CC BY
16
RELIABILITY SAMPLING PLAN FOR GENERALIZED INVERTED EXPONENTIALLY DISTRIBUTED UNDER PROGRESSIVE TYPE-II
CENSORED DATA
S. Singh1, A. Kaushik2 •
1,2Department of Statistics, Banaras Hindu University, Varanasi, India. arundevkauhsik@gmail.com Corresponding Author Email: 1statsshubham@bhu.ac.in
Abstract
This article aims to explore a sampling strategy designed to assess the reliability of products that exhibit lifetimes following a GIED. Considered sampling approach has been specially constructed for a Type-II progressive censoring scheme, which includes binomial removals as part of its methodology. Its core objectives is to find out acceptance constant and the optimum sample size. To facilitate practical implementation, the article presents a tabulated form of the sampling plan for the selected specification, as per the considered censoring scheme. To validate the dependability and precision of the suggested sampling approach, we perform a Monte Carlo experiment under various scenarios.
Keyword : Generalized Inverted Exponential Distribution (GIED); OC-Curve; Reliability Sampling Plan; Sinulation; Progressive Censoring.
1. Introduction
In life testing and reliability studies, direct observation of the exact lifetime of a specific event of interest for all tested units is often impractical. This situation arises in various scenarios, such as clinical trials and in engineering where individuals may remain alive or disease-free beyond the study period. To streamline costs and time, some units may be randomly withdrawn from the experiment, resulting in censored data. It is essential to assess the effect of censoring on reliability and determine whether it provides meaningful information or not.
In the field of statistics, a variation of the exponential distribution, termed the one-parameter inverse exponential or inverted exponential distribution (IED), has been advanced. This distribution exhibits an inverted bathtub hazard rate. The utilization of IED in survival analysis has been advocated by several researchers, as exemplified by [22] and [23]. A two-parameter extension of the inverted exponential distribution (IED), called the generalized inverted exponential distribution (GIED), was proposed by [24] and demonstrated that GIED fits real datasets better than IED, based on K-S statistics and likelihood ratio tests. Furthermore, [25] conducted a study on reliability estimation based on progressive Type-II censored samples under the classical paradigm. A common way to evaluate the quality of a product is to check if it meets certain specifications related to its reliability and lifetime. Some standard sampling plans, such as MIL — STD — 414 and MIL — STD — 105 as discussed by [1], can be used to compare the results with predefined criteria. However, these plans may not be suitable for situations where observing all failures is too expensive or time-consuming, especially for products with high reliability. In such cases,
censored tests are often used. Censoring is one of the main feature of lifetime study or reliability study. Censoring desirably or undesirably occurred in the experiment. There are several type of censoring schemes discussed by [2] and [9]. Now a days practitioners and researchers have advocated for a versatile censoring scheme known as progressive Type-II censoring. Progressive Type-II censoring is a method of reliability sampling that involves removing a certain number of units that have not failed at each failure time. This method can reduce the cost and time of testing, but it also introduces some challenges in the analysis.
The progressive Type-II censoring scheme is an extension of Type-II censoring, which incorporates the removal of units from a life-test at predetermined or random inspection times. In this scheme, out of the initial total of (n) units simultaneously placed on a life test, only (m) units are fully observed, while the remaining (n — m) units are withdrawn from the experiment at different time points. Some of the researchers who have developed reliability sampling plans with progressive Type-II censoring are [6], [8] and [7]. A reliability sampling plan by [8] focused on the exponential distribution, while [7] considered the Log-normal and Weibull distributions. Also, [10], [12], and [11] also studied the exponential, Weibull, and Log-normal distributions, respectively, but with different assumptions on the number of units removed at each failure. A comprehensive review of progressive Type-II censoring and its applications provided by [9]. Progressive Type-II censoring is a complex process that requires careful planning and analysis.
An example of the application of progressive censoring in evaluating the performance of electronic components provided by [13]. In such cases, certain test units may require removal due to factors like excessive heat, resulting in situations that fall under the purview of Type-II PCR. Furthermore, [16] conducted extensive investigations into issues related to parameter estimation and the expected duration of experiments under Type-II PCR censoring. In numerous practical scenarios, managing removals presents a formidable challenge, rendering the assumption of fixed and known removals impractical. In acknowledgment of this constraint, [3] advocates the adoption of random removals, emphasizing its practical viability. Therefore, the implementation of Type-II censoring with random removals becomes a more pragmatic choice. In this approach, a suitable distribution, such as the binomial distribution, can be employed to model the removal pattern. To the best of our knowledge, the utilization of reliability sampling plans for Type-II censoring with binomial removals has not been previously documented. In current study, our primary focus lies in the development of a reliability sampling plan for the GIED under Type-II progressive censoring with random removals. This entails that the number of removals at each failure is subject to a binomial distribution. In Section 2, we will introduce our proposed model and establish the maximum likelihood estimators (MLEs) for the model parameters. In Section 3, we present the Operating Characteristic (O.C.) curve, providing insights into the performance of our sampling plan. Section 4 delves into an in-depth examination of the sampling plan's design. Finally, in Section 5, we offer our concluding remarks and provide a succinct summary of the key findings derived from this study.
A generalisation of the one parameter IED is a two parameter GIED having PDF and CDF as follows:
2. Methods
t > 0, v > 0, n > 0.
t > 0, v > 0, n > 0.
(1)
Where, v is the shape parameter and n is the scale parameter. Let tp be the pth percentile of the GIED, it is given by,
On simplification, we get
t = -n
p ln(1 - (1 - p)1/v) •
and median of the distribution is given by
md =
ln(l — (0.5)1/v)'
the reliability function is given by
S(t)= (^1 — exp (—t > 0, (q,v) > 0. The failure rate function of the GIED(q, v) is given by
h(t) = § = fY—ehf; > >0, (n, v) > 0
For simplicity point of view, let us make a transformation Z = ln(t).
ty(z) = vq exp (—z — q exp(— z)) (1 — exp ( — q exp(—q exp(— z))))
v-1
(3)
; z > 0, v > 0, q > 0.
and its distribution function is given by
Y(z) = 1 — (1 — exp (—q exp(—z)))v;z > 0, v > 0, q > 0. (4)
Let's consider the following transformations: p = ln v and a = 1. It simplifies our analysis to work with the model represented by equation 3. Now, we have a set of m ordered log-failure times, denoted as Z! < Z2 < ... < Zm, selected from a pool of n items. The value of m is pre-determined, indicating the number of failures that occur before the testing concludes.
At the ith failure event, a random removal of ri items takes place from the testing pool. The number of items removed, ri, follows a binomial distribution characterized by parameters (n — m) and removal probability (pr). In the context of a Type II progressive censoring (Type II PCR), we define the likelihood function as follows:
L(t; p, a) = L1 (t; ц, a)PR.
where,
and
Li (t; p. a) = n № )(1 - f(zl ))r i=1
PR = P(Rm-1 = rm-11Rm-2 = rm-2. Rm-3 = rm-3~R1 = r1) xP(Rm-2 = Tm-2\Rm-3 = Tm-3, Rm-4 = fm-4»^ = r1)...P(R2 = = H)•
= (n - m)\ j rj ( )(m-1)(n-1)-Em=11 (m-j)rj
m—1 v ' r ' r
nm=1 ri !(n — m — jl1)!
Where, Ci = n — T!j=\ (rj + 1). To obtain the maximum likelihood estimators (MLEs) of p and a, the likelihood have been maximized at MLEs of the parameters, for more details see [28] and [27]. In this type of censoring scheme, the number of failures is predetermined before the experiment begins. The experiment is terminated once the desired number of failures is observed. Assuming that the number of failures is fixed as m, we denote ti as the time at the ith removal, and ri as the number of the random removals of the ith component.
m
3. OC curve
The OC curve is a tool used to evaluate the effectiveness of a sampling plan. It does so by charting the likelihood of accepting a lot against the proportion of non-conforming items within that lot. This evaluation relies on the principles of asymptotic distribution theory. In the context of this evaluation, we utilize the following equations: First, we have:
T'~ - ka) - N(0,1)
The standardized variate is expressed as:
(T' - (p - ka))^n
W
Vv
To construct the OC curve, which represents the probability of accepting a lot, denoted as L(p), we use the following equation:
L(p) = Pr[T' > L']= 1 - 3>
a(up + ki )^n
TV
In this equation, up stands for the quantile of the standard logistic distribution that corresponds to the given proportion of non-conforming items, denoted as p. $(•) represents the standard normal distribution function.
Therefore, to determine an optimal sampling plan for specific points on the OC curve, denoted as (pa, 1 — a) and (p^, ft), the following equations need to be solved for the variables k and n:
g(upa + ki )Vn
za TV _0
(5)
a(up„ + ki )y/n
--TV— _0
where, upa and up^ denote the quantiles of the standard normal distribution and za and zp
denotes the quantiles of the log-life distribution. Thus on solving equation (5), we get
k _ zpa — zpp ua (6)
ux — ui-ß
and
ux — ui—ß \ iff1
zpa zpß J \ n
ff2 \ — (Yii(n, pr, fc ) — 2kYi2 (n, pr, fc ) + Y22(n, pr, fc )) J (7)
n
4. Layout of sampling plan
4.1. Sampling Plan
In our study, we adopt the methodology originally proposed by [17] to evaluate the acceptability of a batch. Specifically, we concentrate on variable sampling plans with one-sided specification limits. Let's consider a lot of size n randomly drawn from a larger population. The log-lifetimes of the items in this lot follow a distribution characterized by Equation (3). This distribution is defined by a set of unknown parameters, denoted as v and y. We seek to obtain the maximum likelihood estimators for these parameters, denoted as v and n
In this context, we have a lot with a proportion of non-conforming items, denoted as p0 (where p0 < pa), which is considered acceptable and should be approved with a probability of
at least (1 — a). Here, pa represents the proportion of non-conforming items that corresponds to the desired probability of acceptance, denoted as (1 — a) on the operating characteristic (OC) curve. We define L' as the quantile of the Y(-) given in the equation (4) that corresponds to the proportion of non-conforming items for the chosen probability of acceptance (1 — a). This is calculated as L' = Y—1(pa). The decision to accept or reject the lot hinges on the comparison of the estimate (p — ka) with the value of L'. If (p — ka) is greater than or equal to L', the lot is accepted; otherwise, it is rejected. The acceptance constant, denoted as k, is a pivotal factor in making this decision.
The key focus is on specifying the optimal sample size (n) and the pertinent acceptance constant (k) within the framework of the proposed censoring scheme. One can note that, the distribution of the variable (p — ka) will be AN ((p — ka), ^(yn(n, pr, fc) — 2kyu(n, pr, fc) + 722(n, pr, fc))) •
Here, 7n (n, pr, fc), 712(n, pr, fc), and 722(n, pr, fc) are elements of the asymptotic dispersion matrix. You can find detailed expressions for these in the Appendix provided by [26]. Here, pr represents the removal probability, and fc denotes the censoring fraction. Choose two points, (pa,1 — a) and (pp, p), on the OC curve suggested by [7]. To calculate these points, we use
the formulas: yT = Y—1 (t) = — ln (—1 ln (1 — (1 — U)and ua = 1 (a), where Y(-) and $(•) are the cumulative distribution functions (CDF) of the log-GIED and the standard normal distribution, respectively. To determine the acceptance constant (k) and the sample size (n) for a given pair of points, (pa, 1 — a) and (pp, p), on the OC curve, along with specified censoring fraction (fc) and removal probability (pr), we solve equations (5).
In Tables 1 and 2, we present the results for various removal probabilities (pr = 0.1, 0.3, and 0.5) and censoring fractions (fc = 0.2, 0.3, 0.4, 0.5, 0.6, and 0.7). The selection of values for pa and pp aligns with the criteria set by MIL-STD-105D. Additionally, we include results for a limit case of standard Type-II censoring, where pr = 0.00001, allowing for a comparison with the findings of [7]. For the computation of the terms 7n(n, pr, fc), Jn(n, pr, fc), and 722(n, pr, fc), we employ a Monte Carlo simulation, generating progressive Type-II censored samples initially.
Specifically, we calculate the moments based on 2000 simulations, assessing the average values of the terms 7n(n, pr, fc), Y12(n, pr, fc), and 722(n, pr, fc) for various values of n. The outcomes are detailed in Table 1 and Table 2 for p = 0.05 and p = 0.10, respectively. The results reveal that, when maintaining a constant pr, the optimum value of n decreases as fc declines, irrespective of the acceptance constant (k). A lower fc implies lesser dropouts, resulting in fewer accurate lifetime observations. Consequently, a larger sample size is necessary to compensate for the loss of information when assessing lot acceptability.
On the other hand, when the censoring fraction fc is held constant, and the same acceptance constant k is used, the sample size does not exhibit a consistent pattern with respect to the removal probability. This discrepancy arises because removal shifts the observations toward the tail of the lifetime distribution, improving the accuracy of lifetime parameter estimation but leading to a loss of information due to dropouts. However, an excessive number of dropouts early in the process diminishes this advantage. Hence, for higher values of pr, the sample size increases as pr, such as pr = 0.5, increases. Conversely, for low to moderate values of pr, the sample size decreases as pr increases due to the impact of a significant number of dropouts.
Table 1: Type-II PCR reliability sampling plan for pa and pft to match with MIL — STD — 105D for 1 — a _ 0.95, ft _ 0.10.
n
pa pß fr ^ 0.7 0.6 0.5 0.4 0.3 0.2 k
pr = 0.i
0.0004i 0.0i840 i4 i0 9 9 8 8 i.i352
0.00284 0.03ii0 24 i9 i8 i7 i6 i6 0.9834
0.00654 0.04260 32 27 26 25 24 23 0.898i
0.0i090 0.05350 40 34 33 3i 30 29 0.8372
0.02090 0.07420 53 47 45 43 4i 39 0.7482
0.03i90 0.09420 65 58 55 53 50 48 0.68i6
pr = 0.3
0.0004i 0.0i840 i4 i2 ii i0 9 8 i.i352
0.00284 0.03ii0 26 23 2i i9 i8 i7 0.9834
0.00654 0.04260 36 33 30 28 25 24 0.898i
0.0i090 0.05350 45 4i 37 35 32 30 0.8372
0.02090 0.07420 6i 56 5i 47 44 4i 0.7482
0.03i90 0.09420 74 68 63 58 54 50 0.68i6
pr = 0.5
0.0004i 0.0i840 i6 i3 i2 i0 9 9 i.i352
0.00284 0.03ii0 29 26 23 20 i8 i7 0.9834
0.00654 0.04260 40 36 32 29 26 24 0.898i
0.0i090 0.05350 50 45 40 36 33 3i 0.8372
0.02090 0.07420 68 6i 55 50 45 42 0.7482
0.03i90 0.09420 83 75 67 6i 56 5i 0.68i6
pr = : 0.000i
0.0004i 0.0i840 50 i5 9 8 8 7 i.i352
0.00284 0.03ii0 66 23 i7 i6 i5 i5 0.9834
0.00654 0.04260 75 30 24 23 23 22 0.898i
0.0i090 0.05350 8i 37 3i 30 29 28 0.8372
0.02090 0.07420 9i 49 44 42 4i 39 0.7482
0.03i90 0.09420 99 6i 56 54 5i 49 0.68i6
4.2. Simulated sampling plan
It is worth noting that in the discussion of the distribution of (f — ka), asymptotic distribution theory is applied, and the derived sampling plans are based on this approximation. However, it is essential to investigate the finite sample behavior of these sampling plans by conducting a Monte Carlo simulations to assess the true probability of acceptance.
In this research, we employ a Monte Carlo simulation to compare the expected probability of acceptance with the actual probability when a designed sampling plan is put into practice within a specific censoring framework. We investigate various scenarios, incorporating removal probabilities (pr) of 0.1, 0.3, and 0.5, as well as censoring fractions (fc) of 0.3, 0.5, and 0.7. Additionally, we consider fixed producer's and consumer's risk (v, ft) settings at (5%, 10%) and (5%, 5%). For each combination of these parameters, we conduct 2000 Monte Carlo simulations to provide precise estimates of the probability of acceptance, denoted as L(p).
To obtain estimates for the parameters k and L(p), we employ the bias-corrected maximum likelihood estimators (MLEs), as detailed by [9]. The obtained results are presented in Table 3 through Table 11, covering various removal probabilities pr and diverse levels of censoring proportions.
Table 2: Progressive Type-Il reliability sampling plan with random removals aligned with the requirements of MIL — STD — W5D for 1 — a = 0.95 and ß = 0.05.
n
Pa Pß fr ^ 0.7 0.6 0.5 0.4 0.3 0.2 k
Pr = 0.1
0.00041 0.01840 19 31 41 50 67 82 1.1664
0.00284 0.03110 13 25 35 43 60 74 1.0066
0.00654 0.04260 12 23 33 41 57 70 0.9180
0.01090 0.05350 11 22 31 40 54 67 0.8553
0.02090 0.07420 11 21 30 37 51 63 0.7641
0.03190 0.09420 10 20 28 36 49 61 0.6962
Pr = 0.3
0.00041 0.01840 19 33 46 57 76 93 1.1664
0.00284 0.03110 16 29 41 52 70 86 1.0066
0.00654 0.04260 14 26 37 47 64 78 0.9180
0.01090 0.05350 13 25 35 44 60 74 0.8553
0.02090 0.07420 11 22 32 40 54 67 0.7641
0.03190 0.09420 11 21 29 37 51 63 0.6962
Pr = 0.5
0.00041 0.01840 20 37 52 64 86 106 1.1664
0.00284 0.03110 17 32 45 56 76 93 1.0066
0.00654 0.04260 15 29 40 51 69 85 0.9180
0.01090 0.05350 13 26 37 46 63 77 0.8553
0.02090 0.07420 12 23 33 42 57 70 0.7641
0.03190 0.09420 11 21 30 38 52 64 0.6962
Pr = = 0.0001
0.00041 0.01840 79 101 112 118 128 135 1.1664
0.00284 0.03110 21 32 41 49 64 79 1.0066
0.00654 0.04260 11 21 30 39 55 71 0.9180
0.01090 0.05350 10 19 28 36 52 66 0.8553
0.02090 0.07420 10 19 28 36 51 64 0.7641
0.03190 0.09420 9 19 27 35 48 60 0.6962
Table 3: Simulated probabilities of acceptance for GIED with pr = 0.1 and 70% censoring.
Sampling plan Probability of acceptance
Pr = 0.1 n m k L( p)
a = 0.05, ß = 0.10 15 4 1.1352 0.5278
24 7 0.9834 0.3812
32 9 0.8981 0.3025
39 11 0.8372 0.2403
52 15 0.7482 0.1518
64 19 0.6815 0.0833
a = 0.05, ß = 0.05 18 5 1.1664 0.6235
30 9 1.0066 0.4398
40 12 0.9180 0.3424
49 14 0.8553 0.2751
66 19 0.7641 0.1797
81 24 0.6962 0.1083
Table 4: Simulated probabilities of acceptance for GIED with pr = 0.1 and 50% censoring. .
Sampling plan Probability of acceptance
pr = 0.1 n m k L( P)
a = 0.05, ß = 0.10 10 5 1.1352 0.5284
19 9 0.9834 0.3869
26 13 0.8981 0.2988
33 16 0.8372 0.2393
46 23 0.7482 0.1520
57 28 0.6815 0.0842
a = 0.05, ß = 0.05 12 6 1.1664 0.6150
23 11 1.0066 0.4357
33 16 0.9180 0.3416
42 21 0.8553 0.2755
58 29 0.7641 0.1803
71 35 0.6962 0.1084
In Table 3, with a constant removal probability pr, varying consumer's risk (ft) and fixed producer's risk (v), and fixed censoring proportion fc, as the sample size n rises, the corresponding value of the number of the failures m also rises. However, the probability of acceptance L(p) diminishes. Similar pattern have been observed from Table 3 to Table 11 for the different values of the censoring proportion fc. From the Table 3 and Table 4, one can study the effect of the change of censoring proportion fc. Here, with a constant removal probability pr, consumer's risk (ft) and fixed producer's risk (v), size of the sample n decreases as the censoring proportion fc decreases. A similar patterns has been observed for the rest of the tables for different values of the censoring fraction fc and removal probability pr, so one can conclude the same in general.
5. Conclusion
Our study has delved into the challenges and intricacies of Type-II Progressive Censoring (Type-II PCR), a common practical scenario where the number of removals is uncertain. Our primary focus has been the development of optimum reliability sampling plans for the GIED lifetime distribution within the framework of Type-II PCR. We have rigorously examined a range of scenarios involving removal probabilities and censoring fractions, shedding light on their influence on these sampling plans.
Table 5: Simulated probabilities of acceptance for GIED with pr = 0.1 and 30% censoring. .
Sampling plan Probability of acceptance
pr = 0.1 n m k L( p)
a = 0.05, ß = 0.10 8 5 1.1352 0.4955
17 11 0.9834 0.3853
24 16 0.8981 0.3014
30 21 0.8372 0.2399
41 28 0.7482 0.1508
51 35 0.6815 0.0838
a = 0.05, ß = 0.05 11 7 1.1664 0.6245
21 14 0.9745 0.4376
30 21 0.9180 0.3421
38 26 0.8553 0.2755
52 36 0.7641 0.1798
64 44 0.6962 0.1086
Table 6: Simulated probabilities of acceptance for GIED with pr = 0.3 and 70% censoring..
Sampling plan Probability of acceptance
pr = 0.3 n m k L( P)
a = 0.05, ß = 0.10 14 4 1.1352 0.5104
26 7 0.9834 0.3833
36 10 0.8981 0.3012
45 13 0.8372 0.2415
60 18 0.7482 0.1513
74 22 0.6815 0.0840
a = 0.05, ß = 0.05 18 5 1.1664 0.6163
33 9 1.0066 0.4396
46 13 0.9180 0.3444
56 16 0.8553 0.2750
76 22 0.7641 0.1800
93 27 0.6962 0.1089
Table 7: Simulated probabilities of acceptance for GIED with pr = 0.3 and 50% censoring. .
Sampling plan Probability of acceptance
pr = 0.3 n m k L( P)
a = 0.05, ß = 0.10 11 5 1.1352 0.5438
21 10 0.9834 0.3973
30 15 0.8981 0.3130
37 18 0.8372 0.2485
51 25 0.7482 0.1590
63 31 0.6815 0.0907
a = 0.05, ß = 0.05 14 7 1.1664 0.6225
27 13 1.0065 0.4410
38 19 0.9180 0.3436
47 23 0.8553 0.2746
65 32 0.7641 0.1803
80 40 0.6962 0.1092
Table 8: Simulated probabilities of acceptance for GIED with pr = 0.3 and 30% censoring.
Sampling plan Probability of acceptance
pr = 0.3 n m k L( p)
a = 0.05, ß = 0.10 9 6 1.1352 0.5106
18 12 0.9834 0.3838
25 17 0.8981 0.2983
32 22 0.8372 0.2400
44 30 0.7482 0.1518
54 37 0.6815 0.0837
a = 0.05, ß = 0.05 12 8 1.1664 0.6311
23 16 1.0066 0.4418
32 22 0.9180 0.3421
40 28 0.8553 0.2742
55 38 0.7641 0.1794
68 47 0.6962 0.1087
Table 9: Simulated probabilities of acceptance for GIED with pr = 0.5 and 70% censoring..
Sampling plan Probabilities of acceptance
pr = 0.5 n m k L( P)
a = 0.05, ß = 0.10 16 4 1.1352 0.5240
29 8 0.9834 0.3841
40 12 0.8981 0.3006
50 15 0.8372 0.2407
67 20 0.7482 0.1510
83 24 0.6815 0.0842
a = 0.05, ß = 0.05 20 6 1.1664 0.6224
37 11 1.0066 0.4414
51 15 0.9180 0.3436
63 18 0.8553 0.2756
85 25 0.7641 0.1799
104 31 0.6962 0.1088
Table 10: Simulated probabilities of acceptance for GIED with pr = 0.5 and 50% censoring.
Sampling plan Probability of acceptance
pr = 0.5 n m k L( P)
a = 0.05, ß = 0.10 12 ,6 1.1352 0.5217
23 11 0.9834 0.3845
32 16 0.8981 0.2995
41 20 0.8372 0.2415
55 27 0.7482 0.1508
68 34 0.6815 0.0836
a = 0.05, ß = 0.05 15 7 1.1664 0.6213
29 14 1.0066 0.4402
41 20 0.9180 0.3434
51 25 0.8553 0.2750
70 35 0.7641 0.1800
86 43 0.6962 0.1088
Table 11: Simulated probabilities of acceptance for GIED with pr = 0.5 and 30% censoring.
Sampling plan Probability of acceptance
pr = 0.5 n m k L( P)
a = 0.05, ß = 0.10 10 7 1.1352 0.5259
19 13 0.9833 0.3834
27 18 0.8981 0.3010
34 23 0.8372 0.2405
46 32 0.7482 0.1507
57 39 0.6815 0.0836
a = 0.05, ß = 0.05 12 8 1.1664 0.6128
24 16 1.0066 0.4396
34 23 0.9180 0.3427
43 30 0.8553 0.2759
58 40 0.7641 0.1791
72 50 0.6962 0.1088
Our key findings, as evident in the parameters of sample size (n) and the acceptance constant (k), underscore the crucial role of an increasing censoring fraction (fc) in necessitating a larger sample size. Generally, the optimal sample size (n) exhibits stability across varying removal probabilities (pr). Nonetheless, it is of paramount importance to highlight the pivotal role played by the removal probability (pr) in shaping the overall test duration. [16] has convincingly demonstrated that an escalation in the removal probability (pr) leads to a significant extension of the test duration. In such cases, an increased sample size becomes imperative to effectively mitigate the extended testing period. These insights emphasize the practical significance of our research in addressing real-world challenges related to reliability testing under Type-II PCR.
Declaration of conflicting interest: All authors have no conflict of interests in the publication of
the manuscript.
Funding : This research received no specific grant from any funding agency in the public,
commercial, or not-for-profit sectors.
Data: No such data are provided in this manuscript. Only simulation study has been performed.
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