1. Introduction
The development of new probability distributions plays a crucial role in statistical modeling, particularly in reliability engineering, survival analysis, and actuarial science. In many real-world applications, existing distributions fail to adequately capture the complexity of observed data, leading to the need for more flexible and robust models (see [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12]). One such challenge arises in studying lifetime data, where traditional distributions may not fully accommodate various hazard-rate behaviors. This motivates the derivation of new probability models that better describe real-world phenomena.
For the PCT-II approach, ref. [
2] created an algorithm to simulate general PCT-II samples from uniform or other continuous distributions. Some procedures for the estimation of parameters from different lifetime distributions based on PCT-II were developed by many authors, which include [
1,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29]. Recently, ref. [
28] looked at the A and D optimal censoring plans in the PCT-II scheme order statistics. Ref. [
26] suggested a new test statistic based on spacings to see if the samples from the general PCT-II scheme are from an exponential distribution. Ref. [
30] estimated a Burr distribution’s unknown parameters, reliability, and hazard functions under the PCT-II sample.
Ref. [
31] showed that when a controlled stepwise sample of the second type was available, parametric (classical and Bayes) point estimation procedures could be used to determine reliability characteristics, such as the reliability function and the mean time to failure for the Xgamma distribution, based on all observations.
To demonstrate the PCT-II scheme analyzing methods, the examiner gives different but identical units to the life test. There are
n units to be tested at the beginning of the experiment, and the test ends when the
mth
unit fails. Once one unit fails, the time
is recorded and
units are randomly selected from the remaining survival units
. Again, when the second unit fails, the time
will be recorded, and
units are then randomly withdrawn from the remaining
survival units. This experiment stops at the
mth failure, which is determined in advance, at time
, and
. The likelihood function based on a PCT-II sample
is given by (see, [
1]):
where
C is defined as
and
and
represent the cumulative distribution function (Cdf) and probability density function (Pdf), respectively. For the representation of the PCT-II scheme,
Figure 1 illustrates the PCT-II scheme.
This article introduces a new probability distribution using the concept of an induced distribution. Let
X be a continuous random variable with Pdf
, Cdf
, and expectation
. We define an induced Pdf
as:
It is well known that:
Thus, the induced Pdf
can be rewritten as:
The assumed
is the Xgamma distribution which was proposed by [
16] with Pdf and Cdf, respectively, given by:
and
Many of the statistical characteristics of this distribution are easily studied, and it exhibits a strong relationship with its parent distribution, i.e., the Xgamma distribution. The Xgamma distribution is chosen for its flexibility in modeling lifetime data, combining gamma and exponential properties, and adapting well to progressive Type-II censoring (PCT-II), and it exhibits similar properties to the Lindley distribution. Using the approach described in Equation (
2), where
is the mean of the random variable
, we derive the Pdf of the new distribution, which we term the induced Xgamma distribution (iXgamma). Hence, the Pdf of the iXgamma distribution is obtained by substituting
from Equation (
4) into Equation (
2). It is expressed as:
The Cdf of the iXgamma distribution can be derived by integrating Equation (
5) concerning
x. It is given by:
The newly proposed iXgamma distribution can also be expressed as a form of the following mixture:
where
and
The Pdf and Cdf plots for the iXgamma distribution with different values of
are shown in
Figure 2. These plots exhibit characteristics similar to those of the exponential distribution.
The novelty of our research is defined in three key aspects. First, we derived a new distribution based on the concept of induced distributions, as given in Equation (
2), utilizing
, leading to the proposed iXgamma distribution. Second, our proposed distribution is not merely an extension of the Xgamma distribution but rather a hybrid approach that integrates two fundamental distributions in mathematical statistics and reliability analysis—the Gamma and exponential distributions.
The limitations and challenges of this work are primarily related to the complexity of parameter estimation, computational intensity, limited sample sizes, model power, prediction, and inference difficulties. Advanced statistical methods and thorough evaluation of the data details and the censoring scheme are frequently needed to address these problems. Due to their costly numerical evaluations and complicated outcomes, new extended-gamma (iXgamma) lifetime models receive little attention from researchers. To our knowledge, no attempt has been made to estimate the iXgamma parameters based on the PCT-II scheme since the iXgamma distribution was presented in the literature, except for the work of [
32], who used a complete sample to study Bayesian inferences based on non-informative priors.
The rest of the article is organized as follows:
Section 2 introduces properties of the proposed iXgamma distribution. We discussed the maximum likelihood estimation and maximum product spacing estimation in
Section 3. In
Section 4, we present the Bayesian inferences and in
Section 5, the confidence intervals estimations are presented. Simulation investigations and discussion are highlighted in
Section 6. Three real datasets are analyzed in
Section 7. In
Section 8, point prediction is addressed using the maximum likelihood predictor and the Bayesian predictor. Finally, various observations are made in
Section 9.
4. Bayesian Estimation
In this section, we derive the Bayesian estimation (BE) of the parameter
for the iXgamma distribution under PCT-II samples, assuming a Gamma prior distribution. The Gamma prior is specified as follows:
Thus, the posterior distribution is given by:
where
is the likelihood function given in Equation (
7) based on the PCT-II samples (x̠). The integral in the denominator ensures that the posterior distribution is normalized. Thus, the posterior distribution can be written
A commonly used symmetrical loss function is the square error loss (SEL), which assigns equal losses to overestimation and underestimation. If
is estimated by an estimator
, then the SEL function is defined as
Therefore, the BE of any function of
, say
under the SEL function, can be obtained as
Ref. [
28] considered a LINEX (linear-exponential) loss function
for a parameter
is given by
This loss function is suitable for situations where overestimation is more costly than underestimation. Ref. [
29] discussed BE and prediction using the LINEX loss. Hence, under the LINEX loss function, the BE of a function
is
The BE of a function of
, concerning the general entropy loss (GEL) function, is given by
All the above estimators in Equations (
15)–(
17) are the form of the ratio of two integrals for which simplified closed forms are not available. The Tierney–Kadan method is a numeric integration technique in such cases, and the Markov Chain Monte Carlo (MCMC) method finds the BEs.
4.1. MCMC Method
The MCMC methodology is one of the most effective numerical approaches in Bayesian inference. Moreover, calculating the normalization constant is unnecessary when summarizing the posterior distribution using MCMC methods. MCMC techniques have been extensively utilized in Bayesian statistical inference [
32,
33,
34,
35,
36]. An algorithm, the Metropolis–Hastings (MH) algorithm, can be utilized as a sampler for the MCMC approach. A critical aspect of this method involves selecting an appropriate proposal distribution that satisfies two key criteria: (1) it should be easy to simulate, and (2) it should closely approximate the posterior distribution of interest. Once such a proposal distribution is identified, the acceptance/rejection rule is employed to generate random samples from the target posterior distribution. This enables efficient approximation of the Bayesian posterior summaries.
After obtaining a set of
M samples from the posterior distribution, discarding a portion of the initial samples (known as the burn-in samples), and retaining the remaining samples for further analysis is common practice. Specifically, the BEs of the parameter
, using the SEL and LINEX, GEL functions, are provided as follows:
where
represents the number of burn-in samples.
MCMC Convergence
The Gelman–Rubin statistics are available to assess the convergence of the samples produced by the posterior distribution using MCMC. The goal of this approach is to confirm that MCMC samples have reached the target distribution that was planned for them. In this work, we use several Markov chains to evaluate convergence using the Gelman–Rubin statistic, which is sometimes referred to as the potential scale reduction factor (PSRF). The following formula is used to determine the PSRF:
where
is the average within-chain variance, and the term in the numerator combines the between-chain variance
with the within-chain variance
to estimate the marginal posterior variance of the parameter. The number of iterations of each chain is shown here by
M.
It is likely that the chains have converged if the PSRF value is near 1. The Gelman–Rubin statistic is extremely valuable as it uses several chains to provide an independent measure of convergence, eliminating the potential of erroneous integration that usually exists when using only a single chain. For more information, see [
37].
4.2. Tierney–Kadane Method
Here, in this method, the ratios of the integrals from Equations (
15)–(
17) are represented in two forms shown below:
The equation takes the form of
where
and
maximize
and
, respectively, and
and
are the negatives of the inverse Hessian of
and
at
and
, respectively. Here, Equation (
25) is used to obtain the Bayes estimators for the parameter
, which is an approximation form. Now, differentiating Equation (
24) with respect to
, we obtain,
Now, the BE of
can be computed as follows:
4.2.1. Under SEL Function
If
, then the Equation (
22) becomes
The first derivatives of
with respect to
is
Again differentiating with respect to
, we obtain
Therefore,
Putting the value in Equation (
25), we obtain the Bayes estimator using the TK method under the SEL function.
4.2.2. Under the LINEX Function
If
, then the Equation (
22) becomes
The first derivative of
with respect to
is
Again differentiating with respect to
, we obtain
Therefore,
Putting the value in Equation (
25), we obtain the Bayes estimator using the TK method under the LINEX function.
4.2.3. Under the GEL Function
If
, then the Equation (
22) becomes
The first derivative of
with respect to
is
Again differentiating with respect to
, we obtain
Therefore,
Putting the value in Equation (
25), we obtain the Bayes estimator using the TK method under the GEL function.
6. Simulation Study and Discussion
In this section, Monte Carlo simulation studies are conducted to evaluate the performance of various estimation methods, including MLE, MPS, and BE, within the framework of the PCT-II scheme for the iXgamma distribution. We produce 1000 data points from the iXgamma distribution with parameters
and 3. The choice of parameter values for the iXgamma distribution covered different shapes and behaviors of the distribution, allowing for an analysis of its behavior under these values. The PCT-II scheme can be assumed according to a given value of
n and
m and a different pattern for removing items
, where
, as shown in
Table 2.
This implies that the scheme corresponds to a Type-II censoring scheme as a particular case, with the number of failure items given by . Moreover, complete sampling is treated as a special case of the PCT-II scheme when for .
6.1. Monte Carlo Simulation Process
Generate random samples for the iXgamma distribution with parameter
using the assumed schemes of PCT-II in
Table 2 employing the algorithm proposed by [
2].
Estimate the parameter using non-Bayesian methods (Non-BE), specifically MLE and MPS. From the MLE, the variance of the parameter can also be derived. When calculating MLEs, the initial estimate values are assumed to coincide with the true parameter values.
Estimate the parameter using Bayesian methods (BE) as follows:
- (a)
Assume the informative prior case, where the hyper-parameters are proposed and fixed at and . These values are plugged into the posterior density.
- (b)
Consider three loss functions: SEL, LINEX (with ), and GEL (with ). We have experimented with different values to achieve the best possible results through these functions, ultimately arriving at the current values.
- (c)
Estimate the parameter using the BE methods adopted in the research, initially through the TK method and MCMC.
- (d)
For the MCMC method, using the MH algorithm:
The initial values for the MH algorithm are set to the MLE estimates with their variances.
Proposal samples for are generated from a normal distribution: , where is the MLE of , and is the observed Fisher information matrix of .
A total of 10,000 posterior samples are generated, with the first 2000 samples discarded as burn-in to enhance the convergence of the MCMC estimation.
The final estimate of is obtained by computing the average based on the chosen loss functions.
The acceptance rate in our MCMC implementation was approximately 65%.
Estimate the CIs for the parameter , which are the NA, NL, and HPD intervals.
Perform steps 1 to 4 iteratively for a total of 1000 repetitions, storing all resulting estimates. Subsequently, compute the average (AV) and root mean square error (RMSE) for the point estimate. Additionally, determine the mean values of the lower and upper bounds of the CIs, calculate the average interval length (AIL), and evaluate the coverage probability (CP) as a percentage (%).
In addition to the specific PCT-II patterns in
Table 2, a general case of complete sampling has been added to estimates where
and did not involve the exclusion of units from the experiment.
6.2. Comments on Results
In
Table 3,
Table 4 and
Table 5,we obtained point estimates for the parameter
at values 0.5, 1.5, and 2.5, respectively, for all estimation methods. Meanwhile, in
Table 6,
Table 7 and
Table 8, we obtained interval estimates for the same parameter values in the same sequence. In general, from the results, we observe that as
m increases, there is an improvement in the estimates converging closer to the assumed parameter value. Additionally, we observe a decrease in RMSE and, finally, a decrease in AILs for all estimation methods. Furthermore, CP is also observed to be constrained within specific bounds greater than 94%. There are also specific observations regarding the patterns of PCT-II and estimation methods, which can be summarized as follows:
With respect to the Non-BE methods, it is observed that the MLE method outperforms the method of percentile score MPS in scenarios involving PCT-II schemes, , , and , whereas the opposite is observed in schemes, and .
In the BE methods, specifically, the TK method, we observe that the LINEX loss function performs best among the three loss functions when the parameter is less than 1, whereas SEL appears to perform better when is greater than 1.
As for the MCMC method, there is variability in the performance of the loss functions and different PCT-II schemes, and we could not determine a clear preference for monitoring patterns or loss functions. However, it can be said that in most cases, the SEL tends to have the highest preference, followed by the GEL.
Regarding interval estimation, we observe that the highest efficiency is with the HPD method, followed by NA, and finally, NL intervals for . For , the NA interval has the smallest AILs.
For checking of the convergence of the MCMC, assuming
and the PCT-II pattern is
, we may create three separate chains using a number of iterations of 10,000 samples to confirm the convergence of the MCMC for the two-parameter
. We were able to use Equation (
21) to estimate the PSRF. According to the results, all PSRF estimations are near to one, demonstrating that the chains have converged to the target distribution and implying that the MCMC method’s sampling is stable and trustworthy for more research.
9. Concluding Remarks
In this research, a new life distribution, named the iXgamma distribution, has been proposed and studied based on induced distributions. The PCT-II scheme is implemented for the proposed distribution. Several mathematical properties of the iXgamma distribution, including its reliability properties, moments, order statistics, and stress-strength reliability, have been derived. Various Non-BE methods, such as MLE and MPS, have been explored. Additionally, Bayesian procedures have been examined using numerical techniques (MCMC with the MH algorithm) and approximation methods, such as the Tierney–Kadane method based on the loss functions SEL, LINEX, and GEL. Interval estimation is also studied for MLE and BE, using Asy-CI and HPD intervals, respectively. The effectiveness of the proposed distribution is demonstrated through the analysis of three actual lifetime datasets. Our proposed distribution is shown to be superior to the Gamma, Rayleigh, exponential, and Xgamma distributions. Furthermore, Monte Carlo simulation studies were conducted to assess the performance of both non-Bayesian and BEs using the PCT-II scheme for the iXgamma distribution. Among the point estimation methods, MLE performed well in non-BE, while the Terry–Kande method excelled in BE. The HPD interval estimation method proved to be the most efficient among different interval estimation techniques. After our study, several predictor functions for the iXgamma distribution under the PCT-II scheme were derived, including the maximum likelihood predictor, best-unbiased predictor, and Bayesian predictor. Future work can explore extending the study to acceptance sampling plans, investigating informative and non-informative priors in BE, and studying causes of failure.