these basic indices are not reliable [12–16]. For non-normal processes, the normality-based classical PCI may produce unreliable and misleading interpretations [17]. Pearn and Chen [18], Tong and Chen [19], and Senvar and Kahraman [17] all studied non-normal distributions. Still, we were unable to depend on one method that functions efficiently in all situations [11]. Furthermore, for various non-normal distributions, it is also claimed that each method performed differently [10]. Thus, for statisticians and practitioners, evaluating various PCIs under non-normal distributions is a crucial research topic. The widespread usage of PCIs in manufacturing requires a thorough analysis of proposed structures of PCIs in the literature using different estimating techniques and distributions.A PCI evaluates the amount of variation process experiences in relation to specification limitations. Point and interval estimations are often emphasized while analyzing PCIs. However, there is a possibility that point estimate will not conclude a realistic assessment of a process' capabilities and confidence intervals (CIs) are considered more useful in determining the variability of an estimate [20]. Hsiang and Taguchi [21] initiated the first move toward constructing CIs for the PCI. Since then, various methods for constructing CIs have been developed. One of these methods is bootstraping, which is unconstrained by distributional assumptions and makes complex inferences in simplified manners. In reality, insufficient or limited data is evaluated using a basic re-sampling method owing to time and budgetary restrictions. Franklin and Wasserman [22] initially used the bootstrap approach to evaluate the PCI-Cpk. Recently, significant advances have been made in the literature about bootstrap CIs and how they behave for processes in absence of normality assumption. Some studies in this regard include Kashif et al. [23], Rao et al. [24, 25], Peng [26], Dey and Saha [27, 28], Dey et al., [29], Saha et al. [30, 31], and Weber et al. [32].

Most statistical studies rely on one of two estimating categories: Bayesian or frequentist. PCIs for both normal and non-normal processes were developed using Bayesian and classical perspectives. Several statisticians, including Chan et al. [2], Cheng and Spiring [33], and Shiau et al. [34, 35], have justified and emphasized the advantages of the Bayesian method. Ramos et al. [36] provided a brief comparison of conventional and Bayesian estimation methods.

In present work, we are intended to develop PCIs named Cpy and CNpmk for Frechet distribution (FD) utilizing different classical methods provided in Ramos et al. [36] listed as, maximum likelihood, least square, Cramer-von-Mises and Anderson-Darling. Additionally, we are using weighted least square, right-tail Anderson-Darling estimators and Bayesian estimators using reference prior along Jeffrey prior for evaluation of PCIs.

The FD is a special case of the generalized extreme value distribution, used to model the maximum of large numbers of independent positive random variables. FD is equivalent to the inverse exponential distribution for the scale parameter β = 1; it is the inverse gamma distribution when β < 1 and for β = 2; it is the inverse Rayleigh distribution. The FD may be utilized in reliability analysis, especially when examining the quality characteristic of fatigue lifespan (the marketing life expectancy of pharmaceutical, automobile radiators, textile fatigue, and electron tubes, among other devices and variables). It provides flexible models for non-normal and asymmetric quality characteristics. Most suited to right-skewed characteristics where small values are most likely but rare extreme values exist. The probability density function (PDF), cumulative distribution function (CDF) and quantile function (QF) are given by

f(x∣α,β)=(αβ)(βx)α+1exp[-(β/x)α];x,α,β>0 F(x∣α,β)=exp[-(β/x)α] Q(γi∣α,β)=β[-(log(γi))-1α]

Kanwal and Abbas [37] evaluated PCIs Spmk and Cs for FD. Kanwal and Abbas [38] provided comparative analysis of quantile based control charts for FD. It is important to note that while the quality characteristic X specifically follows FD, no effort has been taken to investigate PCIs particularly Cpy and CNpmk. The research aims to bridge the gap. Our objective is to provide recommendations for selecting the most effective PCI estimation technique. It is hoped this study will benefit quality engineers, management professionals, and statisticians. In continuation to the introductory section, the remainder of the article is structured as follows. Section 2 comprises of introduction to PCI Cpy and PCI CNpmk. In Section 3 sensitivity analysis is performed, Section 4 presents several classical estimation methods for Cpy and CNpmk whereas bootstrap confidence intervals (BCIs) are covered in Section 5. Section 6 includes a simulation study to evaluate the estimators' effectiveness in estimating Cpy and CNpmk in terms of mean squared error (MSE), coefficient of skewness (β1), coefficient of kurtosis (β2), and BCIs width. Section 7 analyzes the real-life data set as an illustration. Finally, the findings of the present study are shown in Section 8.

 2 PCIs <italic>C</italic><sub><italic>py</italic></sub> and <italic>C</italic><sub><italic>Npmk</italic></sub>

The detail of PCIs Cpy and CNpmk is given below.

2.1 PCI <italic>C</italic><sub><italic>py</italic></sub>

A generalized PCI Cpy has been proposed by Maiti et al. [39]. It includes continuous and discrete random variables, both normal and non-normal, and is either directly or indirectly associated with most PCIs in literature. It can be defined as:

Cpy=F(USL)-F(LSL)F(UDL)-F(LDL)=pp0

where p is the process output, p0 is the desired output, LDL and UDL denotes the lower and upper desirable limits, and USL and LSL are upper and lower specifications respectively. Where, LDL = μ−3σ and UDL = μ+3σ. PCI Cpy may be expressed as (p/0.9973) for process following a normal distribution. The Cpy for FD quality characteristic can be expressed as

Cpy=1p0[exp[-(β/USL)α]-exp[-(β/LSL)α]] 2.2 PCI <italic>C</italic><sub><italic>Npmk</italic></sub>

Vannman [40] combined these four indices Cp, Cpk, Cpm and Cpmk into one super-structure, defined as

Cp(u,v)=d-u|μ-m|3σ2+v(μ-T)2, u,v≥0

where d = (USL − LSL)/2 and m = (USL + LSL)/2. In particular, we have, Cp(0, 0) = Cp, Cp(1, 0) = Cpk, Cp(0, 1) = Cpm and Cp(1, 1) = Cpmk. These indices have been developed when underlying distribution is normal. Later, Chen and Pearn [16] generalized Vannman [40] work and developed a new quantile-based PCI index CNp (u, v) for any underlying distribution, defined as

CNp(u,v)=d-u|M-m|3(F99.865-F0.1356)2+v(M-T)2,

where Fα is the α-th percentile, M is the median. Whereas, median M for is a more robust measure skewed distributions than the process mean μ. Setting (u, v)= (1, 1) leads to CNpmk, for any distribution, given as

CNpmk=min{CNpmkU,CNpmkL}

where

CNpmkU=USL-M3(Q(γ3∣α,β)-Q(γ1∣α,β)6)2+(Q(γ2∣α,β)-T)2

and

CNpmkL=M-LSL3(Q(γ3∣α,β)-Q(γ1∣α,β)6)2+(Q(γ2∣α,β)-T)2.

where, γi − th are quantiles of the two parameter FD with parameters (α, β). The CNpmk for FD quality characteristic can be written as

CNpmk=min(USL-M,M-LSL)3(β(-log(34)-(1α)-log(14)-(1α))6)2+(M-T)2 3 Sensitivity analysis

For a given PCI, the net sensitivity (NS) analysis for distribution function is proposed by Maiti et al. [39], and is defined as

NS=1p0limϵ→0[{F(U)-F(L)}-{F(U-ϵ)-F(L-ϵ)}ϵ]       =f(U)-f(L)p0,

If the NS values are low (in absolute) with respect to the PCI, the distribution is less sensitive and more robust. However, if the NS values are positive, the considered distribution is more sensitive (or less robust) at USL than at LSL; if the NS values are negative, the converse is true. The unit of measurement for the NS values is defectives per million (dpm). The NS values for the following distributions are presented in Table 1. It is apparent from Table 1 that overall FD is more robust as compare to Weibul and log-logistic distribution and Figure 1 shows that Weibull, log-logistic and FD all are less sensitive (or more robust) at USL for the considered values of (L; U) = (1;4) and p0 = 0.95, respectively.

Sensitivity analysis for FD.

Distribution f(U) f(L) NS (dpm)
Frechet (Shape = 2.1, Scale = 1.6 ) 0.09986 0.19780 −103096.5
Weibul ( Shape = 2.1, Scale = 1.6) 0.003810 0.53915 −563515.3
Log-logistic ( Shape = 2.1, Scale = 1.6) 0.058361 0.41536 −375785.9

Sensitivity for FD.

Bar chart showing net sensitivity values for three distributions: Frechet (-103096), Log-logistic (-375786), and Weibull (-563515). The bars are blue, labeled as “LSL Sensitive.” The x-axis shows distribution types and the y-axis displays net sensitivity values. 4 Different methods of estimation for <italic>C</italic><sub><italic>py</italic></sub> and <italic>C</italic><sub><italic>Npmk</italic></sub>

Here, we describe eight different estimators, namely, maximum likelihood estimator (MLE), ordinary least square estimator(LSE) and weighted least squares estimator (WLSE), Cramer-von-Mises estimator (CVME), Anderson-Darling estimator (ADE), and right tail Anderson-Darling estimator (RTADE), Bayesian estimator under reference prior (BERP), and Jeffrey prior (BEJP) to find out the estimates of the parameters α and β for FD and the corresponding estimators of Cpy and CNpmk respectively. The non-linear equations can be solved by using nleqslv package which is available in the R-software [41].

4.1 Maximum likelihood estimator

Let x1, x2, …, xn be a random sample of size n taken from Equation 1.1. Then the log-likelihood function is

l=nlog(α)-nαlog(β)-(α+1)∑i=1nlog(xi)-∑i=1n(βxi)α

The partial derivatives are

∂l∂α=nα+nlog(β)-∑i=1nlog(xi)-∑i=1n(βxi)αlog(βxi)=0 ∂l∂β=nαβ-(αβ)∑i=1n(βxi)α=0

The solutions to these equations do not result in closed form for MLEs. However, these equations are solved simultaneously using a numerical approach. Substituting the MLEs in Equations 2.2, 2.3, we can get the estimators of Cpy and CNpmk as

ĈpyMLE=1p0[exp{-(β^MLE/USL)α^MLE}-exp{-(β^MLE/LSL)α^MLE}] ĈNpmkMLE=min(USL-M,M-LSL)3(β^MLE(-log(34)-(1α^MLE)-log(14)-(1α^MLE))6)2+(M-T)2. 4.1.1 Ordinary and weighted least squares estimator

Swain et al. [42] proposed a least squares approach to minimize the distance in Johnson's translation system. Suppose F(x(i:n)|α, β) denotes the CDF of ordered random variables x1 ≤ x2 ≤ ... ≤ xn of size n from a distribution function F(.|α, β). The least squares estimators of α^LSE and β^LSE can be obtained by minimizing the function

S(α,β)=∑i=1n[F(x(i:n)|α,β)-in+1]2,

differentiating with respect to α and β. The estimators can be obtained by solving the following

∑i=1n[F(x(i)|α,β)-in+1]ξ1(xi|α,β)=0, ∑i=1n[F(x(i)|α,β)-in+1]ξ2(xi|α,β)=0,

where

ξ1(xi|α,β)=-exp{-(βxi)}α(βxi)αlog(βxi)=0,

and

ξ2(xi|α,β)=-exp{-(βxi)}αα(βxi)α-1xi=0.

Similarly, the weighted least squares estimator of α^WLSE and β^WLSE can be obtained by minimizing the function by Erto and Genesis [43].

W(α,β)=∑i=1n(n+1)2(n+2)i(n-i+1)[F(x(i:n)|α,β)-in+1]2=0,

WLSEs can be obtained by solving the following

∑i=1n(n+1)2(n+2)i(n-i+1)[F(x(i:n)|α,β)-in+1]ξ1(xi|α,β)=0, ∑i=1n(n+1)2(n+2)i(n-i+1)[F(x(i:n)|α,β)-in+1]ξ2(xi|α,β)=0.

Where ξ1(xi|α, β) and ξ2(xi|α, β) are shown in Equations 4.8, 4.9. The least squares and weighted least squares estimators of PCI-Cpy for FD are

ĈpyLSE=1p0[exp{-(β^LSE/USL)α^LSE}-exp{-(β^LSE/LSL)α^LSE}] ĈpyWLSE=1p0[exp{-(β^WLSE/USL)α^WLSE}                -exp{-(β^WLSE/LSL)α^WLSE}]

Similarly, the least squares and weighted least squares estimators of PCI-CNpmk for FD are

ĈNpmkLSE=min(USL-M,M-LSL)3(β^LSE(-log(34)-(1α^LSE)-log(14)-(1α^LSE))6)2+(M-T)2 ĈNpmkWLSE=min(USL-M,M-LSL)3(β^WLSE(-log(34)-(1α^WLSE)-log(14)-(1α^WLSE))6)2+(M-T)2. 4.1.2 Cramer-von-Mises estimator

The Cramer-von-Mises test was proposed by Cramer [44] and von-Mises [45], which measures the distance between empirical distribution function and CDF of assumed model. Moreover, Boos [46] studied its properties, which can be expressed as

CVM(α,β)=112n+∑i=1n[F(x(i:n)|α,β)-2i-12n]2,

The Cramer-von-Mises estimator of α^CVME and β^CVME can be incurred by solving the following

∑i=1n[F(x(i:n)|α,β)-2i-12n]ξ1(xi|α,β)=0, ∑i=1n[F(x(i:n)|α,β)-2i-12n]ξ2(xi|α,β)=0.

Where ξ1(xi|α, β) and ξ2(xi|α, β) are mentioned in Equations 4.8, 4.9. The corresponding estimators of PCIs-Cpy and CNpmk are

ĈpyCVME=1p0[exp{-(β^CVME/USL)α^CVME}                 -exp{-(β^CVME/LSL)α^CVME}] ĈNpmkCVME=min(USL-M,M-LSL)3(β^CVME(-log(34)-(1α^CVME)-log(14)-(1α^CVME))6)2+(M-T)2. 4.1.3 Anderson-Darling estimator

Anderson-Darling [47] statistic belong to the class of quadratic empirical distribution function [48]. The Anderson-Darling test (ADT) converges rapidly toward the asymptote [49], which is

ADT(α,β)=-n-1n∑i=1n(2i-1)[log(F(x(i)|α,β))                          +log(1-(F(x(1-i+n)|α,β)))]

The Anderson-Darling estimator (ADE), α^ADE and β^ADE can also be obtained by solving the following non-linear equations

∑i=1n(2i-1)[-exp{-(βxi)α}log(βx(i))(βx(i))α+exp{-(βx(1-i+n))α}log(β(1-i+n))(βx(1-i+n))α1-exp{-(βx(1-i+n))α}exp{-(βxi)α}+log[1-exp{-(βx(1-i+n))α}]] ∑i=1n(2i-1)[-exp{-(βxi)α}α(βx(i))-1+αxi+exp{-(βx(1-i+n))α}α(βx(1-i+n))-1+α(1-exp{-(βx(1-i+n))α})x(1-i+n)exp{-(βxi)α}+log(1-exp{-(βx(1-i+n))α})]

The right tail Anderson-Darling estimator, α^RTADE and β^RTADE are obtained by minimizing with respect to α and β, the function

RTADE(α,β)=n2-2∑i=1nF(x(i)|α,β)-1n∑i=1n(2i-1)                              log(1-(F(x(1-i+n)|α,β))),

These estimators can also be obtained by solving the following non-linear equations

-2∑ni=1exp{-(βx(i))α}log(βx(i))(βx(i))α1-exp{-(βx(i))α}-1n∑ni=1exp{-(βx(i))α-(βx(1-i+n))α}log(βx(1-i+n))(βx(1-i+n))α(1-exp{-(βx(i))α})2=0, -2∑i=1nexp{-(βx(i))α}α(βx(i))-1+α(1-exp{-(βx(i))α})x(i)-1n∑i=1nexp{-(βx(i))α-(βx(1-i+n))α}α(βx(1-i+n))-1+α(1-exp{-(βx(i))α})2x(1-i+n)=0.

The ADE and RTADE of PCI Cpy for FD are

ĈpyADE=1p0[exp{-(β^ADE/USL)α^ADE}              -exp{-(β^ADE/LSL)α^ADE}] ĈpyRTADE=1p0[exp{-(β^RTADE/USL)α^RTADE}                   -exp{-(β^RTADE/LSL)α^RTADE}]

Similarly, the corresponding ADE and RTADE of PCI CNpmk for FD are

ĈNpmkADE=min(USL-M,M-LSL)3(β^ADE(-log(34)-(1α^ADE)-log(14)-(1α^ADE))6)2+(M-T)2 ĈNpmkRTADE=min(USL-M,M-LSL)3(β^RTADE(-log(34)-(1α^RTADE)-log(14)-(1α^RTADE))6)2+(M-T)2. 4.2 Bayesian estimator

This section considers the BERP and BEJP for estimation of unknown parameters of the FD. Abbas and Tang [50] derived Jeffrey's and reference priors for FD, which are as follows:

πJ(α,β)∝|I(α,β)|∝1βπR(α,β)∝|I(α,β)|∝1αβ

The joint posterior distributions using Jeffrey's and reference prior can be written as

πJ(α,β∣x)=αnβnα-1∏i=1nxi-(α+1)exp[-∑i=1n(β/xi)α]∫0∞∫0∞αnβnα-1∏i=1nxi-(α+1)exp[-∑i=1n(β/xi)α]dαdβ. πR(α,β∣x)=αn-1βnα-1∏i=1nxi-(α+1)exp[-∑i=1n(β/xi)α]∫0∞∫0∞αn-1βnα-1∏i=1nxi-(α+1)exp[-∑i=1n(β/xi)α]dαdβ.

The Bayesian estimators of model parameters employing squared error loss function are obtained using Laplace approximation. Which is accessible through the LearnBayes package [51]. The BERP and BEJP of PCI-Cpy are given as

ĈpyBERP=1p0[exp{-(β^BERP/USL)BERPα^}                -exp{-(β^BERP/LSL)α^BERP}] ĈpyBEJP=1p0[exp{-(β^BEJP/USL)BEJPα^}                -exp{-(β^BEJP/LSL)α^BEJP}]

Similarly, the corresponding BERP and BEJP of PCI CNpmk for FD are

ĈNpmkBERP=min(USL-M,M-LSL)3(β^BERP(-log(34)-(1α^BERP)-log(14)-(1α^BERP))6)2+(M-T)2 ĈNpmkBEJP=min(USL-M,M-LSL)3(β^BEJP(-log(34)-(1α^BEJP)-log(14)-(1α^BEJP))6)2+(M-T)2. 5 BCIs for <italic>C</italic><sub><italic>py</italic></sub> and <italic>C</italic><sub><italic>Npmk</italic></sub>

Franklin and Wasserman [22] developed the bootstrap approach, which is typically employed to obtain the empirical distribution of different PCIs. We evaluated standard BCIs for Cpy and CNpmk. The basic steps for bootstrapping are as follows:

A random sample (x1, x2, …, xn) of size n, is drawn from FD(α, β), followed by n bootstrap samples (with replacement) taken from the original sample with a mass of 1/n at each. Classical and Bayesian estimates of (α^,β^) are obtained and then R-th bootstrap estimator are calculated for Cpy & CNpmk, i.e.,

 Ĉpy(R)=Ĉpy(x1,x2,⋯,xn)&ĈNpmk(R)=ĈNpmk(x1,x2,⋯,xn)

A total of nn re-samples exist, and for each sample, Ĉpy(R) is obtained, resulting in a full bootstrapped distribution {Ĉpy*(J);J=1(1)B} and is denoted by Ĉpy*(1)≤Ĉpy*(2)≤⋯≤Ĉpy*(B), and same process is done for ĈNpmk(R).

 6 Simulation study

The present section deals with the simulation experiment for performance assessments of the PCIs namely Cpy and CNpmk for FD based on aforementioned estimation methods. Different parametric combinations (α, β) = [(1, 3.5)(1, 2)(1, 4)(1.2, 2.2)] for Cpy and (1, 2.5), (2.1, 1.6), [1.6, 2.1, (1.6,1.7)] for CNpmk against sample size n = 10, 15, 20, 25, 30, 50 are evaluated and the LSL and USL are set as 1 and 4, respectively whereas, target value is 2.5 and p0 = 0.95. For each design, 95% BCIs were evaluated, along the difference of both the upper and lower confidence bounds, referred as the average width, (β1), (β2) , and MSE, to compare the results shown in Tables 2–5 for PCI Cpy and Tables 6–9 for PCI CNpmk with 10,000 replications using the R-software. Results are evaluated based on indices with reduced average width and minimum MSE.

Estimates of Cpy when α = 1.2 and β = 2.2.

n Method Cpy↓ Ĉpy β1 β2 BCI Width MSE
10 MLE 0.56605 0.57012 0.05505 2.9867 (0.53870, 0.60170) 0.06300 2.751e-04
LSE 0.56371 0.17548 4.2374 (0.49364, 0.63604) 0.14240 1.188e-03
WLSE 0.57643 –0.03607 3.0715 (0.39799, 0.74646) 0.34847 7.639e-03
CVME 0.58262 0.65055 4.0189 (0.54003, 0.64051) 0.10048 8.963e-04
ADE 1.03800 –2.97280 9.9895 (0.94149, 1.05040) 0.11087 2.239e-01
RTADE 0.49594 –0.58147 3.1611 (0.35001, 059320) 0.24319 8.936e-03
BEJP 0.57407 0.11641 3.0179 (0.54335, 0.60675) 0.06340 3.271e-04
BERP 0.56837 0.13821 3.0136 (0.53857, 0.60074) 0.06217 2.533e-04
15 MLE 0.56605 0.57032 0.03828 2.9638 (0.54483, 0.59622) 0.05139 1.910e-04
LSE 0.56379 0.16613 3.9124 (0.50983, 0.62016) 0.11033 7.429e-04
WLSE 0.57640 -0.07305 3.0157 (0.43520, 0.71360) 0.27840 5.141e-03
CVME 0.58255 0.54148 3.7550 (0.54752, 0.62795) 0.08043 6.814e-04
ADE 1.03890 -2.43410 8.5448 (0.97456, 1.05220) 0.07765 2.242e-01
RTADE 0.49726 -0.51604 3.1970 (0.38269, 0.58303) 0.20035 7.358e-03
BEJP 0.57421 0.11159 3.0300 (0.54916, 0.60052) 0.05136 2.389e-04
BERP 0.56854 0.10858 3.0362 (0.54421, 0.59437) 0.05016 1.721e-04
20 MLE 0.56605 0.57018 0.13914 3.0398 (0.54822, 0.59340) 0.04518 1.484e-04
LSE 0.56346 0.10367 3.5655 (0.51661, 0.61272) 0.09610 5.718e-04
WLSE 0.57662 –0.05736 2.9682 (0.45567,0.69306) 0.23739 3.835e-03
CVME 0.58254 0.47585 3.3830 (0.55135, 0.62085) 0.06950 5.850e-04
ADE 1.03800 –2.02890 6.7161 (0.99031, 1.05210) 0.06176 2.233e-01
RTADE 0.49773 –0.42974 3.0861 (0.40093, 0.57569) 0.17476 6.668e-03
BEJP 0.57402 0.06770 2.9974 (0.54668, 0.59058) 0.04390 1.899e-04
BERP 0.56826 0.07117 3.0529 (0.55250, 0.59630) 0.04380 1.310e-04
25 MLE 0.56605 0.57027 0.05150 2.9647 (0.55042, 0.59060) 0.04018 1.240e-04
LSE 0.56347 0.05371 3.7666 (0.52062, 0.60881) 0.08819 4.688e-04
WLSE 0.57577 –0.03274 2.9772 (0.46858, 0.68117) 0.21259 3.042e-03
CVME 0.58259 0.42236 3.5440 (0.55479, 0.61695) 0.06215 5.217e-04
ADE 1.03830 –1.91420 6.5133 (0.96746, 1.05200) 0.08451 2.235e-01
RTADE 0.49810 –0.42952 3.2787 (0.41468, 0.56691) 0.15223 6.169e-03
BEJP 0.57406 0.09228 3.0006 (0.54885, 0.58821) 0.03936 1.642e-04
BERP 0.56850 0.04138 2.9985 (0.55522, 0.59375) 0.03853 1.069e-04
30 MLE 0.56605 0.57035 0.06506 2.9802 (0.55250, 0.58902) 0.03652 1.058e-04
LSE 0.56306 0.04006 3.4653 (0.52458, 0.60266) 0.07809 3.872e-04
WLSE 0.57631 -0.05145 3.0012 (0.48017, 0.67325) 0.19308 2.551e-03
CVME 0.58250 0.37966 3.3320 (0.55668, 0.61267) 0.05599 4.756e-04
ADE 1.03840 –1.74070 5.9311 (0.98076, 1.05180) 0.07108 2.235e-01
RTADE 0.49719 –0.36023 3.1541 (0.41987, 0.56143) 0.14156 6.064e-03
BEJP 0.57399 0.04908 3.0142 (0.55628, 0.59201) 0.03573 1.466e-04
BERP 0.56831 0.05177 2.9213 (0.55074, 0.58646) 0.03572 8.828e-05
50 MLE 0.56605 0.57034 0.06676 3.0517 (0.55645, 0.58479) 0.02834 7.065e-05
LSE 0.56339 0.05829 3.3026 (0.53315, 0.59361) 0.06047 2.393e-04
WLSE 0.57583 –0.06247 3.0646 (0.49931, 0.65055) 0.15124 1.593e-03
CVME 0.58243 0.33132 3.2584 (0.56253, 0.60581) 0.04328 3.897e-04
ADE 1.03850 –1.34450 4.7413 (1.00518, 1.05160) 0.04638 2.234e-01
RTADE 0.49749 –0.27802 3.0563 (0.43892, 0.54865) 0.10974 5.495e-03
BEJP 0.57389 0.07766 3.0868 (0.55443, 0.58262) 0.02819 1.119e-04
BERP 0.56841 0.07173 3.0823 (0.56011, 0.58774) 0.02764 5.596e-05

Estimates of Cpy when α = 1 and β = 3.5.

n Method Cpy↓ ĈPY β1 β2 BCI Width MSE
10 MLE 0.40702 0.40530 0.09179 2.9434 (0.37231, 0.43978) 0.06748 3.047e-04
LSE 0.25497 0.11220 3.0198 (0.11768, 0.40687) 0.28920 2.887e-02
WLSE 0.32178 0.05405 3.1500 (0.17873, 0.46602) 0.28728 1.262e-02
CVME 0.28388 0.18455 3.0487 (0.13357, 0.44327) 0.30969 2.142e-02
ADE 0.97504 –1.06205 4.1864 (0.82950, 1.04720) 0.21773 3.264e-01
RTADE 0.15256 0.25335 2.9100 (0.03756, 0.29251) 0.25494 6.926e-02
BEJP 0.41245 0.13347 3.0518 (0.37974, 0.44734) 0.06760 3.241e-04
BERP 0.40897 0.07903 2.9902 (0.37598, 0.44328) 0.06729 2.945e-04
15 MLE 0.40702 0.40521 0.13431 2.9619 (0.37804, 0.43426) 0.05622 2.052e-04
LSE 0.25507 0.09782 2.9428 (0.13955, 0.37868) 0.23913 2.694e-02
WLSE 0.32151 0.07469 3.0600 (0.20734, 0.44310) 0.23575 1.087e-02
CVME 0.28366 0.07301 2.9725 (0.15744, 0.41294) 0.25550 1.945e-02
ADE 0.97533 –0.89276 3.9387 (0.85502, 1.04380) 0.18882 3.256e-01
RTADE 0.15225 0.18680 2.8800 (0.05417, 0.26454) 0.21036 6.788e-02
BEJP 0.41234 0.08246 3.0949 (0.38490, 0.44060) 0.05570 2.275e-04
BERP 0.40855 0.08261 2.9546 (0.38198, 0.43645) 0.05447 1.967e-04
20 MLE 0.40702 0.40576 0.05108 3.0039 (0.38212, 0.42997) 0.04785 1.499e-04
LSE 0.25520 0.04696 2.9397 (0.15094, 0.36219) 0.21125 2.601e-02
WLSE 0.32273 0.07678 3.1424 (0.22413, 0.42797) 0.20384 9.752e-03
CVME 0.28288 0.05092 2.9926 (0.17317, 0.39429) 0.22112 1.864e-02
ADE 0.97583 –0.70435 3.4445 (0.87686, 1.04130) 0.16446 3.254e-01
RTADE 0.15168 0.18961 2.9300 (0.06528, 0.24658) 0.18129 6.739e-02
BEJP 0.41213 0.05399 3.0860 (0.38828, 0.43639) 0.04810 1.732e-04
BERP 0.40839 0.08971 2.9578 (0.38561, 0.43199) 0.04638 1.428e-04
25 MLE 0.40702 0.40553 0.07135 2.9954 (0.38493, 0.42681) 0.04188 1.192e-04
LSE 0.25569 0.07415 2.9542 (0.16253, 0.35166) 0.18913 2.522e-02
WLSE 0.32271 0.04750 3.0077 (0.23407, 0.41455) 0.18047 9.199e-03
CVME 0.28334 0.07169 3.0002 (0.18825, 0.38419) 0.19594 1.783e-02
ADE 0.97504 -0.65186 3.3719 (0.88733, 1.03650) 0.14920 3.242e-01
RTADE 0.15215 0.11416 2.9100 (0.07299, 0.23595) 0.16295 6.673e-02
BEJP 0.41201 0.07108 2.9755 (0.39134, 0.43362) 0.04282 1.401e-04
BERP 0.40850 0.07053 2.9804 (0.38831, 0.43016) 0.04186 1.157e-04
30 MLE 0.40702 0.40571 0.07454 2.9292 (0.38654, 0.42562) 0.03908 1.017e-04
LSE 0.25631 0.03988 2.9900 (0.17071, 0.34454) 0.17383 2.471e-02
WLSE 0.32137 0.07787 3.0556 (0.24143, 0.40638) 0.16495 9.092e-03
CVME 0.28245 0.02977 3.0077 (0.19400, 0.37411) 0.18011 1.763e-02
ADE 0.97525 –0.57235 3.3980 (0.89689, 1.0342) 0.13728 3.242e-01
RTADE 0.15208 0.12479 3.0000 (0.08043, 0.22961) 0.14918 6.646e-02
BEJP 0.41239 0.04366 2.9108 (0.38942, 0.42815) 0.03873 1.266e-04
BERP 0.40843 0.08236 3.0145 (0.39319, 0.43186) 0.03867 9.867e-05
50 MLE 0.40702 0.40536 0.07219 2.9969 (0.39027, 0.42082) 0.03055 6.365e-05
LSE 0.25586 0.06688 2.9600 (0.18972, 0.32464) 0.13492 2.403e-02
WLSE 0.32239 0.06065 2.9593 (0.25938, 0.38662) 0.12724 8.217e-03
CVME 0.28327 0.04923 2.9766 (0.21283, 0.35567) 0.14284 1.663e-02
ADE 0.97452 –0.45220 3.2079 (0.91546, 1.02230) 0.10685 3.228e-01
RTADE 0.15261 0.09032 3.0010 (0.09588, 0.21205) 0.11617 6.561e-02
BEJP 0.41239 0.05742 3.0480 (0.39410, 0.42406) 0.02996 8.781e-05
BERP 0.40884 0.10077 3.0134 (0.39768, 0.42743) 0.02975 6.117e-05

Estimates of Cpy when α = 1 and β = 2.

n Method Cpy↓ Ĉpy β1 β2 BCI Width MSE
10 MLE 0.49602 0.49701 –0.03060 2.8877 (0.46787, 0.52574) 0.05787 2.222e-04
LSE 0.56196 0.08620 3.0879 (0.40218, 0.72906) 0.32688 1.132e-02
WLSE 0.52948 0.18783 3.0937 (0.39098, 0.67552) 0.28454 6.473e-03
CVME 0.53739 –0.41261 3.8640 (0.34077, 0.69931) 0.35854 9.703e-03
ADE 1.01710 –1.86390 6.5701 (0.85332, 1.05210) 0.19877 2.750e-01
RTADE 0.34039 –0.19153 2.8225 (0.19262, 0.47794) 0.28532 3.004e-02
BEJP 0.49976 –0.02051 3.0222 (0.47126, 0.52805) 0.05679 2.224e-04
BERP 0.49484 -0.05949 2.8928 (0.46590, 0.52240) 0.05649 2.115e-04
15 MLE 0.49602 0.49683 0.00178 2.9213 (0.47292, 0.52046) 0.04754 1.475e-04
LSE 0.56262 0.07329 3.0148 (0.43180, 0.69814) 0.26634 9.076e-03
WLSE 0.53050 0.17292 3.1391 (0.41711, 0.65022) 0.23311 4.756e-03
CVME 0.53787 –0.32895 3.5243 (0.37901, 0.67590) 0.29689 7.224e-03
ADE 1.01680 –1.39545 4.7228 (0.90012, 1.05190) 0.15175 2.734e-01
RTADE 0.34021 –0.18954 2.9669 (0.21063, 0.45521) 0.24458 2.822e-02
BEJP 0.49976 –0.01053 3.0071 (0.47633, 0.52348) 0.04716 1.580e-04
BERP 0.49445 -0.01627 2.9659 (0.47141, 0.51684) 0.04543 1.400e-04
20 MLE 0.49602 0.49732 –0.01473 2.9575 (0.47715, 0.51762) 0.04047 1.099e-04
LSE 0.56165 0.05548 3.0004 (0.44911, 0.67787) 0.22876 7.721e-03
WLSE 0.52884 0.15966 3.1078 (0.43344, 0.63187) 0.19844 3.670e-03
CVME 0.53905 –0.30669 3.2706 (0.40199, 0.65576) 0.25377 5.884e-03
ADE 1.01630 –1.25788 4.6636 (0.92397, 1.05160) 0.12764 2.724e-01
RTADE 0.33932 –0.18655 2.9635 (0.22767, 0.44118) 0.21351 2.754e-02
BEJP 0.49975 -0.04226 2.9773 (0.47433, 0.51491) 0.04058 1.203e-04
BERP 0.49448 -0.00757 2.9855 (0.47956, 0.51967) 0.04012 1.079e-04
25 MLE 0.49602 0.49702 –0.02031 2.9575 (0.47907, 0.51506) 0.03600 8.613e-05
LSE 0.56165 0.04774 3.0970 (0.46030, 0.66525) 0.20495 7.016e-03
WLSE 0.52922 0.15166 3.0231 (0.44195, 0.62382) 0.18187 3.245e-03
CVME 0.53798 –0.25070 3.3359 (0.41721, 0.64190) 0.22468 4.987e-03
ADE 1.01650 –1.14438 4.3247 (0.92265, 1.05140) 0.12880 2.723e-01
RTADE 0.33914 –0.14414 2.9208 (0.24134, 0.42927) 0.18793 2.693e-02
BEJP 0.49967 0.00180 3.0083 (0.48170, 0.51739) 0.03568 9.649e-05
BERP 0.49445 –0.01019 2.9687 (0.47672, 0.51209) 0.03537 8.436e-05
30 MLE 0.49602 0.49706 –0.01300 2.9251 (0.48046, 0.51350) 0.03304 7.340e-05
LSE 0.56140 0.03890 3.0190 (0.46892, 0.65733) 0.18841 6.607e-03
WLSE 0.52865 0.15259 3.0194 (0.45009, 0.61419) 0.16410 2.803e-03
CVME 0.53882 -0.25250 3.3717 (0.42905, 0.63682) 0.20777 4.573e-03
ADE 1.01640 –0.99069 3.7646 (0.93525, 1.05130) 0.11603 2.720e-01
RTADE 0.33980 –0.12655 2.9507 (0.25281, 0.42237) 0.16956 2.632e-02
BEJP 0.49982 –0.05797 2.9793 (0.47822, 0.51101) 0.03279 8.300e-05
BERP 0.49445 0.02200 3.0249 (0.48354, 0.51569) 0.03216 7.249e-05
50 MLE 0.49602 0.49696 0.02456 2.9154 (0.48419, 0.50987) 0.02568 4.442e-05
LSE 0.56150 0.04580 3.0235 (0.49037, 0.63516) 0.14479 5.663e-03
WLSE 0.52988 0.10008 3.1144 (0.46792, 0.59535) 0.12743 2.202e-03
CVME 0.53858 –0.23460 3.1342 (0.45456, 0.61347) 0.15892 3.456e-03
ADE 1.01620 –0.78818 3.6470 (0.95791, 1.05080) 0.09287 2.713e-01
RTADE 0.33990 –0.12557 2.9999 (0.27112, 0.40491) 0.13379 2.555e-02
BEJP 0.49989 –0.00213 3.0138 (0.48715, 0.51239) 0.02525 5.765e-05
BERP 0.49471 0.02875 3.0059 (0.48220, 0.50739) 0.02519 4.322e-05

Estimates for Cpy when α = 1 and β = 4.

n Method Cpy↓ Ĉpy β1 β2 BCI Width MSE
10 MLE 0.36796 0.36568 0.11034 2.9586 (0.33273, 0.40048) 0.06775 3.063e-04
LSE 0.19080 0.17514 2.9241 (0.07213, 0.32634) 0.25421 3.591e-02
WLSE 0.25480 –0.19327 2.9046 (0.13651, 0.36179) 0.22528 1.620e-02
CVME 0.20682 0.03828 2.7957 (0.08103, 0.33429) 0.25326 3.017e-02
ADE 0.99052 –1.37873 4.8897 (0.84687, 1.04620) 0.19937 3.906e-01
RTADE 0.09641 0.46132 3.1155 (0.00000, 0.21365) 0.21365 7.679e-02
BEJP 0.37330 0.13874 3.0160 (0.34056, 0.40848) 0.06792 3.260e-04
BERP 0.37055 0.09407 2.9998 (0.33802, 0.40461) 0.06659 2.993e-04
15 MLE 0.36796 0.36562 0.13610 2.9744 (0.33872, 0.39428) 0.05556 2.059e-04
LSE 0.19137 0.15969 3.0533 (0.08399, 0.30325) 0.21926 3.424e-02
WLSE 0.25555 –0.13407 2.8802 (0.16126, 0.34356) 0.18231 1.487e-02
CVME 0.20533 0.05215 2.8837 (0.10439, 0.30876) 0.20437 2.929e-02
ADE 0.99049 –1.19568 4.6797 (0.88114, 1.04380) 0.16265 3.896e-01
RTADE 0.09675 0.32440 2.9335 (0.019291, 0.19226) 0.17297 7.558e-02
BEJP 0.37317 0.10219 3.1256 (0.34625, 0.40150) 0.05526 2.270e-04
BERP 0.37017 0.10855 2.9891 (0.34366, 0.39853) 0.05487 2.012e-02
20 MLE 0.36796 0.36613 0.05804 2.9748 (0.34253, 0.39057) 0.04804 1.514e-04
LSE 0.19096 0.12178 3.0157 (0.09858, 0.28651) 0.18792 3.361e-02
WLSE 0.25608 -0.13123 2.9617 (0.17517, 0.33128) 0.15611 1.415e-02
CVME 0.20567 0.02509 2.9131 (0.11638, 0.29785) 0.18147 2.853e-02
ADE 0.99109 –0.97555 3.9295 (0.90026, 1.04090) 0.14065 3.898e-01
RTADE 0.09682 0.27134 2.9437 (0.02343, 0.17752) 0.15409 7.503e-02
BEJP 0.37292 0.07705 3.0909 (0.34919, 0.39725) 0.04806 1.721e-04
BERP 0.36997 0.09578 2.9808 (0.34707, 0.39352) 0.04645 1.450e-04
25 MLE 0.36796 0.36595 0.07873 2.9938 (0.34534, 0.38737) 0.04203 1.212e-04
LSE 0.19174 0.13076 2.9465 (0.11190, 0.27817) 0.16626 3.286e-02
WLSE 0.25662 –0.13167 2.9892 (0.18289, 0.32521) 0.14232 1.372e-02
CVME 0.20572 0.02878 2.8879 (0.12642, 0.28764) 0.16122 2.804e-02
ADE 0.99102 –0.85147 3.6568 (0.91095, 1.03850) 0.12751 3.894e-01
RTADE 0.09614 0.25230 2.9488 (0.03261, 0.16856) 0.13595 7.511e-02
BEJP 0.37281 0.07700 2.9670 (0.35213, 0.39436) 0.04223 1.396e-04
BERP 0.37011 0.08075 2.9726 (0.34987, 0.39172) 0.04185 1.188e-04
30 MLE 0.36796 0.36614 0.08563 2.9404 (0.34702, 0.38620) 0.03919 1.035e-04
LSE 0.19195 0.10708 3.0941 (0.116963, 0.27152) 0.15456 3.251e-02
WLSE 0.25564 –0.10124 3.0373 (0.18900, 0.31931) 0.13031 1.371e-02
CVME 0.20520 0.00810 2.9982 (0.13087, 0.27997) 0.14910 2.795e-02
ADE 0.99068 –0.79111 3.7523 (0.91983, 1.03660) 0.11677 3.888e-01
RTADE 0.09630 0.20794 3.0644 (0.03863, 0.15966) 0.12102 7.479e-02
BEJP 0.37322 0.05016 2.9017 (0.35378, 0.39289) 0.03911 1.269e-04
BERP 0.37002 0.08875 3.0109 (0.35099, 0.38996) 0.03897 1.013e-04
50 MLE 0.36796 0.36576 0.07466 3.0422 (0.35092, 0.38151) 0.03058 6.590e-05
LSE 0.19182 0.06632 3.0537 (0.133439, 0.2521) 0.11866 3.195e-02
WLSE 0.25594 -0.09146 2.9899 (0.20423, 0.30540) 0.10116 1.321e-02
CVME 0.20511 0.04087 2.9279 (0.14840, 0.26474) 0.11635 2.741e-02
ADE 0.99113 –0.65469 3.4911 (0.93607, 1.03040) 0.09428 3.889e-01
RTADE 0.09678 0.19481 2.9655 (0.05066, 0.14786) 0.09720 7.416e-02
BEJP 0.37320 0.06430 3.0493 (0.35844, 0.38832) 0.02989 8.706e-05
BERP 0.37042 0.10585 3.0258 (0.35582, 0.38566) 0.02984 6.430e-05

Estimates of CNpmk when α = 1 and β = 2.5.

n Method CNpmk↓ ĈNpmk β1 β2 BCI Width MSE
10 MLE 0.09455 0.13953 0.36470 3.1193 (0.01376, 0.28910) 0.27533 0.00704
LSE 0.15226 0.78429 3.9585 (0.01193, 0.36018) 0.34825 0.01102
WLSE 0.27343 1.66631 4.2491 (0.01456, 0.90664) 0.89208 0.09107
CVME 0.14713 0.33556 3.0434 (0.01209, 0.30671) 0.29461 0.00855
ADE 0.66038 1.03112 4.0391 (0.04223, 1.84680) 1.80454 0.56854
RTADE 0.16432 1.91069 5.6468 (0.01631, 0.50793) 0.49162 0.02020
BEJP 0.14120 0.38805 3.0782 (0.01599, 0.29734) 0.28134 0.00734
BERP 0.13859 0.36549 3.1273 (0.01479, 0.28694) 0.27215 0.00688
15 MLE 0.09455 0.13932 0.28261 3.0187 (0.03607, 0.26155) 0.22548 0.00531
LSE 0.15161 0.74879 4.2557 (0.03215, 0.31622) 0.28407 0.00859
WLSE 0.27543 1.29449 4.9068 (0.04192, 0.75112) 0.70920 0.07149
CVME 0.14754 0.25259 2.9692 (0.03438, 0.27495) 0.24057 0.00656
ADE 0.66514 0.85420 3.7272 (0.08776, 1.60610) 1.51831 0.49357
RTADE 0.16235 1.61139 5.4367 (0.03254, 0.41462) 0.38208 0.01472
BEJP 0.14183 0.27228 3.0392 (0.03531, 0.26153) 0.22621 0.00567
BERP 0.13853 0.30617 3.0872 (0.03499, 0.25871) 0.22371 0.00523
20 MLE 0.09455 0.13880 0.23714 2.9623 (0.04723, 0.24261) 0.19538 0.00450
LSE 0.15107 0.63144 3.8170 (0.04592, 0.28949) 0.24358 0.00711
WLSE 0.27609 1.08803 4.2963 (0.05741, 0.68140) 0.62399 0.06191
CVME 0.14654 0.24878 3.0479 (0.04862, 0.25746) 0.20884 0.00552
ADE 0.66295 0.68044 3.3094 (0.12926, 1.45640) 1.32714 0.44743
RTADE 0.16334 1.32521 5.2102 (0.04839, 0.37206) 0.32368 0.01223
BEJP 0.14088 0.23669 2.9742 (0.04881, 0.24612) 0.19731 0.00469
BERP 0.13850 0.25072 3.0102 (0.04769, 0.24126) 0.19357 0.00440
25 MLE 0.09455 0.13879 0.24068 3.1166 (0.05692, 0.22977) 0.17385 0.00395
LSE 0.15205 0.53035 3.5591 (0.05452, 0.27410) 0.21958 0.00645
WLSE 0.27727 1.03943 4.1911 (0.07113, 0.64561) 0.57448 0.05733
CVME 0.14680 0.19582 2.9780 (0.05844, 0.24370) 0.18526 0.00500
ADE 0.66551 0.63101 3.4219 (0.16243, 1.34770) 1.18527 0.42458
RTADE 0.16445 1.25068 5.0149 (0.05621, 0.36619) 0.30997 0.01126
BEJP 0.14176 0.30045 3.0557 (0.05955, 0.23802) 0.17847 0.00431
BERP 0.13891 0.21879 3.0615 (0.05681, 0.23020) 0.17340 0.00393
30 MLE 0.09455 0.13882 0.21285 3.1236 (0.06233, 0.22116) 0.15883 0.00364
LSE 0.15187 0.45674 3.3196 (0.06296, 0.26255) 0.19959 0.00592
WLSE 0.27621 0.93131 4.0583 (0.07769, 0.60792) 0.53023 0.05281
CVME 0.14719 0.22773 2.9944 (0.06673, 0.23637) 0.16964 0.00468
ADE 0.66206 0.54354 3.2394 (0.19099, 1.29500) 1.10398 0.40351
RTADE 0.16383 1.10622 4.5156 (0.06379, 0.34576) 0.28197 0.00987
BEJP 0.14191 0.22424 3.0324 (0.06602, 0.22748) 0.16147 0.00394
BERP 0.13852 0.21154 3.0490 (0.06357, 0.22116) 0.15759 0.00357
50 MLE 0.09455 0.13856 0.17490 2.9501 (0.07919, 0.20378) 0.12459 0.00297
LSE 0.15174 0.36482 3.2641 (0.08096, 0.23697) 0.15601 0.00484
WLSE 0.27535 0.74077 3.6606 (0.10917, 0.51851) 0.40934 0.04429
CVME 0.14728 0.17589 2.9332 (0.08456, 0.21594) 0.13139 0.00391
ADE 0.66504 0.42852 3.1434 (0.27343, 1.14640) 0.87302 0.37583
RTADE 0.16320 0.84069 3.7693 (0.08159, 0.29251) 0.21091 0.00777
BEJP 0.14196 0.14256 3.0218 (0.08166, 0.20587) 0.12421 0.00327
BERP 0.13900 0.15142 3.0044 (0.07997, 0.20281) 0.12283 0.00297

Estimates of CNpmk when α = 1.6 and β = 1.7.

n Method CNpmk↓ ĈNpmk β1 β2 BCI Width MSE
10 MLE 0.06594 0.70153 0.04900 2.8845 (0.53020, 0.87813) 0.34793 0.41186
LSE 0.76866 2.10380 6.1934 (0.52897, 1.37259) 0.84362 0.54022
WLSE 1.18990 1.92985 5.0223 (0.55272, 3.10470) 2.55200 1.82390
CVME 0.88657 2.94090 5.3842 (0.54136, 2.52560) 1.98425 0.98903
ADE 2.11950 0.63642 3.2982 (0.80043, 4.01100) 3.21050 4.92480
RTADE 0.80604 1.55667 5.3036 (0.54629, 1.32340) 0.77706 0.58650
BEJP 0.70932 0.06076 2.9270 (0.53447, 0.89010) 0.35563 0.42212
BERP 0.69954 0.06605 2.9406 (0.53110, 0.87279) 0.34170 0.40911
15 MLE 0.06594 0.70155 0.07823 3.0091 (0.56182, 0.84667) 0.28485 0.40923
LSE 0.77023 1.71376 6.5295 (0.56242, 1.21072) 0.64831 0.52730
WLSE 1.18630 1.53920 5.5949 (0.59654, 2.76760) 2.17100 1.62250
CVME 0.88474 2.50740 4.5150 (0.57800, 1.96300) 1.38504 0.88311
ADE 2.13660 0.57467 3.3212 (0.97284, 3.69920) 2.72640 4.77420
RTADE 0.80764 1.17270 4.7734 (0.58485, 1.19430) 0.60946 0.57517
BEJP 0.70788 0.03705 2.9200 (0.56611, 0.85217) 0.28606 0.41748
BERP 0.69916 0.04418 2.9470 (0.56104, 0.84057) 0.27954 0.40608
20 MLE 0.06594 0.70235 0.02737 2.9104 (0.58063, 0.82588) 0.24525 0.40895
LSE 0.76874 1.52761 5.8188 (0.58247, 1.13963) 0.55716 0.51734
WLSE 1.19090 1.32374 4.7077 (0.62395, 2.54830) 1.92440 1.54620
CVME 0.88485 2.08890 4.2080 (0.59518, 1.68810) 1.09290 0.82612
ADE 2.12760 0.48419 3.1855 (1.10305, 3.44750) 2.34440 4.60790
RTADE 0.80761 1.00244 4.4125 (0.60046, 1.13680) 0.53632 0.56917
BEJP 0.70929 0.05036 2.9331 (0.58521, 0.83369) 0.24849 0.41788
BERP 0.69966 0.04989 2.9509 (0.57950, 0.82272) 0.24322 0.40546
25 MLE 0.06594 0.70261 0.06433 3.0153 (0.59606, 0.81409) 0.21803 0.40846
LSE 0.76715 1.40514 5.5511 (0.59732, 1.10873) 0.51141 0.51002
WLSE 1.18760 1.19803 4.4456 (0.64949, 2.35030) 1.70080 1.48190
CVME 0.88257 1.91560 4.5144 (0.60868, 2.04250) 1.43384 0.79120
ADTE 2.12010 0.43055 3.1391 (1.19456, 3.26940) 2.07490 4.50210
RTADE 0.80806 0.91228 4.0493 (0.62462, 1.10080) 0.47618 0.56576
BEJP 0.70824 0.00779 3.0144 (0.59604, 0.82048) 0.22443 0.41578
BERP 0.69908 0.04375 2.9620 (0.59201, 0.80796) 0.21595 0.40392
30 MLE 0.06594 0.70260 0.00100 3.0736 (0.60123, 0.80618) 0.20495 0.40800
LSE 0.77008 1.21663 4.8143 (0.60656, 1.08910) 0.48255 0.51133
WLSE 1.19920 1.09693 4.2720 (0.66685, 2.24140) 1.57450 1.47740
CVME 0.87902 1.74380 3.9542 (0.61608, 1.84880) 1.23268 0.76143
ADE 2.12450 0.38145 3.1384 (1.25815, 3.18710) 1.92890 4.47830
RTADE 0.80702 0.91331 4.2439 (0.63572, 1.07120) 0.43552 0.56199
BEJP 0.70799 0.06095 2.9868 (0.60941, 0.81023) 0.20083 0.41488
BERP 0.69952 0.04044 2.9881 (0.60175, 0.79953) 0.19779 0.40396
50 MLE 0.06594 0.70260 0.00375 2.9379 (0.62288, 0.78021) 0.15733 0.40693
LSE 0.76966 0.92481 3.9304 (0.63230, 0.99682) 0.36452 0.50438
WLSE 1.19470 0.82282 3.6438 (0.72147, 1.98080) 1.25940 1.38740
CVME 0.88226 1.32040 4.5029 (0.64110, 1.45050) 0.80938 0.72926
ADE 2.12650 0.29756 3.1288 (1.43097, 2.92490) 1.49400 4.39110
RTADE 0.80974 0.67315 3.7464 (0.66469, 1.00980) 0.34512 0.56114
BEJP 0.70827 0.03773 2.9884 (0.63086, 0.78671) 0.15586 0.41418
BERP 0.69893 0.02658 2.9961 (0.62219, 0.77623) 0.15404 0.40222

Estimates of CNpmk when α = 1.6 and β = 2.1.

n Method CNpmk↓ ĈNpmk β1 β2 BCI Width MSE
10 MLE 0.06447 0.76520 –0.05003 2.9246 (0.57345, 0.95087) 0.37742 0.50045
LSE 0.81850 0.89867 4.2260 (0.56583, 1.21000) 0.64418 0.59582
WLSE 1.13010 2.18072 6.4322 (0.59444, 2.83610) 2.24160 0.50530
CVME 0.80843 0.28310 3.1917 (0.59317, 1.05497) 0.46179 0.56711
ADE 2.65660 0.96581 4.1162 (1.12640, 5.25460) 4.12830 0.86150
RTADE 0.77954 0.04965 2.9953 (0.57525, 0.98999) 0.41473 0.52244
BEJP 0.77160 -0.02646 2.9416 (0.57862, 0.96259) 0.38397 0.50966
BERP 0.76141 –0.01763 2.8989 (0.57168, 0.94891) 0.37722 0.49506
15 MLE 0.06447 0.76685 –0.04684 2.9497 (0.60518, 0.91555) 0.31037 0.49955
LSE 0.81620 0.74592 3.9361 (0.60310, 1.11770) 0.51459 0.58268
WLSE 1.11940 1.73378 6.3472 (0.64183, 2.31550) 1.67370 0.54150
CVME 0.81051 0.18594 3.1284 (0.63405, 1.00302) 0.36897 0.56553
ADE 2.64490 0.71706 3.5806 (1.31630, 4.59160) 3.27530 0.49640
RTADE 0.78041 0.06998 2.8889 (0.61646, 0.95028) 0.33382 0.52005
BEJP 0.77384 –0.02554 3.0099 (0.61491, 0.92996) 0.31506 0.50969
BERP 0.76110 –0.01516 2.9562 (0.60984, 0.91730) 0.30747 0.49156
20 MLE 0.06447 0.76474 –0.01665 2.9672 (0.62627, 0.90196) 0.27570 0.49536
LSE 0.81837 0.56310 3.4250 (0.62534, 1.07250) 0.44713 0.58137
WLSE 1.12390 1.47282 5.3546 (0.66532, 2.16600) 1.50070 0.49820
CVME 0.80825 0.20891 3.1193 (0.65435, 0.97936) 0.32501 0.55991
ADE 2.65500 0.72309 3.6753 (1.45220, 4.39540) 2.94320 0.49570
RTADE 0.77960 0.03223 2.9538 (0.63365, 0.92695) 0.29330 0.51703
BEJP 0.77350 -0.03561 2.9413 (0.63708, 0.90752) 0.27044 0.50759
BERP 0.76109 –0.01433 2.9773 (0.62683, 0.89429) 0.26746 0.48994
25 MLE 0.06447 0.76595 –0.04300 2.9863 (0.64311, 0.88684) 0.24373 0.49519
LSE 0.81662 0.54491 3.3650 (0.64402, 1.04300) 0.39898 0.57617
WLSE 1.12120 1.28389 4.7477 (0.68549, 2.06630) 1.38080 0.49310
CVME 0.80927 0.14946 3.0232 (0.66974, 0.95898) 0.28924 0.54017
ADE 2.65860 0.62683 3.5024 (1.54360, 4.18580) 2.64220 0.49030
RTADE 0.78030 0.08275 3.0305 (0.65132, 0.91397) 0.26265 0.51693
BEJP 0.77202 –0.02624 2.9754 (0.64868, 0.89473) 0.24604 0.50450
BERP 0.76147 –0.00103 2.9677 (0.64232, 0.88090) 0.23858 0.48951
30 MLE 0.06447 0.76601 –0.01087 2.9115 (0.65454, 0.87549) 0.22095 0.49541
LSE 0.81842 0.51678 3.4372 (0.65736, 1.02710) 0.36976 0.57732
WLSE 1.12170 1.22006 4.6331 (0.70872, 1.96350) 1.25480 0.48970
CVME 0.80873 0.13459 2.9765 (0.68134, 0.94577) 0.26443 0.55842
ADE 2.65750 0.53615 3.2677 (1.62500, 4.00850) 2.38350 0.48940
RTADE 0.78037 0.07493 3.1438 (0.66205, 0.90231) 0.24025 0.51629
BEJP 0.77432 –0.01194 3.0289 (0.66404, 0.88545) 0.22141 0.50711
BERP 0.76121 –0.01693 2.9651 (0.65142, 0.86941) 0.21799 0.48854
50 MLE 0.06447 0.76573 0.03407 2.9812 (0.68060, 0.85301) 0.17241 0.49367
LSE 0.81800 0.41427 3.3362 (0.68925, 0.97190) 0.28265 0.57318
WLSE 1.12640 0.97227 4.0802 (0.75837, 1.77420) 1.01580 0.48901
CVME 0.80859 0.13744 3.0367 (0.71136, 0.91330) 0.20195 0.55638
ADE 2.65370 0.43342 3.2279 (1.82660, 3.66830) 1.84170 0.48940
RTADE 0.77986 0.06125 2.9552 (0.68746, 0.87480) 0.18734 0.51404
BEJP 0.77307 –0.05154 3.0901 (0.68593, 0.85740) 0.17146 0.50405
BERP 0.76142 -0.02131 2.9737 (0.67605, 0.84538) 0.16933 0.48761

Estimates of CNpmk when α = 2.1 and β = 1.6.

n Method CNpmk↓ ĈNpmk β1 β2 BCI Width MSE
10 MLE 0.47655 0.52159 0.41323 3.2662 (0.38109, 0.69146) 0.31038 0.23090
LSE 0.52237 0.43211 3.3533 (0.38299, 0.69148) 0.30848 0.23159
WLSE 0.73258 3.20300 3.8283 (0.39077, 2.6616) 2.27087 0.89789
CVME 0.52955 0.39481 3.2154 (0.38489, 0.70508) 0.32018 0.23883
ADE 0.81543 2.58380 6.2601 (0.39641, 2.3826) 1.98622 0.91452
RTADE 0.53082 0.48359 3.3701 (0.38389, 0.7166) 0.33271 0.24066
BEJP 0.52417 0.39720 3.1479 (0.3834, 0.69498) 0.31158 0.23348
BERP 0.52103 0.40900 3.1885 (0.38383, 0.69076) 0.30693 0.23025
15 MLE 0.47655 0.52163 0.36142 3.1844 (0.4056, 0.65967) 0.25407 0.22883
LSE 0.52445 0.36119 3.1353 (0.40878, 0.66411) 0.25533 0.23156
WLSE 0.74171 2.51340 6.5179 (0.41653, 1.986) 1.56946 0.77982
CVME 0.52867 0.38010 3.2944 (0.41127, 0.66843) 0.25716 0.23569
ADE 0.81296 2.18060 8.3435 (0.43139, 1.9793) 1.54787 0.80397
RTADE 0.53097 0.40413 3.1628 (0.40792, 0.68294) 0.27502 0.23856
BEJP 0.52362 0.35439 3.1544 (0.40812, 0.66204) 0.25392 0.23082
BERP 0.52140 0.32376 3.1292 (0.40521, 0.65774) 0.25254 0.22860
20 MLE 0.47655 0.52329 0.30568 3.1316 (0.42129, 0.64088) 0.21959 0.22948
LSE 0.52395 0.32337 3.2239 (0.42157, 0.64472) 0.14108 0.23011
WLSE 0.73482 2.14570 6.4165 (0.4334, 1.6568) 1.22340 0.68507
CVME 0.52997 0.31707 3.1633 (0.42624, 0.65139) 0.22515 0.23587
ADE 0.81264 1.79040 6.3876 (0.45231, 1.7421) 1.28976 0.74580
RTADE 0.53137 0.36075 3.3029 (0.42238, 0.66036) 0.23798 0.23770
BEJP 0.52447 0.29892 3.0931 (0.42105, 0.6422) 0.22114 0.23053
BERP 0.52122 0.28793 3.1155 (0.42022, 0.63795) 0.21773 0.22736
25 MLE 0.47655 0.52211 0.23235 3.0287 (0.42975, 0.62682) 0.19708 0.22764
LSE 0.52414 0.30262 3.2097 (0.43216, 0.6303) 0.22314 0.22960
WLSE 0.73886 1.91220 6.6826 (0.44341, 2.1052) 1.66176 0.65213
CVME 0.52959 0.24340 3.0745 (0.43585, 0.63508) 0.19923 0.23486
ADE 0.80790 1.70400 6.3697 (0.46904, 1.8337) 1.36469 0.70554
RTADE 0.53154 0.32597 3.0502 (0.43608, 0.64457) 0.20849 0.23706
BEJP 0.52451 0.21781 3.0358 (0.42853, 0.62985) 0.20132 0.23000
BERP 0.52084 0.26573 3.1086 (0.43039, 0.62451) 0.19412 0.22636
30 MLE 0.47655 0.52222 0.23819 3.0753 (0.43771, 0.61769) 0.17998 0.22732
LSE 0.52343 0.24792 3.0193 (0.43993, 0.61942) 0.19814 0.22847
WLSE 0.73980 1.75380 5.9867 (0.45367, 1.9053) 1.45165 0.62549
CVME 0.52952 0.20622 3.0336 (0.44321, 0.62435) 0.18114 0.23436
ADE 0.81393 1.50990 5.6278 (0.47725, 1.7014) 1.22411 0.69531
RTADE 0.53137 0.27704 3.0533 (0.44112, 0.63541) 0.19429 0.23645
BEJP 0.52330 0.21565 3.0883 (0.43894, 0.61768) 0.17873 0.22831
BERP 0.52107 0.24305 3.0738 (0.43795, 0.61535) 0.17741 0.22617
50 MLE 0.47655 0.52254 0.13895 2.9262 (0.45583, 0.59356) 0.13772 0.22679
LSE 0.52413 0.22964 3.2282 (0.45667, 0.59775) 0.17949 0.22830
WLSE 0.73327 1.41750 5.1888 (0.47236, 1.3974) 0.92502 0.55478
CVME 0.52948 0.19328 3.0662 (0.46251, 0.6029) 0.14039 0.23346
ADE 0.81536 1.18070 4.6248 (0.51149, 1.4187) 0.90719 0.65512
RTADE 0.53153 0.20504 2.9924 (0.4602, 0.60997) 0.14977 0.23559
BEJP 0.52374 0.20226 3.1459 (0.45686, 0.59659) 0.13973 0.22792
BERP 0.52074 0.19038 3.0383 (0.45519, 0.59291) 0.13772 0.22505

The results presented in Tables 2–9, all demonstrate that with increase in sample size, both average width and MSE are reduced for all PCIs against all estimation methods, suggesting that a small sample size, n < 25 and moderately large (30, 50) sample produces better results.

If we look more closely, we can find that the BERP has minimum MSEs for small to moderate sample sizes when compared to MLE, LSE, WLSE, CVME, ADE, RTADE, and BEJP. Moreover, the MSE are least for BERP for both PCIs Cpy and CNpmk. Furthermore, in this investigation BERP outperformed other estimating methods by producing the least width of estimates for the configurations under consideration.

 7 Real data illustration

This section examines the actual dataset comprising of the failure timings of 20 electric carts, as presented by Rao et al. [24], listed in Table 10. To evaluate the BCIs of PCIs the estimation methods ML, LS, WLS, CVM, AD, RTAD, BEJP, and BERP are utilized. Initially, we evaluate if the analyzed data set originates from a FD by examining the p-value of the Kolmogorov-Smirnov (KS) test, Anderson-Darling (AD) and Cramer-von Mises (CVM) test. The point estimates for the data set are α = 0.9070176, β = 5.2836813. The statistic along with p-values of corresponding goodness of fit test are summarized in Table 11 and complemented the analysis with graphical diagnostics, density and PP plots in Figure 2. Which are evident that this data set follows two parameter FD quite well. To estimate the PCIs, the USL and LSL, were set to 53.0 and 0.90 respectively and p0 = 0.95. The mean of the data set, T = 26.95, is set as the target value. Table 12 shows the BCI widths of the PCIs Cpy and CNpmk based on the aforementioned estimation techniques for FD. It is clear from Table 12 that BERP has a narrow BCI than its counterparts. BERP is a more effective method for employing PCI interval estimations. For all PCIs, BERP outperformed MLE and BEJP and all aforementioned methods in terms of BCI width. The distribution shape of PCIs is approximately normal. This study suggests that BERP is more effective for examining observed PCIs.

Lifetime (in months) for failure of 20 electric carts.

0.90 1.50 2.30 3.20 3.90 5.00 6.20 7.50 8.30 10.40
11.10 12.60 15.00 16.30 19.30 22.60 24.80 31.50 38.10 53.00

The goodness-of-fit test for FD.

Goodness-of-fit criteria Statistic p-value
KS test D = 0.13296 0.8264
CVM test W2 =0.08647 0.6604
AD test A2 = 0.55865 0.6858

PP and density plots of FD.

P-P plot and density plot. The P-P plot shows empirical CDF against fitted FD CDF with data points aligning along the diagonal. The PDF chart displays failure time in months with a red curve over gray bars, illustrating a decreasing trend in probability density.

Estimates of PCIs.

Method Ĉpy BCI Width β1 β2
Cpy
MLE 0.92078 (0.89945, 0.93904) 0.03959 –0.13548 2.77005
LSE 0.91166 (0.63039, 1.05263) 0.42224 –0.74486 2.98523
WLSE 0.89512 (0.68421, 1.05263) 0.36842 –0.52125 3.09433
CVME 0.87545 (0.62719, 1.05059) 0.42340 –0.85247 3.48811
ADE 1.02326 (0.99105, 1.04822) 0.05717 –0.13970 2.50095
RTADE 0.71724 (0.52361, 0.90628) 0.38267 –0.13970 3.05548
BEJP 0.92516 (0.90316, 0.93872) 0.03556 –0.57699 3.08739
BERP 0.91571 (0.89684, 0.93173) 0.03489 –0.34767 2.34959
CNpmk
MLE 0.23669 (0.33183, 0.17723) 0.15461 0.78127 3.63986
LSE 0.24974 (0.37872, 0.17924) 0.19948 0.99275 3.88410
WLSE 0.25439 (0.38106, 0.17974) 0.20131 0.87826 3.51100
CVME 0.25045 (0.37489, 0.17886) 0.19604 0.97198 4.09305
ADE 0.25037 (0.37252, 0.17733) 0.19518 0.96169 4.09897
RTADE 0.25016 (0.36858, 0.18292) 0.18566 0.89554 3.61039
BEJP 0.23583 (0.32502, 0.17629) 0.14873 0.75081 3.55995
BERP 0.23581 (0.32371, 0.17853) 0.14518 0.85753 3.94472
8 Conclusion

This article examines two PCIs: Cpy and CNpmk, using various estimating approaches for FD. In general, we used the MLE, LSE, WLSE, CVME, ADE, RTADE, BERP, and BEJP to produce estimates and BCIs for the aforementioned PCIs. Extensive simulation studies were conducted to compare these approaches with various small to moderately large sample sizes and parametric values. The average estimates for these PCIs, along with their coefficient of skewness, kurtosis, BCIs, and corresponding MSEs, have been acquired. The MSE of CNpmk is significantly less than that of Cpy. Furthermore, simulation findings indicated that BERP outperformed the other estimators for MSEs and the width of PCIs. The examination of the real data set validated this finding. This study aims to help engineering professionals and applied statisticians.

 Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

 Author contributions

TK: Formal analysis, Methodology, Software, Writing – original draft, Conceptualization. KA: Project administration, Supervision, Software, Writing – review & editing. MZ: Resources, Writing – review & editing. ZH: Writing – review & editing, Resources, Methodology.

 Funding

The author(s) declare that no financial support was received for the research and/or publication of this article.

 Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

 Generative AI statement

The author(s) declare that no Gen AI was used in the creation of this manuscript.

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