1. Introduction
The progressive type-II censoring (PCS-T2) method is the most popular scheme in reliability and survival analysis. Compared with the traditional type-II censoring method, it is better. Progressive censoring is advantageous in a variety of real-world applications, including business, medical research, and therapeutic settings. Up until the test’s conclusion, it permits the removal of any remaining experimental units. Assume that n units are used in a life test and that it is not desirable to record every failure because of financial and time constraints. Consequently, only a portion of unit failures are seen. A sample like this is known as a censored sample. Assume that one of the units was accidentally damaged after the test started but before they all burned out. This unit needs to be taken out of the life test if the experiment is still going on. In this situation, a framework for analyzing this kind of data is provided by the progressive censoring scheme. A few examples of primary references are [
1,
2].
PCS-T2 has drawn a lot of attention in the literature as a very flexible censoring system (see [
3] for further details). When testing
independent units at a time
, the failure number to be noticed
and the progressive censored samples,
, where
, are specified. When the initial failure is seen (suppose that
), the other surviving units
are chosen at random, and
of those units is disqualified from the test. Similarly, at the moment of the second failure (suppose that
),
of
are selected at random and deleted from the test, and so on. At the time of the
failure (suppose that
), every survivor unit still present
is removed from the experiment.
Whenever the test units are particularly reliable, the major drawback of this censoring is that it could take longer to finish the progressively type-II hybrid censored samples (PHCS-T2). The authors of [
4] proposed a progressive type-I hybrid censored strategy (PHCS-T1) as a remedy for this issue. This method combines PCS-T2 with conventional type-I censoring. Under PHCS-T1, the trial period is stopped at
, maximum likelihood estimators (MLEs) were not always available due to the fact that relatively a few failures might occur before time
in PHCS-T1. To resolve this issue, [
5] presented the PHCS-T2 scheme. At
, the experiment comes to an end under PHCS-T2. It can take some time until such
failures are really observed, despite the fact that PHCS-T2 promises a fixed number of failures.
It could take a while to gather the needed failures, even though the PHCS-T2 ensures an effective number of observable failures. Thus, [
6] devised the generalized progressive type-II hybrid censoring (GPHC-T2). Assume that the thresholds
and 2, as well as the integer
, are preassigned in such a way that
and
.
and
represent the overall number of failures up to periods
and
, respectively. Then, at
,
of
are arbitrarily excluded from the test, followed by
of
, and so on.
The experiment is over, and all remaining units are deleted at
. If
failures are observed without any further withdrawals up until time
(Case-I); if
, the test is terminated at time
(Case-II); or, if not, the test is terminated at time
(Case-III). Keep in mind that the GPHCS-T2 modifies the PHCS-T2 by guaranteeing that the test is completed at the scheduled time
.
demonstrates the longest period of time the researcher is willing to let the experiment continue. As a result, one of the following three data types will be visible to the experimenter:
Figure 1 indicates the cases of generalized type-II progressive hybrid sample as follows:
Assume that in a distribution with a cumulative distribution function (cdf)
F(.), and probability density function (pdf)
f(.), the variables
and
represent the respective lifetimes. As a result, the GPHCS-T2 likelihood function is expressed as follows:
where
stand in for Case-I, Case-II, and Case-III, respectively, and
is a combination form of dependability functions.
Table 1 displays the GPHCS-T2 notations from Equation (1). Many censoring techniques can also be inferred as particular examples from Equation (1), including
With setting to 0, use PHCS-T1.
. by setting PHCS-T2.
You may do hybrid type-I censoring by setting .
, can be used to do hybrid type-II censoring.
To do type-I censoring, set
A type-II censored sample is produced by setting
On the basis of GPHCS-T2, more studies have been conducted. For instance, Ref. [
7] investigated the prediction issue of forthcoming Burr-XII distribution failure rates. The authors of [
8] created the Weibull distribution with little data with an objective Bayesian analysis. The authors of [
9] addressed the competing risks from exponential data, and [
10] more recently examined both the point and interval estimations of the Burr-XII parameters. Last but not least, [
11] addressed the Fréchet distribution’s optimality under generalized censoring schemes. In this paper, the KMKu model under generalized censoring samples is studied. Where the KMKu model was initially proposed by [
11]. Also, they found that the Kumaraswamy model’s and KMKu shape forms in the pdf for different parameter values are comparable. It may be asymmetric, unimodal, increasing, or decreasing. In addition, the bathtub, U-shape, J-shape, or increasing shapes of the hazard rate function (hrf) for the KMKu model are all possible. But suppose that
is the lifespan random variable of a test item adheres to the KMKu distribution, denoted by the notation
, where
are the shape parameters. Therefore, it is supplied by its pdf, cdf, reliability function (RF),
, and hrf, all represented by the letters
,
, and
accordingly:
and
Although the KMKu model has a lot of flexibility because of its different shapes of hrf and pdf, to our knowledge, no studies have yet been done under censorship. Particularly, the generalized type-II progressively hybrid censoring scheme has not produced any data for the new KMKu lifetime model’s survival traits and model parameters. To fill this gap, the following are the objectives of this study: Firstly, the probability inference for any function of the unknown KMKu parameters, such as
R(
t) or
h(
t), is derived. The second objective is to derive independent gamma priors from the squared error (SE) loss and produce Bayes estimates for the same unknown parameters, employing the provided estimation procedures, such as classical and Bayesian approaches. The unknown parameters of the KMKu distribution are discovered using the approximation confidence intervals (ACIs) and highest posterior density (HPD) interval estimators. The acquired estimates are computed using the R programming language’s “maxLik” and “coda” packages because the theoretical findings of
and
obtained by the suggested estimation techniques cannot be represented in closed form. [
12,
13] offered these packages. Using four optimality criteria, the ultimate aim is to develop the most efficient progressively censored sample technique. The effectiveness of the different estimators is investigated using a Monte Carlo simulation with the entire sample size, which can be combined in a variety of ways, effective sample size, threshold timings, and progressively censored samples. We compare the average confidence lengths (ACLs), mean relative absolute biases (MRABs), and simulated root mean squared errors (RMSEs) of the derived estimators. The optimal censoring tactic should be chosen after evaluating how effectively the given techniques will function in practice. The remaining portions of this study are structured as follows: The maximum likelihood, Bayes inferences, and reliability functions of the unknown parameters are presented in
Section 2 and
Section 3, respectively. The credible and asymptotic intervals are built into
Section 4.
Section 5 goes into depth about the results of the Monte Carlo simulation. The optimal methods for progressive censoring are discussed in
Section 6. Two actual data sets are indicated in
Section 7. Finally, the conclusion and discussion are given in
Section 8.
2. Likelihood Estimation
Assume that the representation of a GPHCS-T2 sample of size
taken from
is
. The probability function of GPHCS-T2 may be represented by substituting
for
in Equation (1) and adding Equations (2) and (3); for more information, see [
14].
where
and .
The proper log-likelihood function for Equation (6) is
as follows:
where
, and
By partially differentiating Equation (7) with reference to
and
, the subsequent two findings are produced. After being equal to zero, likelihood equations must be simultaneously solved in order to create the MLEs.
and
where
and
, respectively, we have
,
According to Equations (8) and (9), it is necessary to simultaneously satisfy a system of two nonlinear equations in order to derive the MLEs of and in the KMKu model. As a result, for and , there is not, and cannot be computed, an analytical closed-form solution. Thus, it may be estimated for each specific GPHCS-T2 data set using numerical techniques like the Newton-Raphson iterative method. When the estimates of and are derived by replacing them with and , the MLEs and , respectively, may be easily computed.
3. Bayes Estimator
The HPD intervals for the Bayes estimators of , R(t), and h(t) are developed using the SE loss function. To do this, it is assumed that the KMKu parameters and , respectively, have independent gamma priors of the forms and .
The normal distribution can be a standard choice for data if the domain of that distribution is from −∞ to ∞, and the beta distribution can be a standard choice for data if the domain of that distribution is from 0 to 1. Similarly, the gamma distribution can be a standard choice for non-negative continuous data if the domain of the gamma distribution is from 0 to ∞. This is one of the most important reasons, but there are other reasons as follows:
We believe the main motivation for the gamma prior is usually to constrain the random variables to positive values.
The gamma distribution is considered one of the most important and well-known statistical distributions because it is compatible with many engineering, mathematical, statistical, and medical applications.
The gamma distribution is one of the most famous distributions that is used in mathematical solutions (integrations), especially when the data are from 0 to ∞.
In previous studies, the gamma distribution was the most popular prior distribution and was associated with the best statistical results.
Gamma priors should be considered for a variety of reasons, including the fact that they are (1) adjustable, (2) offer diverse shapes based on parameter values, and (3) fairly basic and brief and might not generate a solution to a challenging estimation problem. Then, the combined previous density of
and
is determined; for more details on this topic, see [
15,
16].
If it is anticipated that for
are known. The joint posterior pdf of
and
Equations (6) and (10), when combined, results.
The Bayes estimate,
, of
and
respectively, under SE loss,
is what is meant by the posterior expectation of Equation (11), which is given.
It is clear from Equation (11), that it is impossible to explicitly express the marginal pdfs of
and
. In order to accomplish this, we recommend creating samples from Equation (11) utilizing Bayes MCMC methods to calculate the joint Bayes estimates and supplying their HPD intervals. The complete conditional pdfs of
and
are provided for the MCMC sampler from Equation (11) to be performed as intended.
and
The Metropolis-Hastings (M-H) approach is considered to be the best solution to this problem because no analytical method exists to reduce the posterior pdfs of
and
in Equations (12) and (13), respectively, to any known distribution (for further information, see [
17,
18]. The sampling method of the M-H algorithm is implemented according to:
First, establish the starting points, and .
Set S = 1 after that.
Thirdly, from and , respectively, create and .
The fourth step: Obtaining and
Fifth, use the uniform distribution to generate the samples and .
Sixth: Set and respectively, if and are both smaller than and , respectively. Set and , correspondingly, if not.
Seventh: Establish that S equals S + 1.
Eighth: Repeating steps three through seven a number of times will give you the values for and for .
Ninth: To calculate the RF in Equation (4) and hrf in Equation (5), use
and
for
, respectively, for a given mission period
.
and
The convergence of the MCMC sampler must be ensured, and starting,
and
values must be eliminated. The first simulated variants, let us say
, are removed as burn-ins. Therefore, using the remaining
samples of
or
, (let us suppose
), the Bayesian estimates are computed. On the basis of the SE loss function, the Bayes MCMC estimates of
are shown.
5. Optimal PCS-T2 Designs
The experimenter may want to pick the “best” censoring scheme out of a collection of all accessible censoring schemes in order to provide the most details about the unknown parameters under investigation, especially in the context of dependability. First, [
1] examined the problem of deciding which censoring strategy is most appropriate under various circumstances. However, a number of optimality criteria,
, where
have been proposed, and several assessments of the top censoring strategies have been made. The precise values of
(total test units),
(effective sample), and
(ideal test thresholds) are picked in advance according to the accessibility of the units, the accessibility of the experimental settings, and cost factors (see [
22]). A number of articles in the literature have addressed the topic of contrasting two (or more) different censoring techniques. For examples, see [
23,
24]. To help us choose the best censoring strategy,
,
Table 2 offers a variety of widely used measures.
It is advised that the observed Fisher information,
values for
, be maximized. For criterion
and
, we also wish to reduce the determinant and trace of
. The best censoring strategy for multi-parameter distributions may be selected using scale-invariant criteria. While dealing with unknown multi-parameter distributions makes it more challenging to compare the two Fisher information matrices, dealing with single-parameter distributions allows for the use of scale-invariant criteria to compare a variety of criteria
. The logarithmic MLE of the
quantile,
, tends to have a variance that is minimized by the p-dependent criterion
. As a result, the logarithm of the KMKu distribution for time
may be calculated using
By using the delta technique to solve for Equation (4), the estimate of the variance for the
of the KMKu distribution is given as
where
while
The maximum value of the
criterion and the lowest value of
correspond to the best censoring. On the other hand, the greatest value of the
criterion and the lowest value of the
criterion correspond to the best censoring.
6. Simulation
Using different combinations of (threshold points), n (sample size), s (size of censored sample), and R (censored removal), Monte-Carlo (MC) simulations were carried out to assess the true performance of the acquired point and interval estimators of and . To establish this goal, for KMKu(1.4, 1.5), KMKu(1.4, 0.5), and KMKu(0.4, 0.5), we replicated the GPHCS-T2 mechanism 1000 times. Taking , two different choices of n and s were used as (n = 30, 50, 100), and the choices of s were used as (s = 20, 25) at n = 30, (s = 35, 45) at n = 50, and (s = 70, 90) at n = 100. At = 0.6, the true values of and were 0.4278 and 1.4899, respectively. At = 0.85, the true values of and were 0.2526 and 3.3106, respectively.
Additionally, by utilizing the binomial elimination distribution and taking into account different censoring schemes for each combination of
and
, the following is conducted: according to the following probability mass function, the number of units removed at each failure time is expected to follow a binomial distribution.
Additionally, assume that for any
,
is independent of
. In light of this, the likelihood function can be written as follows:
where
That is,
where the GPHCS-T2-based KMKu distribution’s parameters do not affect the binomial parameter
(Independent). We chose the binomial parameter
with varied values of 0.3 and 0.8.
The MLEs and 95% ACI estimates of
and
were assessed after 1000 GPHCS-T2 samples had been gathered using R 4.2.2 programming software and the “maxLik” library. We simulated 12,000 MCMC samples and omitted the first 2000 iterations as burn-in to obtain the Bayes point estimates along with their HPD interval estimates of the same unknown parameters using the “coda” library in the R 4.2.2 programming language. The estimates and their variances were equated with the Fisher information matrix of
and
to produce the ML estimator, which it denoted as elective hyper-parameters, and this was contributed by [
25]. This process allowed for the extraction of the hyper-parameters of the informative priors.
The key general finding is that the suggested values for and performed well.
All estimations of R(t), and h(t) functioned satisfactorily as n(or s) grew.
In most cases, the MSE, Bias, and WCI of all unknown parameters fell while their CPs grew as (T1, T2) increased.
Due to the gamma information, the Bayes estimates of and behaved more predictably than the other estimates. Regarding credible HPD intervals, the same statement might be made.
When the parameter of binomial r was increased, the proposed estimates of and performed better in most cases.
7. Application
The data set, which has been examined by [
11], had 30 assessments of the tensile strength of polyester fibers. The following details are included in the data set: “0.023, 0.032, 0.054, 0.069, 0.081, 0.094, 0.105, 0.127, 0.148, 0.169, 0.188, 0.216, 0.255, 0.277, 0.311, 0.361, 0.376, 0.395, 0.432, 0.463, 0.481, 0.519, 0.529, 0.567, 0.642, 0.674, 0.752, 0.823, 0.887, 0.926”. For data on the strength of polyester fibers, where the Kolmogorov-Smirnov distance is 0.0569 with a
p-value of 0.9999, [
11] explores the MLE of this model using several measures of goodness-of-fit. The Kolmogorov-Smirnov test findings showed that the KMKu distribution fits the data on polyester fiber strength.
Two GPHCS-T2 samples with
s = 20 and 25 were produced from the tensile strength of polyester fibers data in order to explain the proposed estimation methodology. The binomial removal has been used to obtain the GPHCS-T2 samples with different parameters of
p = 0.2, 0.5, and 0.8.
Table 6 lists the computed
R(
t) and
h(
t) at
t = 0.6 and 0.85 by maximum likelihood estimates (MLE) and Bayesian estimation, respectively, along with their standard error (SE). By repeating the MCMC sampler 12,000 times and disregarding the first 2000 times as burn-in, the Bayes estimates (with their SE) were evaluated using incorrect gamma priors and are also provided in
Table 4 because there was no prior knowledge about the unknown KMKu parameters
and
from the given data set. In order to estimate unknown hyperparameters for the computational logic, elective hyperparameters were employed. In terms of the minimum standard error and interval width values, it is evident from
Table 6 that the MCMC estimates of
and
performed better than the others.
Figure 2,
Figure 3 and
Figure 4 were created to examine the maximum values of the estimators by profile likelihood as well as the existence and uniqueness of the log-likelihood function by contour plot with regard to different d and q options based on GPHCS-T2 samples with
s = 20 and distinct
p = 0.2, 0.5, and 0.8, respectively.
Figure 5 clearly shows that the MCMC technique converged favorably and that the recommended size of the burn-in sample was adequate to completely nullify the impact of the recommended beginning values.
Figure 5 demonstrates that the estimated estimates of
and
were roughly symmetrical for each sample when
s = 20.
Figure 6,
Figure 7 and
Figure 8 were created to examine the maximum values of the estimators by profile likelihood as well as the existence and uniqueness of the log-likelihood function by contour plot with regard to different d and q options based on GPHCS-T2 samples with
s = 25 and distinct
p = 0.2, 0.5, and 0.8, respectively.
Figure 9 clearly shows that the MCMC technique converged favorably and that the recommended size of the burn-in sample was adequate to completely nullify the impact of the recommended beginning values.
Figure 9 demonstrates that the estimated estimates of
and
were roughly symmetrical for each sample when
s = 25.