Statistical inference of the Birnbaum-Saunders model using adaptive progressively hybrid censored data and its applications

1.   Introduction

Birnbaum and Saunders ^[1] proposed a two-parameter fatigue life distribution for simulating the fatigue life of metals subjected to periodic stress. They also established a natural physical rationale for the Birnbaum-Saunders (BS) model through fatigue failure produced by cyclic stress. It is also one of these distributions that was created by performing a monotone transformation on a typical normal random variable. Despite the Weibull model being the most commonly used failure time distribution, the BS population has lately attracted a lot of interest, owing to the varied forms of its density and the non-monotonicity of the nature of its failure rates. Since the work of Birnbaum and Saunders ^[1], extensive work has been done on this model, providing different interpretations, generalizations and inferential methods. See, for example, Ng et al. ^[2], Lemonte et al. ^[3], Pradhan and Kundu ^[4] and Xiuyun et al. ^[5]. Recently, Balakrishnan and Kundu ^[6] reviewed the same model in the context of various inferential approaches. Suppose $Y$ is a random variable that follows the BS distribution, symbolized by $Y\sim\mbox{BS}({\mathit{\boldsymbol{\Sigma}}})$, where ${\mathit{\boldsymbol{\Sigma}}} = (\gamma, \mu)^{\top}$. Define the following quantities:

The probability density function (PDF) (say, $f(\cdot)$) and cumulative distribution function (CDF) (say, $F(\cdot)$) of the BS distribution can be given, respectively, as follows:

and

where $\Phi(\cdot)$ is the CDF of the standard normal distribution, $\gamma > 0$ is the shape parameter, and $\mu > 0$ is the scale parameter. From (1.2) and (1.3), the BS distribution's reliability function (RF) and hazard rate function (HRF) (at a mission time $t > 0$), symbolized by $R(\cdot)$ and $h(\cdot)$, are given, respectively, by

and

Figure 1 depicts the density and failure rate shapes of the BS distribution. It exhibits that the BS density is always unimodal for all values of $\gamma$ and $\mu$. It also indicates that the shape of the HRF of the BS is an upside-down shape for all values of $\gamma$, as proved by Kundu et al. ^[7].

Experimenters frequently gather Type-Ⅱ progressively censored (T2-PC) data in life-testing and reliability studies to decrease testing time and expenses. Balakrishnan and Aggarwala ^[8] presented the definition, mathematical features, and applications of the T2-PC sample. Kundu and Joarder ^[9] proposed a more broad censorship technique known as progressive Type-Ⅰ hybrid censoring scheme. The problem with this technique is that the usable sample size is random and may turn out to be very few or zero for dependable goods. The statistical inference methods will be inefficient as a result. To deal with this shortcoming, Ng et al. ^[10] offered an adaptive Type-Ⅱ progressive hybrid censoring (AT2-PHC) technique to improve statistical investigation efficiency. Let $k$, $T$, and $R_1, R_2, \dots, R_k$ refer to the set amount of failures, the ideal duration on the test, and the progressive censoring plan, respectively, which are specified before beginning the examination, with the realization that the values for a particular of the $R_i$ can vary as needed throughout the experiment. The AT2-PHC scheme can be distilled in the following manner: Consider that $n$ components are subjected to a life test with $k < n$ then, at the time of $i^{th}$ failure, denoted by $Y_{i:k:n}$, $R_i$ of the remaining components are randomly removed from the test. Following this approach, we have two distinct possibilities. The first is the standard T2-PC scheme, which occurs when the time of the $k^{th}$ failure, indicated by $Y_{k:k:n}$, comes before time $T$, i.e., $Y_{k:k:n} < T$. The other scenario occurs when $Y_{j:k:n} < T < Y_{j+1:k:n}$, where $(j+1 < k)$ and $Y_{j:k:n}$ is the $j^{th}$ failure time. In this case, we are not going to eliminate any living objects from the test by including $R_{j+1} = R_{j+2} = \dots = R_{k-1} = 0$ and $R^{*} = n-k-\sum_{i = 1}^{j} R_{i}$. This schematic makes certain that we finalize the test when we meet the required amount of failures $k$, and that the overall test duration does not deviate significantly from $T$. Let $y_{1:k:n} < \dots < y_{j:k:n} < T < y_{j+1:k:n} < \dots < y_{k:k:n}$ be an AT2-PHC sample from a continuous population with PDF $f(y)$ and CDF $F(y)$, with a progressive censoring plan $\textbf{R} = (R_{1}, \dots, R_{j}, 0, \dots, 0, R^{*})$, then, without the constant term, the likelihood function of the observed data can be presented as

Many studies considered the AT2-PHC scheme; see, for example, Nassar and Abo-Kasem ^[11], Alotaibi et al. ^[12], Ahmad et al. ^[13], Dutta et al. ^[14], and Elshahhat and Nassar ^[15], among others.

According to Balakrishnan and Kundu ^[6], around 200 studies explored various aspects of the BS distribution. Scarce work has been done to estimate the parameters or reliability indices of this model utilizing newly proposed censoring strategies. This may be justified due to the intricate structure of the new censoring techniques as well as the intractable formulas of the BS distribution's PDF and CDF. Therefore, we are motivated in this study to study the estimation problems of the BS distribution using the AT2-PHC scheme. This scheme is chosen because of its flexibility in ending the experiment and its ability to generalize some censoring schemes like Type-Ⅱ censoring and T2-PC schemes. Moreover, the available studies consider only the estimation of the model parameters without saying anything about the estimation of the reliability indices. In order to get point and interval estimates of the shape, scale, RF, and HRF of the BS distribution, we explored two estimating approaches: maximum likelihood and Bayesian methods. First, the maximum likelihood estimates (MLEs) are obtained, followed by two approximate confidence intervals (ACIs) acquired using the normal approximation and log-transformed MLEs. For acquiring Bayesian point estimates, we address the usage of the squared error loss function and the implementation of the Markov chain Monte Carlo (MCMC) process. The Bayes credible intervals (BCIs) and highest posterior density (HPD) credible intervals are also computed. Following that, we undertake a simulation study to compare the various estimates using some statistical criteria to determine which method offers superior estimates. Two applications are addressed from a practical standpoint to demonstrate the usefulness of the proposed methodologies.

The remaining sections of the paper are formatted as follows: Section 2 discusses the MLEs and ACIs. The Bayesian estimation of the BS distribution is presented in Section 3. The discoveries of the simulation investigation are presented in Section 4. In Section 5, we present two applications to real-world data. Finally, some final thoughts are presented in Section 6.

2.   Classical estimation

In this section, we will delve into the estimation of the BS distribution through the AT2-PHC scheme using the maximum likelihood approach. Employing this approach, we aim to derive both point and interval estimates of the unknown parameters as well as reliability indices. Suppose we have an AT2-PHC sample $\underline{\text{y}}$, drawn from a population that acts as the BS distribution, with progressive censoring design $\textbf{R}$. Upon omitting the constant term, the likelihood function can then be stated by combining (1.2), (1.3), and (1.6) as follows:

where $w(y_{i}; \mu)$ and $\acute{w}(y_{i}; \mu)$ are as defined in (1.1). It is possible to express the log-likelihood function of (2.1) as

By differentiating the log-likelihood function in (1.2) with respect to parameters $\gamma$ and $\mu$ and equating them to zero and one, we can get the MLEs of the parameters. The normal equations in this case are given by

and

where $H[y] = \phi(y)/\Phi(-y)$ and $\phi(.)$ is the PDF of the standard normal distribution,

The simultaneous solution of (2.3) and (2.4) is the MLEs of $\gamma$ and $\mu$, denoted by $\hat{\gamma}$ and $\hat{\mu}$, respectively. To find the required MLEs, one must employ a numerical method because (2.3) and (2.4) are nonlinear equations. After obtaining the MLEs $\hat{\gamma}$ and $\hat{\mu}$, we can employ the invariance property of the MLEs to acquire the MLEs of the RF and HRF of the BS distribution as

and

where $\hat{\mathit{\boldsymbol{\Sigma}}} = (\hat{\gamma}, \hat{\mu})^{\top}$.

Along with obtaining the point estimates of the different parameters, it would be useful to get the interval estimates of these parameters. It is established that the MLEs' asymptotic distribution approaches normality, such that

Here, we use the observed Fisher information matrix to estimate the variance-covariance matrix as follows:

with the following elements

and

where $w_{i}\equiv w(y_{i}; \mu)$,

$\acute{H}[y] = yH(y)-H^{2}(y)$, $\acute{\psi}(y_{i}; \mu) = -\frac{1}{4\mu^{2}}\left(\sqrt{\frac{\mu}{y_{i}}} +3\sqrt{\frac{y_{i}}{\mu}} \right),$ and $H^{\circ}[w_{i}/\gamma] = \frac{w_{i}\acute{H}[w_{i}/\gamma]}{\gamma^{3}}-\frac{2H[w_{i}/\gamma]}{\gamma^{2}}.$

Thus, the ACIs of $\gamma$ and $\mu$ can be computed, respectively, as $\hat{\gamma}\pm z_{\alpha/2}\widehat{SE}(\hat{\gamma})$ and $\hat{\mu}\pm z_{\alpha/2}\widehat{SE}(\hat{\mu})$, where $z_{\alpha/2}$ is the percentile $100(1-\alpha/2)\%$ of the standard normal distribution and $\widehat{SE}(.)$ is the standard error of estimate obtained by taking the square root of the estimated variance acquired from (2.5). Building the ACIs for RF and HR is a further important topic. The present issue is to find the variances of $\hat{R}(t)$ and $\hat{h}(t)$ given the variances of $\hat{\gamma}$ and $\hat{\mu}$. To address this issue, a particular approach is the delta method, which can be applied to approximate the required variances.

Assume that $\hat{D}_{1} = (R_\gamma, R_\mu)$ and $\hat{D}_{2} = (h_\gamma, h_\mu)$ are two vectors that are obtained at the MLEs $\hat{\gamma}$ and $\hat{\mu}$ and comprise the first partial derivatives of RF and HRF for $\gamma$ and $\mu$, respectively, with the following elements

The approximate estimated variances of $\hat{R}(.)$ and $\hat{h}(.)$ can be obtained as

As a result, one can get the ACIs of RF and HRF as

with $\widehat{SE}(\hat{R}) = \sqrt{\widehat{var}(\hat{R})}$ and $\widehat{SE}(\hat{h}) = \sqrt{\widehat{var}(\hat{h})}$. It is noteworthy to note at the end of this section that negative lower bounds may result from obtaining the ACIs using the normal approximation (ACI-NA) of the individual parameters. In such cases, one can use the normal approximation of the log-transformed (NL) MLE technique to circumvent this issue. For any parameter, say $\kappa$, the ACI using NL (ACI-NL) can be obtained as follows:

3.   Bayesian estimation

We have covered point and interval parameter estimation using the frequentist approach in the last section. This section examines the Bayesian methodology under the squared error loss function. Achcar ^[16] started by thinking about the Bayesian inference of a BS distribution's parameters. He took into account Jeffreys prior to $\gamma$ and $\mu$ to create the Bayesian inference using the complete sample. Xu and Tang ^[17] have taken into account the non-informative prior for the Bayesian inference of the parameters and have employed the Lindley approximation to get Bayes estimates of the unknown parameters of the BS distribution using the entire sample data. Because of the intricate formulations of the Fisher information matrix, it is difficult to establish the Jeffreys prior for the BS distribution in the case of AT2-PHC data. Therefore, we consider the same Jeffreys prior to $\gamma$ and $\mu$ as suggested by Achcar ^[16] as follows:

where $\zeta(\gamma^{2}) = \left(\frac{1}{\gamma^{2}}+\frac{1}{4} \right)^{0.5}$. It is to be noted that when $\zeta(\gamma^{2}) = 1$, the Jeffreys prior is reduced to the non-informative prior case. Combining the likelihood function in (2.1) with (3.1), one can write the joint posterior distribution over proportionality as

where $\underline{\text{y}}$ is the observed data. Let $\varpi(\mathit{\boldsymbol{\Sigma}})$ be any function of the parameters to be estimated, then under the squared error loss function, the Bayes estimate, denoted by $\tilde{\varpi}(\mathit{\boldsymbol{\Sigma}})$, is the posterior mean acquired as follows:

The integral provided in (3.3) lacks an explicit expression. To overcome this challenge, we consider the use of the MCMC technique to create the Bayes point, BCI, and HPD estimates. The MCMC approach is used to get samples from (3.2) in order to obtain the necessary estimations. The posterior distribution is sampled using the MCMC technique, and the Bayes estimates, BCIs, and HPD intervals are then obtained using these samples. We must first obtain the full conditional distributions of the unknown parameters $\gamma$ and $\mu$ in order to employ the MCMC approach. The necessary full conditional distributions of $\gamma$ and $\mu$ can be obtained, respectively, from (3.2), as

and

Due to the complexity of the conditional distributions in (3.4) and (3.5), which cannot be reduced to any well-known distributions, we employ the Metropolis-Hastings (M-H) algorithm. For sampling from complicated probability distributions, particularly in Bayesian estimation, the M-H algorithm is a popular MCMC technique. It is essential to Bayesian statistics because it helps approximate the posterior distribution by combining observed data and past knowledge; for more detail about the MCMC technique, see Contreras-Reyes et al. ^[18]. To acquire the Bayes estimates, BCIs, and HPD intervals of $\gamma$ and $\mu$, as well as the life parameters, the following processes must be followed with a normal proposal distribution (NPD) in order to obtain the necessary samples as:

Step 1. Set $l = 1$ and start with initial values $\left(\gamma^{(0)}, \mu^{(0)}\right) = (\hat{\gamma}, \hat{\mu}).$

Step 2. Simulate $\gamma^{(l)}$ from $g_{\gamma}(\gamma|\mu, \underline{\text{y}})$ using NPD and follow the M-H steps.

Step 3. Generate $\mu^{(l)}$ from $g_{\mu}(\mu|\gamma, \underline{\text{y}})$ using NPD and follow the M-H steps.

Step 4. Use $(\gamma^{(l)}, \mu^{(l)})$ to compute $R^{(k)}$ and $h^{(k)}$, respectively.

Step 5. Put $l = l+1$.

Step 6. Repeat the process $M$ times to generate a sequence of

After discarding the first $\acute{M}$ acquired samples as a burn-in period, the Bayes estimate of any parameter, say $\eta$, can be computed as

In order to compute the BCI and HPD interval for $\eta$, sort the acquired sample of $\eta^{(l)}, l = \acute{M}+1, \dots, M$ as $\eta^{(\acute{M}+1)}, \dots, \eta^{M}$. Therefore, the $100(1-\alpha)\%$ BCI of $\eta$ can be obtained as

Moreover, the $100(1-\alpha)\%$ HPD interval of $\eta$ is given by

where $l^* = \acute{M}+1, \dots, M$ is determined such that

where $[\varsigma]$ refers to the largest integer less than or equal to $\varsigma$.

4.   Monte Carlo comparisons

Monte Carlo simulations are performed in this part to test the performance of the proposed estimators of $\gamma$, $\mu$, $R(t)$, and $h(t)$ developed in this study. From $\mbox{BS}(\gamma, \mu) = (0.5, 1.5)$, based on various options of $n$ (total experimental units), $k$ (effective sample size), and $T$ (threshold time), we simulate 1,000 AT2-PHC samples. At $t = 0.1$, the plausible values of $R(t)$ and $h(t)$ are 0.98954 and 0.12939, respectively. When $T( = 1, 2)$ and $n(40, 80)$, the number of failed subjects $k$ is specified as a failure percentage $\frac{k}{n}\times100\% = 50\%$ and $75\%$. Briefly, the next procedure reports the generation steps of the AT2-PHC strategy:

Step 1: Simulate a T2-PC sample as:

(a) Create $\vartheta$ independent observations of size $k$ as ${{\vartheta}_{1}}, {{\vartheta}_{2}}, \dots, {{\vartheta}_{k}}$.

(b) Set ${{\varrho}_{i}} = \vartheta_{i}^{{{\left(i+\sum\nolimits_{d = k-i+1}^{k}{{{R}_{d}}} \right)}^{-1}}}, \ i = 1, 2, \dots, k$.

(c) Set ${{u}_{i}} = 1-{{\varrho}_{k}}{{\varrho}_{k-1}}\cdots {{\varrho}_{k-i+1}}$ for $i = 1, 2, \dots, k$.

(d) Set ${{Y}_{i}} = {{F}^{-1}}({{u}_{i}}; \gamma, \mu), \ i = 1, 2, \dots, k$, as the acquired T2-PC sample from $\text{BS}(\gamma, \mu)$.

Step 2: Find $j$ (at ${{Y}_{j}} < T < {{Y}_{j+1}}$) and remove the staying items ${{Y}_{j+2}}, \dots, {{Y}_{k}}$.

Step 3: From ${f\left(y; \gamma, \mu \right)}\left[1-F\left({{y}_{j+1}}; \gamma, \mu \right) \right]^{-1}$, obtain the first $k-j-1$ order statistics with sample size $n-j-\sum\nolimits_{d = 1}^{j}{{{R}_{d}}}-1$ as ${{Y}_{j+2}}, \dots, {{Y}_{k}}$.

Once 1,000 AT2-PHC samples are acquired, using $\textsf{R}$ programming software version 4.2.2, we utilize the following recommended packages:

● A '$\textsf{maxLik}$' package with '$\textsf{maxNR}$' function (by Henningsen and Toomet ^[19]) to implement the Newton-Raphson method in turn to evaluate the MLEs and ACIs of $\gamma$, $\mu$, $R(t)$, and $h(t)$.

● A '$\textsf{coda}$' package with '$\textsf{run_metropolis_MCMC}$' function (by Plummer et al. ^[20]) in turn to calculate the Bayes' and credible estimates.

Following Achcar ^[16], we have evaluated the Bayes estimators based on both Jeffreys and non-informative priors, denoted by Pr[A] and Pr[B], respectively. Using the M-H algorithm, for $\gamma$, $\mu$, $R(t)$, or $h(t)$, we ignore the first 2,000 of the 12,000 simulated Markovian variates as burn-in.

To examine the performance of the removal design $\textbf{R}$, we consider the following censoring schemes (CSs):

Practically, the average estimates (Av.Es) $\gamma$, $\mu$, $R(t)$, or $h(t)$ (say $\phi$) derived from the maximum likelihood (or MCMC) approach are computed by

where $\check{\phi}^{(i)}$ is an estimate of $\phi$ at $i$th sample, ${{\phi }_{1}} = \gamma$, ${{\phi }_{2}} = \mu$, ${{\phi }_{3}} = R(t)$, and ${{\phi }_{4}} = h(t)$.

The performance of the point estimations of $\phi$ is evaluated based on the following two metrics:

ⅰ) Root mean squared-error (RMSE):

ⅱ) Average relative absolute bias (ARAB):

The performance of the $100(1-q)\%$ interval estimations of $\phi$ is evaluated based on the following two metrics:

ⅰ) Average confidence length (ACL):

ⅱ) Coverage percentage (CP):

respectively, where $\mathbb{I}(\cdot)$ is the indicator operator and $(\mathcal{L}(\cdot), \mathcal{U}(\cdot))$ refers to the (lower-bound, upper-bound) of an interval estimate.

In Tables 1–8, the simulated RMSE, ARAB, ACL, and CP values of $\gamma$, $\mu$, $R(t)$, and $h(t)$ are displayed. From Tables 1–8, in regard to the smallest RMSE, ARAB, and ACL values as well as the highest CP values, we report the following comments:

● All offered estimations for the parameters and/or reliability characteristics of the BS model behaved well.

● As $n$(or $k/n$) grows, the accuracy of all estimates becomes satisfactory. The same note is also reached when the sum of removal items ($\sum\nolimits_{i = 1}^{k}{{{R}_{i}}}$) decreases.

● As $T_i, \ i = 1, 2$ increase, all simulated results of $\gamma$, $\mu$, $R(t)$, and $h(t)$ decrease. Also, when $T_i, \ i = 1, 2$ increase, the simulated CPs grow and close to the prespecified nominal level. It is to be noted that when $T$ increases, no additional failures are observed because the number of observed failures $k$ is predetermined. The little improvement observed in the various criteria, like the reduction in RMSEs and ACLs when $T$ increases, may be due to random error.

● All Bayes point estimates as well as the associated interval estimates behave better than those acquired from the likelihood approach.

● Due to the non-informative knowledge on the BS model parameters, the point and interval estimations of $\gamma$, $\mu$, $R(t)$, or $h(t)$ developed by Pr[A] and Pr[B] are close to each other. In general, the Bayes estimates using Pr[B] perform better than those using Pr[A] in terms of minimum RMSEs, ARABs, and ACLs.

● Comparing the interval estimation methods of $\gamma$, $\mu$, $R(t)$, or $h(t)$, it is clear that:

– The BCI and HPD interval estimates of $\gamma$, $\mu$, $R(t)$, or $h(t)$ are quite similar to each other.

– The ACI-NL estimates of $\gamma$ and $R(t)$ are superior to those acquired from ACI-NA estimates.

– The ACI-NA estimates of $\mu$ and $h(t)$ are superior to those acquired from ACI-NL estimates.

– The interval estimates of $\gamma$, $\mu$, $R(t)$, or $h(t)$ offered using both BCI and HPD approaches are better than those acquired from ACI-NA or ACI-NL methods.

● Comparing the proposed CSs 1, 2, and 3, it is clear that:

– The collected estimates of $\gamma$ and $R(t)$ perform better via CS-3 'right-censoring' than others.

– The collected estimates of $\mu$ and $h(t)$ perform better via CS-1 'left-censoring' than others.

● In conclusion, the Bayes' using the M-H framework is recommended for estimating the model parameters or the reliability features of the BS lifespan model.

5.   Data applications

In this section, we will utilize two real datasets from the clinical and chemical sectors to demonstrate the relevance and application of the suggested approaches to real-world occurrences.

5.1.   Clinical data

In this application, we shall use a clinical dataset of 56 blood samples from organ transplant recipients and assay an aliquot of each sample using a standard-recognized method of high-performance liquid chromatography. This dataset was originally given by Hawkins ^[21] and later rediscussed by Nassar et al. ^[22]. In Table 9, for computational convenience, all points of blood samples are divided by ten.

At first, we must determine whether the BS distribution is an acceptable model for blood data. The MLEs of $\gamma$ and $\mu$ are used to estimate the Kolmogorov-Smirnov (K-S) distance and its associated $P$-value. The MLEs $\gamma$ and $\mu$ with associated standard-errors (SEs) are 0.6678 (0.0631) and 20.839 (1.7563), respectively. At the same time, the K-S distance is 0.062 with a $P$-value of 0.982. This means that the BS distribution matches the blood data quite well. Also, based on the full blood dataset, the adaptability of the BS model is highlighted via various graphical tools, namely: (a) Contour; (b) estimated density; and (c) estimated/empirical reliability; see Figure 2. Figure 2(a) shows that the MLEs $\hat{\gamma}\cong{0.6678}$ and $\hat{\mu}\cong{20.839}$ existed and are unique; Figure 2(b) shows that the fitted BS density line reasonably fits the histogram; and Figure 2(c) shows that the fitted BS reliability line captures the empirical reliability line adequately.

Briefly, to show the effectiveness of the BS model based on the blood dataset, the BS$(\gamma, \mu)$ distribution is compared with three popular models, namely: lognormal$(\gamma, \mu)$, Weibull$(\gamma, \mu)$, and gamma$(\gamma, \mu)$ distributions.

To establish this goal, the K-S distance (with its $P$-value) is obtained for each model; see Table 10. The MLEs (with their SEs) of $\gamma$ and $\mu$ are also calculated and reported in Table 10. It indicates that the BS distribution is the best model compared to its competitors. It also indicates that the lognormal distribution is the next best choice among others. Figure 3 displays the histogram of the blood data and the fitted density lines, the empirical and fitted reliability functions, and the probability-probability (PP) lines. Additionally, Figure 3 supports the same fit findings.

Using various choices of $T$ and $\textbf{R}$, we highlight the suggested estimates based on three AT2-PHC samples (with $k = 26$) created from the blood data; see Table 11. Using the proposed M-H steps, we simulate 40,000 MCMC iterations and discard the first 10,000 samples as burn-in. For each S[$i$] for $i = 1, 2, 3,$ in Table 11, the maximum likelihood and Bayes' MCMC estimates (along with their SEs) as well as 95% asymptotic (ACI-NA and ACI-NL methods) intervals and 95% credible (BCI and HPD) intervals (along with their widths) of $\gamma$, $\mu$, $R(t)$, and $h(t)$ (at $t = 10$) are obtained; see Table 12. Both Pr[A] and Pr[B] are also utilized to obtain the MCMC estimates and associated credible interval estimates. The results in Table 12 show that the Bayes point and interval estimates of ${\gamma}$, ${\mu}$, $R(t)$, and $h(t)$ obtained using Pr[B] perform better than other estimates in terms of minimum SEs and interval widths.

Figure 4 shows the log-likelihood functions of $\gamma$ and $\mu$ for all samples S[$i$] for $i = 1, 2, 3,$ which demonstrates that all offered frequentist estimates of $\gamma$ and $\mu$, respectively, existed and are unique. To do any more calculations based on the industrial device data in the future, we recommend using these numbers as starting points. In Table 13, using the staying 30,000 draws (from S[1] as an instance) of $\gamma$, $\mu$, $R(t)$, and $h(t)$, several properties including mean, mode, (1$^{st}$, 2$^{nd}$, and 3$^{rd}$) quartiles denoted by $Q_i, \ i = 1, 2, 3$, standard deviation (SD) and skewness (Skew.) are reported. It supports the same facts reported in Table 12. From S[1] (as an example), to evaluate the convergence status of the staying 30, 0000 MCMC iterations, trace (with Gaussian kernel) and MCMC frequencies plots $\gamma$, $\mu$, $R(t)$, and $h(t)$ are shown in Figure 5. Other MCMC properties and plots based on S[2] and S[3] are reported in the Supplementary File.

The sample mean and HPD interval boundaries are displayed as solid and dotted lines in each panel, respectively. Figure 5 indicates that the MCMC procedure converges very well and that the generated posterior estimates of $\gamma$, $\mu$, and $R(t)$ are fairly symmetric, while those near to are being negatively skewed.

5.2.   Chemical data

This application analyzes a chemical dataset that represents 101 observations of the fatigue life of 6061-T6 aluminum coupons cut parallel to the direction of rolling and oscillated at 18 cycles per second (with a maximum stress per cycle of 31 000 psi); see Ng et al. ^[23]. Table 14 lists the new converted fatigue lifetimes of the given aluminum coupons after dividing each time point by ten, for instance.

From Table 14, the MLEs (SEs) of $\hat{\gamma}$ and $\hat{\mu}$ are 0.1704 (0.0120) and 13.182 (0.2226), respectively, as well as a K-S ($P$-value) becomes 0.0849 (0.459). Thus, the BS distribution is a reasonable model for fitting aluminum data. Again, using the complete aluminum dataset, Figure 6(a) indicates that the MLEs $\hat{\gamma}\cong{0.1704}$ and $\hat{\mu}\cong{13.182}$ existed and are unique; Figure 6(b)–(c) indicates that the fitted BS's density and reliability lines capture the most in the data histogram and the empirical reliability line, respectively. This fact is also supported by Figure 7.

Again, based on the aluminum dataset, the BS distribution is compared to lognormal, Weibull, and gamma distributions; see Table 15. It shows that the gamma distribution is the best choice, and the BS distribution is the third-best choice among others.

Using the complete aluminum data, three AT2-PHC samples (with $k = 31$) are created; see Table 16. Point and 95% interval estimates developed by maximum likelihood and Bayes' approaches of ${\gamma}$, ${\mu}$, $R(t)$, and $h(t)$ (at $t = 10$) are obtained; see Table 17. It demonstrates that Bayes (point/interval) estimates operate similarly to frequentist estimates. The profile log-likelihood functions in Figures 8 showed that the MLEs of ${\gamma}$ and ${\mu}$ existed and are unique. Based on 40,000 MCMC iterations (when the first 10,000 iterations are ignored as 'burn-in'), the acquired Bayes point and 95% BCI/HPD interval estimates are carried out using Pr[A] and Pr[B]. From Table 17, one can see that the Bayes point and interval estimates of the various parameters based on Pr[B] perform better than other estimates in terms of minimum SEs and interval widths.

Using the remaining 30,000 MCMC variates of ${\gamma}$, ${\mu}$, $R(t)$, and $h(t)$ for S[1] (as an example), their trace and posterior histograms are shown in Figure 9. It indicates that the simulated MCMC draws of $\gamma$, $\mu$, $R(t)$, or $h(t)$ are mixed appropriately and behave fairly symmetrically. Again, using S[1], the same properties reported in Table 13 are recomputed and provided in Table 18. All findings reported in Table 18 support the same facts listed in Table 17. As supplementary materials, the same plots (in Figure 9) and the same properties (in Table 18) of ${\gamma}$, ${\mu}$, $R(t)$, and $h(t)$ based on S[2] and S[3] are presented.

Finally, the findings of the study based on two real-world datasets from the clinical or chemical fields revealed that the suggested BS model is useful in highlighting the relevance of the supplied estimating approaches to real-world occurrences. In a similar way, one can easily perform our computational design for any other dataset from other sectors, such as physics, environment, business and fishery.

6.   Concluding remarks

We checked the estimations of the BS distribution, including its unknown parameters, RF, and HRF, in the context of adaptive progressively Type-Ⅱ hybrid censored data. To obtain the MLEs, numerical techniques must be used to solve the normal equations. Two approximate confidence intervals are considered using the asymptotic properties of MLEs. The delta approach is used to approximate the variances of the estimators of the RF and HRF. The MCMC approach has been applied to determine the Bayes estimates for the BS distribution based on the squared error loss function. In addition, the Bayes and highest posterior density credible intervals are considered. We undertake a simulation analysis that takes into account different sample sizes and censoring schemes to evaluate both point estimation approaches and four interval estimations. A lot of conclusions can be drawn from the simulation study's findings, which support the superiority of Bayes estimates over conventional estimates. At last, two real-world datasets were explored to highlight how various suggested approaches could potentially be used in daily life. As a future work, using the proposed censored sample, one can easily expand the proposed maximum likelihood calculations by performing the expectation-maximization algorithm of the BS distribution.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research was funded by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R50), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

The authors would desire to express their thanks to the editor and the four anonymous referees for useful suggestions and valuable comments.

Conflict of interest

The authors declare that they have no conflicts of interest.