CC BY
21
ABHIMANYÜ S YADAV, MAHENDRA SAHA, AMARTYA BHATTACHARYA,
ARINDAM GUPTA RT&A No 4 (76)
ASSESSMENT OF GLPI FOR LD Volume 18' December 2023
ASSESSMENT OF GENERALIZED LIFETIME PERFORMANCE INDEX FOR LINDLEY DISTRIBUTION USING PROGRESSIVE TYPE-II SAMPLES
ABHIMANYU S YADAV1, MAHENDRA SAHA2, AMARTYA BHATTACHARYA3, ARINDAM GUPTA4
•
1 Department of Statistics, Banaras Hindu University, Varanasi, India. [email protected] 2 Department of Statistics, Central University of Rajasthan, Ajmer, India. [email protected] 3 National Atlas and Thematic Mapping Organisation, GoI, Kolkata, India. [email protected] 4 Department of Statistics, The University of Burdwan, Burdwan, India. [email protected]
Abstract
A meaningful subject of discourse in manufacturing industries is the assessment of the lifetime performance index. In manufacturing industries, the lifetime performance index is used to measure the performance of the product. A generalized lifetime performance index (GLPI) is defined by taking into consideration the median of the process measurement when the lifetime of products follow a parametric distribution may serve better the need of quality engineers and scientists in industry. The present study constructs various point estimators of the GLPI based on progressive type II right censored data for the Lindley distributed lifetime in both classical and Bayesian setup. We perform Monte Carlo simulations to compare the performances of the maximum likelihood and Bayes estimates with a gamma prior of Cy (L) under progressive type-II right censoring scheme. Finally, the validity of the model is adjudged through analysis of a data set.
Keywords: Bayesian estimation, Metropolis-Hastings method, Process Capability Index, Maximum likelihood estimator.
1. Introduction
Process Capability Indices (PCIs) have wide use in industries for evaluating a manufacturing process and whether or not it can produce articles within the specified limits. PCIs aim to quantify the capability of a process (X) to meet some specifications related to a measurable characteristic of its produced items. These specifications are determined through the lower specification limit (L), the upper specification limit(U) and the target value (T). PCI is an effective means to measure a process's performance and potential capabilities. In the manufacturing industry, PCIs are utilized to assess whether product quality meets customer expectations. Since capability is typically defined in dictionaries as the ability to carry out a task or achieve a goal, a better process capability implies better product quality. If the process capability is evaluated with product survival lifetime, it is clear that a larger lifetime means better product quality, higher reliability, and the process is capable. Hence, the lifetime of products exhibit the larger- the better quality characteristic of time
ABHIMANYU S YADAV, MAHENDRA SAHA, AMARTYA BHATTACHARYA, RT&A No 4 (76)
ARINDAM OTPIA Volume 18, December 2(02633
ASSESSMENT OF GLPI FOR LD_
orientation. It should be noted that the lifetime of products does not follow a normal distribution. For instance, [14], [34], [6], [24] and [17] pointed out that the product lifetime possesses an exponential distribution. The description of the lifetime by the Weibull distribution was noted by [34], [40], [41] and [16]. In addition, [34] also mentioned that lifetime follows a gamma distribution. Since the lifetime of products exhibits the larger-the-better quality characteristic of time orientation, [35] and [23] recommended the use of the capability index (lifetime performance index) for evaluating the lifetime performance of electronic components, where L is the lower specification limit. Also, there are many different PCIs available in the literature. The hypothesis testing procedures are developed by [20] and [45] using the maximum likelihood estimator of PCI for Pareto distribution under type-II censored sampling and progressive type-I interval censoring, respectively. A hypothesis testing procedure was developed by [42] for the PCI of the Gompertz distribution based on the progressive type-I interval censored sample. The MLE was used by [46] to estimate the PCI of Rayleigh distribution based on the progressive type-I interval censored sample and developed a new hypothesis testing procedure utilizing an asymptotic distribution of this estimator. Some classical estimations and bootstrap confidence interval methods for the PCIs are derived by [36, 37] when the process follows exponentiated exponential and normal distributions, respectively. The classical and Bayes estimates of PCIs are obtained by [12,13] for generalized exponential and normal distributions, respectively. For an expository review, the reader may follow the following articles of bibliography of the literature on PCIs, viz., [25], [38], [48] and [1].
The lifetime performance index (LPI) is defined by [23], denoted by Cl, which mainly originated from the concept of symmetry of lifetime distributions. The uniformly minimum variance unbiased estimator (UMVUE) for CL was obtained by [39] and considered the problem of the hypothesis testing procedure for the exponential distribution. The UMVUE of CL to develop the confidence interval under exponential distribution was obtained by [9]. The maximum likelihood estimator of the lifetime performance index based on first-failure progressive right type-II censored sample for Lindley distribution was obtained by [18]. The maximum likelihood estimates of the lifetime performance index based on progressive first failure censoring scheme Weibull, exponential and two-parameter exponential distributions, were obtained by [2, 3, 4] respectively. The lifetime performance index of products based on progressively Type-II censored for the Pareto samples was evaluated by [5]. The MLE, some Bayesian estimators, and credible intervals were given by [49] for the lifetime performance index of the Pareto distribution based on the general progressive type-II censored data. Approximate and exact parametric bootstrap confidence intervals are proposed by [50] for the process performance index of power-normal distribution. Most of the time, lifetime distributions are not necessarily symmetric. In this case, the median of the process distribution plays an important role than the process mean Therefore, it should be better if the index deals with the distribution's median (^e). If ^ is replaced by the inferential aspects and their property studies will be somewhat complicated. Using the median, [32] proposed a generalized process capability index (GPCI) that is the ratio of the proportion of specification conformance (or process yield) to the proportion of desired (or natural) conformance. In the same tune, [33] defined a GLPI given as
Cy(L) = ^ C)
_ 1 - 2F(L)
= 1 - 2a '
Here F( ) denotes the cumulative distribution function (CDF) of the process distribution and a = P(X < LDL) with LDL being the lower desirable limit (practitioners sometimes take it as a lower tolerance limit). Here (1 — a) is the confidence level close to unity. Statistical inference for CY(L), viz., properties of GLPI order, testing procedure for GLPI and parametric bootstrap confidence intervals based on a complete sample for the Lindley and in particular for exponential distribution have been obtained by [33].
abhimanyu s yadav, mahendra saha, amartya bhattacharya, rt&A No 4 (76)
arindam gufta Volume 18, December 20263) ASSESSMENT OF GLPI FOR LD_
In the case of a complete sample, it is necessary to continue the experiment until the last item (or product) fails. Sometimes, many articles have very long lifetimes, and the experiment continues for a very long period, so that the results may be of little interest or use. Then, it may be desirable to terminate the test before all the items under test fail, and the resulting observations will be called the censored sample. Various types of censored samples exist, including type II, progressive type II, and progressive first-failure censored samples. The testing of the hypothesis problem was proposed by [43] based on the maximum likelihood estimator (MLE) of CL for two-parameter exponential distribution under type-II right censored sample. Based on a type-II right censored sample, the confidence interval using Pareto distribution was obtained by [20, 21]. A hypothesis testing procedure was proposed by [27, 28] based on MLE and UMVUE with the exponential distribution under progressive type-II right censored samples, respectively. The MLE of CL was obtained by [19] under progressive first-failure censored samples from two-parameter exponential distributions. The CL for the exponential lifetime products are evaluated by [29, 30] based on type-II censored data. They obtained Bayes's estimate of CL for the Rayleigh lifetime products based on upper record values, respectively. Recently, [11] assessed the lifetime performance index for Weibull distributed products based on progressive type-II right censored samples.
In this article, we consider a progressive type-II right censoring scheme, which is helpful in a specific fraction of individuals at risk that may be removed from the experiment at each of several ordered failure times. Therefore, a progressive censoring scheme allows us to incorporate the removals before the experiment's termination into analysis, which is a very common situation in life-testing experiments. To the best of our knowledge thus far, an attempt has yet to be made to study the GLPI CY(L) based on a progressive type-II right censoring scheme. Filling up this gap is the aim of the present study. In this article, we consider the GLPI CY(L), introduced by [33] that could be used for either normal or non-normal and either continuous or discrete characteristics and is very simple and could be used comfortably by the practitioners.
The paper is arranged as follows. In section 2, The MLE and the Bayes estimate of Cy(L) are suggested based on progressive type II right censored sample for the Lindley distributed lifetime. In section 3, the testing procedure due to the GLPI is done. In section 4, an extensive Monte Carlo study is carried out to compare the performances of CY(L) based on considered methods of estimation (MLE and Bayes) in terms of their corresponding mean squared errors (MSEs). A real-world application has been discussed to illustrate the proposed index under progressive type-II right censored samples in section 5. A brief concluding remark is made in section 6.
2. Estimation of cy (l) under progressively type-II censored sample for
Lindley products
Suppose that the lifetime of products may be modeled by Lindley distribution and let X denote the lifetime of such product. Hence, the probability density function (PDF) and cumulative distribution function (CDF), specified by Lindley distribution (see, [31]) are given as
a2
f (x) = f^T (1 + x)e-xe, x > 0, 6 > 0 (2)
6 + 1
and
F(x) = 1 - 1 +6 + X° e-x6, x > 0, 6 > 0 (3)
6 +1
ABHIMANYU S YADAV, MAHENDRA SAHA, AMARTYA BHATTACHARYA, RT&A No 4 (76)
ARINDAM OTPIA Volume 18, December 2(02633 ASSESSMENT OF GLPI FOR LD_
respectively, where, 0 is the parameter. Now, for a process whose distribution can be regarded as Lindley, the GLPI is given as
Cy(L) = {^}
1 _ 2(1 - 1+|0+0Le_0L 1 -2a
(4)
Here, in the following subsections, we derived the maximum likelihood estimate (MLE) and the Bayes estimate of Cy(L) under progressively type-II right censoring scheme for Lindley distributed products, respectively.
2.1. Maximum likelihood estimate of Cy (L)
The experimenter may not always observe the lifetimes of all the products (or items) put on tests for conducting life testing experiments. The reason may be time limitation and/or other restrictions (such as money, mechanical or experimental difficulties, material resources, etc.) on data collection. Therefore, censored samples may arise in practice. In an industrial experiment, products (or items) may break accidentally. These lead us into the area of progressive type-II censoring. Under this scheme, n units are placed on test at time zero, and m failures are observed. When the first failure is observed, ri of the surviving units are randomly selected and removed. At the second observed failure, r2 of the surviving units are randomly selected and removed. Termination of the experiment occurs when the m-th failure is observed, and the remaining rm = n _ ^m-1 ry _ m surviving units are all removed. Inferences for the data obtained by progressive censoring have been investigated, among others, by [10], [7], [15], and [44]. So, in this paper, we consider the case of the progressive type-II right censoring.
Let X1:m:n, X2:m:n, ..., xm:m:n be a progressive type-II right censored sample where X1:m:n, X2:m:n, ..., xm :m :n denote the observed failure times and r1, r2, ..., rm denote the corresponding numbers of items removed (withdrawn) from the test. If m be the number of failures observed before termination, then x1:m:n < x2:m:n < ..., < xm:m:n be the observed ordered lifetimes. For convenience, we will write xi:m:n as X(j). Let r denote the number of items removed at the time of the ith failure, 0 < r, < n _ £=1 ry _ i, i = 2, 3, ..., m _ 1 with 0 < r1 < n _ 1 and rm = n _ E^1 ry _ m, where r, 's and m are pres-specified integers [see, Viveros and Balakrishnan (1994)]. The complete sample ( r1 = r2 = ... = rm = 0 ) and type-II right censored samples ( r1 = r2 = ... = rm_1 = 0, rm = n _ m) are special cases of this scheme. For further details and for relevant references the reader may follow the article of [7]. The likelihood function of 0 under progressive type-II right censoring scheme is given by
m m_1
1(0) = A n f (*(o; 0) [1 _ F(*(o; 0)]ri, where A = n(n _ r1 _ 1)...(n _ E ry _ m + 1)
i=1 y=1
02m m m m
= A. (1 + 0)m+Em=1, e_0Em=1 (1+ri)x(i) n(1 + X(i)) n(1 + 0 + 0X(i))ri (5)
Therefore, the log-likelihood function is given by
mm
L(0) = ln 1(0) = k + 2mln0 _ (m + £ ri) ln(1 + 0) _ 0 £(1 + ri)x(i)
i=1 i=1
m
+ £ ri ln(1 + 0 + 0X(i)), (6)
i=1
abhimanyu s yadav, mahendra saha, amartya bhattacharya, rt&A No 4 (76)
arindam gupta Volume 18, December 202633 ASSESSMENT OF GLPI FOR LD_
where k is constant, independent of 6. Now, for MLE of the parameter 6, ^dr = 0
2m m + ri ~9
i + e
de
ra + r )x + r ri(1 + x(i)} - 0 - g(i+ri )x(i) + Ei+j+exo -
(7)
The explicit solution for the parameter 9 through the above non-linear equation is not possible. Hence, to solve this equation for 9, we have to proceed by some numerical method, from the previous equation,
d2L(e) _ 2m m + Ej-i r
de2
-W +
- r
ri(1 + x(i))2
(1 + e)2 i=i (1 + e + e x^)
2
Hence, the Fisher's information I(e) is obtained as;
t (e)- E( d2 L(e))- 2m m + rm-i r> + mrE 1 (e) - ) - 02 - (i + e)2 + r r'E
e2
(i + x(q)2 (i + e + e X{1))2
(8)
Further, after obtaining the solution above mentioned non-linear equation, the MLE of the index CY(L) can be directly computed using invariance property of MLE. Let ! be the solution of the non-linear equation, then the MLE CY(L) of CY(L) is given by
Cy (L)
i - 2 ( i - i+e+eLe-eL
i -2a
(9)
The asymptotic distribution of MLE 6 is normal distribution N(6,1 1 (6)). Hence, the two-sided tail asymptotic 100(1 — a)% confidence interval for the parameter 6 is given by
e t tXii-1 (e) j,
where I(e) - 2m - + rm-1
rj (1+X(i))2
and Ta/2 is the upper a/2-point of standard
62 (\+o)2 1 (l+e+exi{))2 normal deviate. The asymptotic distribution of MLE Cy (L) is also normal distribution N(Cy(L), Var(Cy(L)) with
Var(CY (L))
dCY(L)
de
x Var(e)
-2e L{1 + (1 + e)(1 + L)}e (1 - 2a)(1 + e)2
-eL
x i-1(e).
The asymptotic variance is approximated as
Var(CY (L))
-2d L{1 + (1 + e)(1 + L)}e (1 - 2a) (1 + e)2
-eL
x i-1 (e).
2.2. Bayes estimate of Cy (L)
In this section, we obtain the Bayes estimator of Cy (L) under the assumption that the parameter 6 is random variable and follows some prior distribution. Let the prior distribution of is assumed to be Gamma with parameter (k,a). Then the distribution of 6 is given as
g(e)
r(k)
e-aeek-\ e> o
Now, the posterior distribution of by using Equations (2.5) and (2.9) is given as
g(e | x) a
2m+k-
(i + e)m+rm-i r
m
-e-e(a+r-i(1+ri)x(i)) n(i + e + ex(l)y ; e > o.
i=1
(10)
(11)
2
2
2
k
a
1
abhimanyu s yadav, mahendra saha, amartya bhattacharya, rt&A No 4 (76)
arindam gupta Volume 18, December 2(02633 ASSESSMENT OF GLPI FOR LD_
Hence, the Bayes estimate of the parameter 0 under squared error loss function is obtained by the following Equation:
E(0) = ZI (1 + g)m+Em=1, )x(i)) n(l + 0 + ^(i))"'- do (12)
where Z is a proportionality constant. The computation of the Bayes estimate of the index CY(L) under the same assumption of prior and loss function is not possible directly from the above posterior distribution. Since, the explicit form of the posterior PDF is not available but the associated plot exhibit a more or less assume the shape of normal probability distribution. Thus, the Metropolis-Hastings method with normal proposal distribution is to be used to generate random numbers from respective posterior distribution using the Gibbs algorithm. The following steps are taken to generate the posterior random deviates from the above posterior is as follows:
• Start with an initial guess
• Set t = 1.
• Using the Metropolis-Hastings, generate 0(t) from g(0 | x) with the N ^0(t-1), 1 j proposal distribution.
• Compute CY(L)(t) from Equation (1)
• Set t = t + 1.
• Repeat steps 3-5, T times.
Note that in step 3, we use the Metropolis-Hastings algorithm with q(0(t-1), o2) proposal distribution as follows:
1. Let x = 0(t-1).
2. Generate y from the proposal distribution q.
3. Let p(x,y) = min (l, ÄM^.
'go (x)q(y)y
4. Accept y with the probability p(x, y) or accept x with the probability 1 — p(x, y).
The posterior deviates for CY(L) using the above mentioned steps is simulated using the random deviates of 0 by plug-in principal. Let CY(L)1, CY(L)2, ■ ■ ■ , CY(L)T be the T simulated posterior deviates, then the approximate posterior mean, and posterior variance of CY(L) are given by
1 ^
and
respectively.
Ê(Cy(L)|x) = T £ Cy(L) 1 t=i
1 t 2
MSE(Cy(L)|x) = T £ (Cy(L)f - Cy(L))2 1 t=i
3. Testing procedure for the generalized lifetime performance index
using progressive type-II samples
In this section, following statistical hypothesis testing will be performed to access whether the CY(L) adheres the required level. The proposed hypothesis testing procedure using progressive type-II samples can be performed for CY(L), summarized as follows:
abhimanyu s yadav, mahendra saha, amartya bhattacharya, rt&A No 4 (76)
arindam gupta Volume 18, December 2023 ASSESSMENT OF GLPI FOR LD_
1. Determine the lower specification limit L and the GLPI, C°.
2. Construct the null hypothesis H° : CY(L) < CY against the alternative H : CY(L) > C°.
3. Specify the level of significant a.
4. Compute Cy(L) and MCy(L)) - [—20L{;+(1+)0(i++)L)}e—^]2.1—1 (0).
5. Set critical region w : CY(L) > C^ + r^Var(CY(L)).
4. Simulation and discussion based on progressively type-II censored
sample.
In this section, a comparison study has been carried out through simulation study under progressively type II censoring scheme between maximum likelihood and the Bayes estimate of Cy(L) in terms of their corresponding mean squared errors (MSEs) for the considered set up. All the calculations have been made by using R software (see, [22]). The progressive type-II right censored sample from the considered lifetime distribution for the different variation of the parameter (0), censoring parameters (n, m), censoring schemes (r's) and lower desired limit (L) is generated by following the algorithm suggested by [8] algorithm, as stated below:
1. Generate m independent Uniform(0,1) observations Wi, W2, ..., Wm.
2. Set V = wf/(i+rm+rm-i+...+rm-'-+i) for i = 1,2,..., m.
3. Set Ui = 1 — Vm.Vm_1...Vm_i+1 for i = 1,2,..., m. Then U1, U2,..., Um is the required progressive type-II censored sample from the Uniform (0,1) distribution.
4. Finally, we set Ui = 1 — ^g+f'0 e—Xi0, and solve this equation by Newton-Raphson method to get X' for i = 1,2,..., m. Then X1, X2,..., Xm is the required progressive type-II censored sample from the distribution (2).
The simulated MSE of the MLE and the Bayes estimate of Cy(L) have been presented in Tables 1-6 for different censoring schemes with some particular choices of 0 and L. From the Tables 1-6, it is found that the MSE of each estimator is decreasing with n, the sample size. This verifies the consistency property of all the estimators. It is also observed that the performance of the Bayes estimation is relatively better than the MLE under all the considered choices.
5. Appiications
A real data set is cited to illustrate the MLE, Bayes estimate of LPI CL (see, [23]) and the proposed GLPI Cy(L) for progressive type-II right censoring scheme for the Lindley distributed lifetime. The considered data set is described in detail by [33] (also available in [26]), and the goodness of fit test to Lindley distribution is discussed therein. The data set is primarily fitted to the exponential model in Lawless. [33] have checked the data set with the Lindley distribution and found it to be a better fit. Thus, from the same data set, the progressive type-II censored data are generated for the different values of m = 1°, 15, L=100 and the censoring schemes, and the corresponding MLE and Bayes estimates of Cy(L) is reported in the Table 7. The Bayes estimates for the real data set is computed under non-informative prior.
Now, the proposed testing procedure of the GLPI CY(L) is performed for the above chosen schemes as follows. For the considered data set, the progressive type-II censored data are generated for the same censoring schemes, mentioned in Table 8, respectively. The MLE of the parameter 0 for the Lindley distribution is obtained from the Eqn. (2.7) for all the considered schemes and the same are reported in Table 8. The statistical test for testing the null hypothesis H° : CY(L) < 1 against the alternative hypothesis H1 : CY(L) > 1 has been performed for the
arindam gupta Volume 18, December 202633
ASSESSMENT OF GLPI FOR LD
Table 1: MLEs of Cy (L) and their MSEs with L = 0.1 and L = 0.3, samples generated from the Lindley distribution for 6 = 0.50 under progressively type-II censoring scheme [True Cy (L) = 1.073193 when L = 0.1 and Cy(L) = 0.992842 when L = 0.3].
2[0]*n,m 2[0]*Schemes L=0.1 L=0.3
CY (L)MLE MSE[Cy ( L)mle ] CY (L)MLE MSE[Cy ( L)mle ]
7[0]*10, 8 0*10 1.07058 0.00149 0.98230 0.01431
2, 0*7 1.06040 0.00316 0.99986 0.01086
1,1, 0*6 1.06686 0.00326 0.99877 0.01036
0*7, 2 1.06560 0.00272 0.98734 0.01451
0*6, 1*2 1.07203 0.00191 0.99791 0.00939
1, 0*6, 1 1.06470 0.00424 0.97192 0.01438
0*3,1*2, 0*3 1.06315 0.00349 0.99770 0.00973
7[0]*20, 16 0*20 1.07081 0.00072 0.99550 0.00439
4, 0*15 1.06834 0.00083 1.00409 0.00546
1*4, 0*12 1.06991 0.00108 0.99523 0.00572
0*15, 4 1.07087 0.00106 0.98304 0.00653
0*12, 1*4 1.07045 0.00138 1.00387 0.00463
2,0*14,2 1.06791 0.00096 1.06689 0.00177
0*7, 2*2, 0*7 1.06902 0.00092 0.99288 0.00449
7[0]*30, 24 0*30 1.07523 0.00048 0.99840 0.00301
6,0*23 1.06878 0.00068 0.97993 0.00392
1*6, 0*18 1.06788 0.00077 0.99505 0.00416
0*23, 6 1.06747 0.00088 0.99042 0.00439
0*18, 1*6 1.06858 0.00057 1.00475 0.00349
3, 0*22, 3 1.06891 0.00063 0.99489 0.00358
0*9,1*6, 0*9 1.07182 0.00050 0.98809 0.00379
7[0]*50, 40 0*50 1.07370 0.00028 0.99244 0.00218
10, 0*39 1.06526 0.00050 0.98831 0.00264
2*5, 0*35 1.07372 0.00028 0.98873 0.00256
0*39, 10 1.07123 0.00045 0.99057 0.00281
0*30, 1*10 1.06765 0.00052 0.99992 0.00220
5, 0*38, 5 1.07045 0.00032 0.98705 0.00375
0*18, 2*5, 0*17 1.07152 0.00038 0.98859 0.00329
Note: In the table, Scheme (0 * 3, r) indicates that at 1st, 2nd and 3rd failure, no active unit is withdrawn or removed but at 4th failure, r
active units are drawn or removed.
arindam gupta Volume 18, December 202633
ASSESSMENT OF GLPI FOR LD
Table 2: Bayes estimates ofCy (L) and their MSEs with L = 0.1 and L = 0.3, samples generated from the Lindley distribution for 6 = 0.50 under progressively type-II censoring scheme [True Cy (L) = 1.073193 when L = 0.1 and Cy (L) = 0.992842 when L = 0.3].
2[0]*n,m 2[0]*Schemes L=0.1 L=0.3
CY ( L) Bayes MSE[Cy (L) Bayes ] CY (L) Bayes MSE[Cy (L) Bayes]
7[0]*10, 8 0*10 1.07597 0.00135 0.97546 0.01372
2, 0*7 1.06649 0.00283 0.99215 0.01049
1,1, 0*6 1.07202 0.00310 0.99148 0.00999
0*7, 2 1.07260 0.00245 0.97842 0.01403
0*6, 1*2 1.07847 0.00168 0.98854 0.00911
1, 0*6, 1 1.07092 0.00416 0.96363 0.01415
0*3,1*2, 0*3 1.06895 0.00315 0.99046 0.00958
7[0]*20, 16 0*20 1.07378 0.00066 0.99127 0.00437
4, 0*15 1.07176 0.00077 0.99953 0.00533
1*4, 0*12 1.07317 0.00101 0.99098 0.00568
0*15, 4 1.07475 0.00098 0.97784 0.00658
0*12, 1*4 1.07413 0.00128 0.99871 0.00449
2,0*14,2 1.07175 0.00088 1.06302 0.00169
0*7, 2*2, 0*7 1.07239 0.00085 0.98837 0.00439
7[0]*30, 24 0*30 1.07714 0.00047 0.99551 0.00297
6,0*23 1.07127 0.00063 0.97659 0.00390
1*6, 0*18 1.07023 0.00072 0.99216 0.00409
0*23, 6 1.07036 0.00081 0.98673 0.00420
0*18, 1*6 1.07131 0.00052 1.00128 0.00341
3, 0*22, 3 1.07160 0.00059 0.99127 0.00345
0*9,1*6, 0*9 1.07408 0.00048 0.98499 0.00360
7[0]*50, 40 0*50 1.07486 0.00028 0.99075 0.00209
10, 0*39 1.06679 0.00047 0.98640 0.00240
2*5, 0*35 1.07516 0.00027 0.98682 0.00247
0*39, 10 1.07291 0.00044 0.98830 0.00273
0*30, 1*10 1.06939 0.00049 0.99764 0.00193
5, 0*38, 5 1.07212 0.00030 0.98480 0.00357
0*18, 2*5, 0*17 1.07286 0.00037 0.98670 0.00292
arlNdam gupta Volume 18, December 2023
ASSESSMENT OF GLPI FOR LD
Table 3: MLEs o/Cy (L) and their MSEs with L = 0.1 and L = 0.3, samples generated from the Lindley distribution for 0 = 0.75 under progressively type-II censoring scheme [True Cy(L) = 1.038898 when L = 0.1 and CY(L) = 0.891517 when L = 0.3].
2[0]*n,m 2[0]*Schemes L=0.1 L=0.3
CY (L)MLE MSE[Cy (L) MLE ] CY ( L)MLE MSE[Cy (L) MLE ]
10, ,8 0*10 1.02408 0.00490 0.90298 0.01973
2, 0*7 1.03005 0.00589 0.88015 0.02835
1,1,0*6 1.02637 0.00580 0.90158 0.02385
0*7, 2 1.02057 0.00629 0.91253 0.02451
0*6, 1*2 1.02306 0.00557 0.92649 0.02055
1, 0*6, 1 1.03016 0.00484 0.88536 0.03157
0*3,1*2, 0*3 1.01554 0.00776 0.90067 0.01950
20, 16 0*20 1.03593 0.00230 0.89003 0.01112
4, 0*15 1.03237 0.00296 0.88569 0.01545
1*4, 0*12 1.03649 0.00246 0.90247 0.01130
0*15, 4 1.02421 0.00318 0.89690 0.01192
0*12, 1*4 1.02827 0.00291 0.88864 0.01237
2,0*14,2 1.03430 0.00229 1.03440 0.00274
0*7, 2*2, 0*7 1.03789 0.00200 0.89309 0.01031
30, 24 0*30 1.03208 0.00159 0.88711 0.00637
6,0*23 1.03609 0.00180 0.89093 0.00982
1*6, 0*18 1.03655 0.00173 0.88934 0.00759
0*23, 6 1.03475 0.00164 0.89663 0.00958
0*18, 1*6 1.03212 0.00182 0.89877 0.00774
3, 0*22, 3 1.03816 0.00202 0.89079 0.01046
0*9,1*6, 0*9 1.03469 0.00183 0.89383 0.00753
50, 40 0*50 1.03908 0.00077 0.89749 0.00487
10, 0*39 1.03609 0.00114 0.89273 0.00453
2*5, 0*35 1.03278 0.00128 0.89426 0.00483
0*39, 10 1.03953 0.00110 0.90031 0.00463
0*30, 1*10 1.03555 0.00109 0.89815 0.00530
5, 0*38, 5 1.03578 0.00128 0.89571 0.00574
0*18, 2*5, 0*17 1.03867 0.00089 0.89162 0.00504
arindam gupta Volume 18, December 202633
ASSESSMENT OF GLPI FOR LD
Table 4: Bayes estimates ofCy (L) and their MSEs with L = 0.1 and L = 0.3, samples generated from the Lindley distribution for 6 = 0.75 under progressively Type-II censoring scheme [True Cy (L) = 1.038898 when L = 0.1 and Cy (L) = 0.891517 when L = 0.3].
2[0]*n,m 2[0]*Schemes L=0.1 L=0.3
CY (L) Bayes MSE[Cy (L) Bayes ] CY ( L) Bayes MSE[Cy (L) Bayes ]
10, 8 0*10 1.03297 0.00455 0.89338 0.01833
2, 0*7 1.03916 0.00579 0.87174 0.02609
1,1, 0*6 1.03618 0.00560 0.89210 0.02173
0*7, 2 1.03254 0.00582 0.90036 0.02258
0*6, 1*2 1.03506 0.00509 0.91270 0.01865
1, 0*6, 1 1.04124 0.00444 0.87482 0.02897
0*3, 1*2, 0*3 1.02560 0.00733 0.88980 0.01799
20, 16 0*20 1.04061 0.00223 0.88503 0.01078
4, 0*15 1.03777 0.00285 0.88032 0.01488
1*4, 0*12 1.04167 0.00237 0.89705 0.01079
0*15, 4 1.03095 0.00294 0.88995 0.01139
0*12, 1*4 1.03464 0.00272 0.88194 0.01196
2,0*14,2 1.04039 0.00220 1.02836 0.00293
0*7, 2*2, 0*7 1.04343 0.00193 0.88711 0.00992
30, 24 0*30 1.03536 0.00152 0.88355 0.00628
6,0*23 1.03974 0.00175 0.88705 0.00954
1*6, 0*18 1.04011 0.00168 0.88558 0.00743
0*23, 6 1.03937 0.00157 0.89173 0.00930
0*18, 1*6 1.03660 0.00174 0.89408 0.00747
3, 0*22, 3 1.04224 0.00197 0.88630 0.01015
0*9, 1*6, 0*9 1.03840 0.00177 0.88977 0.00734
50, 40 0*50 1.04110 0.00076 0.89530 0.00479
10, 0*39 1.03837 0.00111 0.89024 0.00445
2*5, 0*35 1.03504 0.00124 0.89184 0.00473
0*39, 10 1.04223 0.00109 0.89728 0.00452
0*30, 1*10 1.03823 0.00106 0.89516 0.00518
5, 0*38, 5 1.03836 0.00125 0.89290 0.00565
0*18, 2*5, 0*17 1.04098 0.00088 0.88919 0.00497
arlNdam gupta Volume 18, December 2023
ASSESSMENT OF GLPI FOR LD
Table 5: MLEs o/Cy (L) and their MSEs with L = 0.1 and L = 0.3, samples generated from the Lindley distribution /or 0 = 1.50 under progressively type-II censoring scheme [True Cy(L) = 0.916334 when L = 0.1 and CY(L) = 0.560892 when L = 0.3].
2[0]*n,m 2[0]*Schemes L=0.1 L=0.3
CY (L)MLE MSE[Cy ( L)mle ] CY (L)MLE MSE[Cy ( L)mle ]
7[0]*10, 8 0*10 0.92111 0.01876 0.59670 0.06439
2, 0*7 0.92292 0.02211 0.58234 0.07653
1,1, 0*6 0.92693 0.01922 0.55731 0.06999
0*7, 2 0.92967 0.01968 0.57833 0.07527
0*6, 1*2 0.93835 0.01868 0.60658 0.06007
1, 0*6, 1 0.92819 0.02075 0.59201 0.06345
0*3,1*2, 0*3 0.91772 0.02080 0.57585 0.06962
7[0]*20, 16 0*20 0.92597 0.00852 0.57983 0.02738
4, 0*15 0.91773 0.01063 0.56602 0.03939
1*4, 0*12 0.92573 0.01097 0.57893 0.03167
0*15, 4 0.91871 0.01025 0.58104 0.03375
0*12, 1*4 0.91498 0.01218 0.59792 0.03509
2,0*14,2 0.91435 0.01054 0.58679 0.03709
0*7, 2*2, 0*7 0.91880 0.00977 0.57989 0.03197
7[0]*30, 24 0*30 0.91462 0.00617 0.56864 0.01894
6,0*23 0.91960 0.00780 0.57341 0.02488
1*6, 0*18 0.92418 0.00739 0.57963 0.02501
0*23, 6 0.92321 0.00829 0.57843 0.02103
0*18, 1*6 0.92471 0.00667 0.56706 0.02135
3, 0*22, 3 0.91766 0.00781 0.57184 0.01984
0*9,1*6, 0*9 0.91798 0.00590 0.55905 0.02281
7[0]*50, 40 0*50 0.91497 0.00387 0.57058 0.01225
10, 0*39 0.91995 0.00436 0.56369 0.01427
2*5, 0*35 0.92503 0.00390 0.56944 0.01577
0*39, 10 0.92130 0.00401 0.56377 0.01520
0*30, 1*10 0.91975 0.00443 0.57915 0.01139
5, 0*38, 5 0.91687 0.00477 0.56287 0.01436
0*18, 2*5, 0*17 0.91364 0.00370 0.55129 0.01093
arindam gupta Volume 18, December 202633
ASSESSMENT OF GLPI FOR LD
Table 6: Bayes estimates ofCy (L) and their MSEs with L = 0.1 and L = 0.3, samples generated from the Lindley distribution for 6 = 1.5 under progressively Type-II censoring scheme [True Cy (L) = 0.916334 when L = 0.1 and Cy (L) = 0.560892 when L = 0.3].
2[0]*n,m 2[0]*Schemes L=0.1 L=0.3
CY ( L) Bayes MSE[Cy (L) Bayes ] CY (L) Bayes MSE[Cy (L) Bayes]
7[0]*10, 8 0*10 0.90365 0.01723 0.58724 0.05271
2, 0*7 0.90402 0.01989 0.57706 0.05956
1,1, 0*6 0.90828 0.01743 0.55465 0.05433
0*7, 2 0.90508 0.01810 0.57052 0.05563
0*6, 1*2 0.91331 0.01689 0.59403 0.04676
1, 0*6, 1 0.90561 0.01861 0.58232 0.05013
0*3,1*2, 0*3 0.89825 0.01857 0.57098 0.05346
7[0]*20, 16 0*20 0.91626 0.00821 0.57459 0.02488
4, 0*15 0.90710 0.01022 0.56201 0.03504
1*4, 0*12 0.91540 0.01042 0.57481 0.02815
0*15, 4 0.90527 0.01001 0.57409 0.02974
0*12, 1*4 0.90220 0.01193 0.59034 0.03104
2,0*14,2 0.90213 0.01034 0.57034 0.03124
0*7, 2*2, 0*7 0.90799 0.00946 0.57521 0.02814
7[0]*30, 24 0*30 0.90805 0.00612 0.56529 0.01775
6,0*23 0.91221 0.00761 0.57023 0.02298
1*6, 0*18 0.91709 0.00714 0.57655 0.02317
0*23, 6 0.91407 0.00807 0.57355 0.01931
0*18, 1*6 0.91574 0.00648 0.56287 0.01980
3, 0*22, 3 0.90917 0.00764 0.56766 0.01829
0*9,1*6, 0*9 0.91040 0.00581 0.55630 0.02116
7[0]*50, 40 0*50 0.91096 0.00385 0.56855 0.01179
10, 0*39 0.91526 0.00428 0.56163 0.01363
2*5, 0*35 0.92053 0.00379 0.56757 0.01504
0*39, 10 0.91552 0.00393 0.56076 0.01451
0*30, 1*10 0.91423 0.00437 0.57613 0.01084
5, 0*38, 5 0.91159 0.00473 0.56055 0.01368
0*18, 2*5, 0*17 0.90905 0.00370 0.54963 0.01048
abhimanyu s yadav, mahendra saha, amartya bhattacharya, arindam gupta
ASSESSMENT OF GLPI FOR LD
pre-chosen level of significance a = 5% and L = 100. Also, the estimate of GLPI Cy (L) and the one-sided 95% confidence interval for Cy(L), i.e., [LB, to) are computed and are reported in Table 8. From the obtained result it can be verified that the value of Cy = 1 does not belong to the one-sided confidence interval, thus the null hypothesis Ho is rejected. Hence, the rejection of the null hypothesis indicates that the GLPI for the considered censored observations meets the required level.
Table 7: Real data estimates o/Cl and Cy (L) for different censoring schemes where L = 100.
n m Schemes C LMLE C LBayes CY ( L) MLE CY (L)B«yes
6[0]*19 3[0]*10 0,1*9 1.29976 1.31230 1.06033 1.06053
1*9,0 1.32879 1.30256 1.07422 1.07421
0*4,3*3,0*3 1.33256 1.32036 1.08983 1.08686
3[0]*15 0*11,1*4 1.26546 1.27359 1.05995 1.05736
1*4,0*11 1.24328 1.25370 1.06700 1.05697
0*6,1*4,0*5 1.25042 1.23734 1.08076 1.07896
Table 8: Testing of hypothesis ofCi, Cy(L)for different censoring schemes based on considered data set.
n m Schemes e C(Qmle L'b Var(C(L)) Cy(L)mle LB Var(CY(L))
6[0]*19 3[0]*10 0,1*9 0.001602602 1.29976 1.145873 0.00032479 1.06033 1.06053 0.00012319
1*9,0 0.001193257 1.32879 1.267304 0.00025823 1.07422 1.07421 0.00004212
0*4,3*3,0*3 0.001438218 1.33256 1.192645 0.00025178 1.08983 1.08686 0.00008373
3[0]*15 0*11,1*4 0.002093421 1.26546 1.238082 0.0000432 1.05995 1.05736 0.00030325
1*4,0*11 0.002393421 1.24328 1.253402 0.00089531 1.06700 1.05697 0.00024422
0*6,1*4,0*5 0.002293421 1.25042 1.210789 0.000045286 1.08076 1.07896 0.00030897
ABHIMANYÜ S YADAV, MAHENDRA SAHA, AMARTYA BHATTACHARYA, ARINDAM GUPTA
ASSESSMENT OF GLPI FOR LD
6. CONCLUSIONS
The present article considers the problem of estimating the GLPI, introduced by [32], under progressive type-II right censored sample where the lower specification limit is given for the Lindley distributed products. The model parameter and the GLPI are obtained by the MLE and Bayes estimation methods, respectively. A comparison study has been carried out through the Monte Carlo simulation study under a progressive type-II censoring scheme between MLE and the Bayes estimate of GLPI in their corresponding MSEs. A real data set is analyzed to study the performance of the proposed index. Though the approach of classical estimation and Bayes estimation are different in direction, assuming the gamma prior and using MCMC method, we discussed the Bayes estimation of Cy(L). The Bayes estimate of the performance index is relatively better than MLE in terms of corresponding MSEs. The proposed procedure can be extended to obtain the confidence interval of Cy (L) based on MLE and Bayes estimate to evaluate whether the product quality meets the required level. In our upcoming course of work, the problem will be attempted. We may use the MLE and the Bayes estimate of Cy (L) based on the progressive type-II right censored sample to draw conclusions about additional lifetime distributions in future.
Declarations
Disclosure of Conflicts of interest/competing interests: The authors declare that they have no conflict of interest.
Authors contributions: Each author has equal contribution. All authors jointly write, review and edit the manuscript.
Funding: The authors received no specific funding for this study.
Data Availability Statements: All cited data analysed in the article are included in References. Data sets are also provided in the article.
Ethical Approval: This article does not contain any studies with human participants performed by any of the authors.
Code availability: Codes are available on request.
References
[1] Ahmed, S., Alatefi, M., Alkahtani, M. and Anwar, S. (2018): Bibliometric analysis for process capability research. Quality Technology and Quantitative Management, 459-477.
[2] Ahmadi, M. V., Doostparast, M. and Ahmadi, J. (2013): Estimating the lifetime performance index with Weibull distribution based on progressive first-failure censoring scheme. Journal of Computational and Applied Mathematics, 239, 93-102.
[3] Ahmadi, M. V., Doostparast, M. and Ahmadi, J. (2015): Statistical inference for the lifetime performance index based on generalised order statistics from exponential distribution. International Journal of Systems Science, 46 (6), 1094-107.
[4] Ahmadi, M. V., Ahmadi, J. and Abdi, M. (2019): Evaluating the lifetime performance index of products based on generalized order statistics from two-parameter exponential model. International Journal of System Assurance Engineering and Management, 10 (2), 251"75.
[5] Ahmadi, M. V., and Doostparast, M. (2021): Evaluating the lifetime performance index of products based on progressively Type-II censored Pareto samples: A new bayesian approach. Quality and Reliability Engineering International, doi:10.1002/qre.3040.
[6] Anderson, D. R., Sweeney, D. J. and Williams, T. A. (1990): Statistics for Business and Economics. Saint Paul, Minnesota, West Publishing Company.
[7] Balakrishnan, N. and Agarwala, R. (2000): Progressive censoring theory, Methods and applications, Birkhauser, Boston.
[8] Balakrishnan, N. and Sandhu, R. A. (1996): Best linear unbiased and maximum likelihood estimation for exponential distributions under general progressive type II censored samples. Sankhya, 58,1-9.
abhimanyu s yadav, mahendra saha, amartya bhattacharya, rt&A No 4 (76)
arindam gupta Volume 18, December 202633 ASSESSMENT OF GLPI FOR LD_
[9] Chen, H. T., Tong, L. I. and Chen, K. S. (2002): Assessing the lifetime performance of electronic components by confidence interval. Journal of the Chinese Institute of Industrial Engineers, 19, 53-60.
10] Cohen, A. C. (1963): Progressively censored samples in life testing. Technometrics, 5, 327-329.
11] Dey, S., Sharma, V. K., Anis, A. Z. and Yadav, B. (2017): Assessing lifetime performance index of Weibull distributed products using progressive type II right censored samples. international Journal of System Assurance Engineering and Management, 8(2), 318-333.
12] Dey, S., M. Saha, and S. Kumar. (2021a): Parametric confidence intervals of Spmk for generalized exponential distribution. American Journal of Mathematical and Management Sciences, 1-22.
13] Dey, S., C. Zhang, and M. Saha. (2021b): Classical and Bayesian estimation of the index Cpmk and its confidence intervals for normally distributed quality characteristic. Journal of Statistical Computation and Simulation, 91(10), 1911"34.
14] Epstein, B. and M. Sobel, M. (1953): Life- Testing. Journal of American Statistical Association, 48, 486-502.
15] Fernandez, A. Z. (2004): On estimating exponential parameters with general type II progressive censoring. Journal of Statistical Planning and Inference, 121,135-147.
16] Giri, B. C., Jalan A. K. and Chaudhuri, K. S. (2000): Economic order quantity model with Weibull deterioration distribution, shortage and ramp-type demand. International Journal of Systems Science, 34, 237-243.
17] Gupta, R., Kishan, R. and Kumar, P. A. (1999): Two non-identical unit parallel system with correlated lifetimes. International Journal of Systems Science, 30,1123-1129.
18] Hassanein, W. A. (2018): Statistical inference of the lifetime performance index for Lind-ley distribution under progressive first failure censoring scheme applied to HPLC data. International Journal ofBiomathematics, 11(5), doi: 10.1142/S1793524518500730.
19] Hong, C. W., Lee, W. C. and Wu, J.W. (2012): Computational procedure of performance assessment of lifetime index of products for the Weibull distribution with the progressive first-failure-censored sampling plan. Journal of Applied Mathematics, doi:10.1155/2012/717184.
20] Hong, C. W., Wu, J. W., Cheng, C. H. (2007): Computational procedure of performance assessment of lifetime index of businesses for the pareto lifetime model with the right type II censored sample. Applied Mathematics and Computation , 184, 336-350.
21] Hong, C. W., Wu, J.W. and Cheng, C. H. (2008): Computational procedure of performance assessment of lifetime index of pareto lifetime businesses based on confidence interval. Applied Soft Computing, 8, 698-705.
22] Ihaka, R. and Gentleman, R (1996): R: A Language for Data analysis and Graphics", Journal of Computational and Graphical Statistics, 5, 299-314,1996.
23] Kane, V. E. (1986): Process capability indices. Journal of Quality Technology, 18, 41-52.
24] Keller, G., Warrack, B. and Bartel H. (1994): Statistics for Management and Economics. Belmont, CA, Duxbury Press.
25] Kotz, S. and Johnson, N. L. (2002): Process Capability Indices - A Review, Journal of Quality Technology, 34(1), 2-19.
26] Lawless, J. F. (2003): Statistical models and methods for lifetime data. Wiley, New York.
27] Lee, W. C., Wu, J. W. and Hong, C. W. (2009): Assessing the lifetime performance index of products with the exponential distribution under progressively type-ii right censored samples. Journal of Computational and Applied Mathematics, 231, 648-656.
28] Lee, W. C., Wu, J. W. and Lei, C. L. (2010): Evaluating the lifetime performance index for the exponential lifetime products. Applied Mathematical Modelling, 34,1217-1224.
29] Lee, H. M., Wu, J. and Lei, C. (2013a): Assessing the lifetime performance index of exponential products with step-stress accelerated life testing data. IEEE Transaction on Reliability, 62, 296304.
30] Lee, W. C., Wu, J. W., Hong, C. W. and Hong, S. F. (2013b): Evaluating the lifetime performance index based on the Bayesian estimation for the Rayleigh lifetime products with the upper record values. Journal of Applied Mathematics, doi:10.1155/2013/547209.
ABHIMANYU S YADAV, MAHENDRA SAHA, AMARTYA BHATTACHARYA, ARINDAM GUPTA
ASSESSMENT OF GLPI FOR LD
[31] Lindley, D. V. (1958): Fiducial distributions and Bayes' theorem. Journal of the Royal Statistical Society, 20, 102-107.
[32] Maiti, S. S., Saha, M. and Nanda, A. K. (2010): On generalizing process capability indices. Journal of Quality Technology and Quantitative Management, 7(3), 279-300.
[33] Maiti, S. S., Bhattarchya, A. and Saha, M. (2021): On generalizing lifetime performance index. Life Cycle Reliability and Safety Engineering, 10, 31-38.
[34] Meyer, P. L. (1965): Introductory Probability and Statistical Applications. Reading, MA, Addison-Wesley.
[35] Montgomery, D. C. (1985): Introduction to statistical quality control. New York, John Wiley and Sons.
[36] Saha, M., Dey, S. and S. Nadarajah, S. (2021a): Parametric inference of the process capability index Cpc for exponentiated exponential distribution. Journal of Applied Statistics, 1"25.
[37] Saha, M., S. Dey, S., Yadav, A. S. and S. Ali. (2021b): Confidence intervals of the index Cpk for normally distributed quality characteristics using classical and Bayesian methods of estimation. Brazilian Journal of Probability and Statistics, 35 (1), 138-57.
[38] Spiring, F., Leung, B., Cheng, S. and Yeung, A. (2003): A Bibliography of Process Capability Papers, Quality and Reliability Engineering International, 19(5), 445-460.
[39] Tong, L. I., Chen, K. S. and Chen, H. T. (2002): Statistical testing for assessing the performance of lifetime index of electronic components with exponential distribution. International Journal of Quality and Reliability Management, 19, 812-824.
[40] Wu, J. W., Lin, C., Tan, B. and Lee, W. C. (2000): An EOQ inventory model with time-varying demand and Weibull deterioration with shortages. International Journal of Systems Science, 31, 677-683.
[41] Wu, J. W. (2002): EOQ inventory model for items with Weibull distribution deterioration, time-varying demand and partial backlogging. International Journal of Systems Science, 33, 323-329.
[42] Wu, S. F. and Hsieh, Y. T. (2019): The assessment on the lifetime performance index of products with Gompertz distribution based on the progressive type I interval censored sample. Journal of Computational and Applied Mathematics, 351, 66-76.
[43] Wu, J. W., Lee, H. M. and Lei, C. L. (2007): Computational testing algorithmic procedure of assessment for lifetime performance index of products with two-parameter exponential distribution. Applied Mathematics and Computation, 190,116-125
[44] Wu, S. J., Chen, D. H. and Chen, S. T. (2006): Bayesian inference for Rayleigh distribution under progressive type censored sample. Applied Stochastic Business and Industry, 22, 269-278.
[45] Wu, S. F., and Lu, J. Y. (2017): Computational testing algorithmic procedure of assessment for lifetime performance index of Pareto products under progressive type I interval censoring. Computational Statistics, 32(2), 647-66.
[46] Wu, S. F. and Chang, W. T. (2021): Power comparison of the testing on the lifetime performance index for Rayleigh lifetime products under progressive type I interval censoring. Communications in Statistics - Simulation and Computation, 1"14.
[47] Viveros, R. and Balakrishnan, N. (1994): Interval estimation of parameters of life from progressively censored data. Technometrics, 36, 84-91.
[48] Yum, B. J. and Kim, K. W. (2011). A bibliography of the literature on process capability indices: 2000-2009. Quality and Reliability Engineering International, 27, 251-268.
[49] Zhang, Y. and Gui, W. (2021). Statistical inference for the lifetime performance index of products with Pareto distribution on basis of general progressive type II censored sample. Communications in Statistics - Theory and Methods, 50(16), 3790-808.
[50] Zhu, J., Xin, H., Zheng, C. and Tsai, T. R. (2022). Inference for the process performance index of products on the basis of power-normal distribution. Mathematics 10(1), 35.
