Analysis of a new jointly hybrid censored Rayleigh populations

1.   Introduction

If the lifespan of particular items is relatively lengthy and/or the sample size $n$ is large, it is challenging to continue the examination until all $n$ observations are recorded (full data). The censored mechanism is a less costly choice for ending the test under predetermined conditions. The hybrid censoring (HC), by Balakrishnan and Kundu ^[1], is a combination of Type-Ⅰ (time) and Type-Ⅱ (failure) censoring methods in which the life-testing experiment is stopped when a pre-defined number $m$ (of $n$) items fail or a certain time $T$ on the test is reached. The key advantage of employing a HC plan over standard Type-Ⅰ or Type-Ⅱ censoring is that it saves time and money by reducing the projected testing time and failures seen.

In reliability studies, to evaluate the performance of the lifetime of two competing units coming from different production lines in the same facility, joint censoring schemes become quite useful for performing comparative life tests. In this scenario, Balakrishnan and Rasouli ^[2] proposed joint Type-Ⅱ censoring; Su ^[3] developed jointly Type-Ⅰ hybrid censoring (JHC-T1) arising from multiple independent exponential populations; Su and Zhu ^[4] proposed jointly generalized Type-Ⅰ HC; Shafay ^[5] proposed jointly Type-Ⅱ HC; and recently, Elshahhat and Abo-Kasem ^[6] proposed a jointly generalized Type-Ⅱ HC plan.

Briefly, the JHC-T1 can be stated as follows: Assume that two items (say, $A$ and $B$) are created in the identical factory by two distinct lines of operation (say, $L_1$ and $L_2$). Assume that $\mathbf{X} = \{{{X}_{1}}, {{X}_{2}}, ..., {{X}_{m}}\}$ and $\mathbf{Y} = \{{{Y}_{1}}, {{Y}_{2}}, ..., {{Y}_{n}}\}$ are two separate samples of $m$ and $n$ pieces of product $A$ and $B$, respectively, that get placed in a life-testing experiment at the same time. Here, we have $N = m+n$. Assume that $\mathbf{X}$ ($\mathbf{Y}$) are independent and identically distributed (iid) variables from a population with cumulative distribution function (CDF), ${{F}_{X}}(\cdot)$ (${{F}_{Y}}(\cdot)$), and probability density function (PDF), ${{f}_{X}}(\cdot)$ (${{f}_{Y}}(\cdot)$). Regarding the manufacturing information on the product of interest, the threshold point $T$ is also assumed to be fixed beforehand. Let ${{G}_{(1)}}\leqslant{{G}_{(2)}}\leqslant\cdots\leqslant{{G}_{(N)}}$ be the order statistics of $N$ random variables $\{\mathbf{X}; \mathbf{Y}\} = \{{{X}_{1}}, {{X}_{2}}, ..., {{X}_{m}}; {{Y}_{1}}, {{Y}_{2}}, ..., {{Y}_{n}}\}$.

To perform a life test that doesn't take more time to complete due to cost constraints, the investigator decides to perform the proposed censoring. As a result, the collected data ($\mathbf{g}$, $\mathbf{z}$) will consist of one of the following data forms:

where $d$ represents the size of observed failures before $T$. More specifically, let $m_D = \sum\nolimits_{i = 1}^{D}{{{z}_{i}}}$ and $n_D = \sum\nolimits_{i = 1}^{D}(1-{{z}_{i}})$ be the number of subjects that failed in $\mathbf{X}$ and $\mathbf{Y}$, respectively. Also, let $D = m^{*}+n^{*}$, where $D\in \{1, 2, ..., N\}$, be the effective sample size. The main advantage of the JHC-T1 is that it can effectively guarantee that the experiment stops the test as soon as a certain number of failures ($r$) or a threshold time ($T$) is first achieved.

If the JHC-T1 failure times of $N$ specimens are coming from two samples of continuous population with PDFs ${{f}_{i}}\, (\cdot)$ and CDFs ${{F}_{i}}\, (\cdot)$ for $i = 1, 2,$ then the joint PDF of $\theta$ given $\mathbf{g}$ can be expressed as

where $\boldsymbol{\theta }$ is a parameter vector, ${{C}_{D}} = \frac{m!n!}{(m-{m^{*}})!(n-{n^{*}})!}$, and ${T}^{*} = {g}_{(r)}(\mbox{or}\ T)$ for Case-1 (or Case-2). In Figure 1, a schematic representation of the proposed plan is depicted.

Remark 1. From (1.1), when $T\to \infty$ and $r = N$, the joint Type-Ⅰ HC (by Su ^[3]) reduced to the joint Type-Ⅱ censoring (by Balakrishnan and Rasouli ^[2]).

In this paper, depending on unit capacity, experimental facilities, and cost restrictions, we explore a new scheme called JHC-T1, in which the experimenter terminates the test once the predetermined time (or the predetermined number of failures) is recorded first. Therefore, to the greatest extent of our understanding, no research has been presented on the estimation problem of population parameters when lifetime data are acquired via the proposed jointly hybrid censoring. Thus, the novelty of this work originates from the fact that, in the presence of new JHC-T1 failure times, it is the first time that two maximum likelihood and Bayesian approaches for the Rayleigh distribution's parameters of life have been compared since its inception.

The Rayleigh distribution is the simplest probability distribution of wind speed to represent the wind resource because it requires only knowledge of the average wind speed. It is also utilized in reliability analysis of radar and microwave networks, image recognition, wind energy modeling, electro-vacuum device design, wave height modeling in oceanography, acoustics, and magnetic resonance imaging, among other applications; see Chattamvelli and Shanmugam ^[7]. From a probability point of view, this is a subset of the Weibull lifespan model, which Rayleigh proposed while studying acoustic problems. For this purpose, we consider the Rayleigh lifetime model, which is commonly used in several areas of statistics. In the context of a hybrid censored sample, the proposed population has been discussed by Kwon et al. ^[8], Asgharzadeh and Azizpour ^[9], and Jeon and Kang ^[10], among others.

Our primary goal of this study is fourfold:

● Compute the maximum likelihood and asymptotic interval estimators for the unknown mean lifespan of the test subjects in two samples based on the Rayleigh population with varied scale parameters.

● Acquire both symmetric and asymmetric Bayes' estimators as well as two bounds of credible interval estimators for the same unknown quantities.

● Using various choices of $n$, $m$, $r$, and $T$, the performance of the proposed methods is compared through a Monte-Carlo simulation.

● Two genuine data set-based applications from the engineering sector highlight the ability of two different Rayleigh populations to accommodate various data types and adapt the given estimation approaches to actual practical situations.

The rest of the article is arranged as follows: In Section 2, the maximum likelihood estimators and the asymptotic intervals are obtained. The Bayesian inference along with the credible intervals are discussed in Section 3, where three loss functions are considered, the square error loss function, the linear exponential loss function, and the general entropy loss function. Numerical simulation with the Monte Carlo technique is implemented in Section 4 to assess the performance of the proposed estimates in terms of root mean square error and average relative absolute biases, and the use of the probability coverage to assess the interval estimation performance. In Section 5, illustrative examples from the engineering field are developed. Last, Section 6 concludes the results of the study.

2.   Likelihood inference

This section presents the maximum likelihood estimators (MLEs) as well as asymptotic interval estimators (ACIs) of two Rayleigh population parameters in the presence of such proposed censored data.

2.1.   Maximum likelihood estimators

Suppose the lifetimes $\mathbf{X}$ (of size $m$ from facility-line ${{L}_{1}}$) are iid random variables from a Rayleigh population with a scale parameter ${\xi }_{1}$ having the following PDF ${{f}_{X}}(x; {{\xi }_{1}})$ and CDF ${{F}_{X}}(x; {{\xi }_{1}})$ as

respectively. Similarly, let the lifetimes $\mathbf{Y}$ (of size $n$ from facility-line ${{L}_{2}}$) are iid random variables from another Rayleigh population with a scale parameter ${\xi }_{2}$ having the following PDF ${{f}_{Y}}(y; {{\xi }_{2}})$ and CDF ${{F}_{Y}}(y; {{\xi }_{2}})$ as

Substituting (2.1) and (2.2) into (1.1), the likelihood function of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ can be written up to proportional as

where ${{q}_{1}} = \sum\nolimits_{i = 1}^{D}{g_{i}^{2}\, {{z}_{i}}}+\, (m-{m^{*}})\, {{T}^{*}}^{2}$ and ${{q}_{2}} = \sum\nolimits_{i = 1}^{D}{g_{i}^{2}\, (1-{{z}_{i}})}+\, (n-{n^{*}})\, {{T}^{*}}^{2}$. The log-likelihood function, $\ell (\cdot)\propto \ln L(\cdot)$, of (2.3) becomes

Through the differentiation of (2.4) with regard to $\xi_1$ and $\xi_2$, then equating each outcome to zero, the MLEs $\hat{\xi}_1$ and $\hat{\xi}_2$ of $\xi_1$ and $\xi_2$ can be easily acquired respectively as

Remark 2. From (2.5), it is obvious that when $\sum\nolimits_{i = 1}^{D}{{{z}_{i}}}$ equals zero (or $D$), the MLE ${{\hat{\xi }}_{1}}$ (or ${{\hat{\xi }}_{2}}$) does not exist. Thus, the existence of the MLEs $\hat{\xi }_{i}, \ i = 1, 2$ is conditioned on $1 \leqslant {m^{*}}\leqslant D-1$ and $1 \leqslant {n^{*}}\leqslant D-1$.

Remark 3. When $T\to \infty$ and $r = N$, we extend the results of Al-Matrafi and Abd-Elmougod ^[11] in the case of jointly Type-Ⅱ censoring to jointly Type-Ⅰ hybrid censoring.

2.2.   Asymptotic intervals

To obtain two-sided $100(1-\omega)\%$ ACIs of ${{\xi }_{1}}$ and ${{\xi }_{2}}$, the asymptotic normality of their MLEs ${{\hat{\xi }}_{1}}$ and ${{\hat{\xi }}_{2}}$ is utilized. The $2\times2$ matrix of a Fisher information (say $\mathbf{I}(\cdot)$) is acquired by the negative-expected of the second-partial derivatives of (2.4) with regard to ${{\xi }_{1}}$ and ${{\xi }_{2}}$ as

Following the invariance property of ${{\hat{\xi }}_{1}}$ and ${{\hat{\xi }}_{2}}$, the asymptotic variance-covariance matrix (say ${\Sigma}(\cdot) = \mathbf{I}^{-1}(\cdot)$) of the parameters ${{\xi }_{1}}$ and ${{\xi }_{2}}$ can be easily offered by locally at ${{\hat{\xi }}_{1}}$ and ${{\hat{\xi }}_{2}}$, respectively, as

where $\widehat{var}({{\hat{\xi }}_{1}}) = \frac{\hat{\xi }_{1}^{2}}{{m^{*}}}$ and $\widehat{var}({{\hat{\xi }}_{2}}) = \frac{\hat{\xi }_{2}^{2}}{{n^{*}}}$. From (2.6), it is clear that ${{\mathbf{I}}_{12}}(\cdot) = {{\mathbf{I}}_{21}}(\cdot) = 0$.

Since the MLEs ${{\hat{\xi }}_{1}}$ and ${{\hat{\xi }}_{2}}$ are asymptotically normally distributed as ${\hat{\xi }_{1}}{\sim}N({{\xi }_{1}}, \widehat{var}({{\hat{\xi }}_{1}}))$ and ${\hat{\xi }_{2}}{\sim}N({{\xi }_{2}}, \widehat{var}({{\hat{\xi }}_{2}}))$, see Lawless ^[12], Thus, from (2.7), the $100(1-\omega)\%$ two-sided ACIs for ${{\xi }_{1}}$ and ${{\xi }_{2}}$ are given, respectively, by

where ${{z}_{{\omega }/{2}\; }}$ denotes the $({\omega }/{2})$th percentage point of the standard normal distribution.

3.   Bayes paradigm

This section deals with obtaining the Bayes' point estimators as well as the corresponding credible intervals of the unknown Rayleigh parameters ${{\xi }_{1}}$ and ${{\xi }_{2}}$ against several loss functions.

3.1.   Loss functions

The loss function is significant in statistical analysis since it focuses on estimating precision. However, the squared-error loss (SEL) is a particularly widely utilized symmetric loss in the literature for creating a Bayes estimator of an unknown parameter of interest. Njomen et al. ^[13] studied Bayesian estimation under different Loss Functions in competing Risks model, Hasan et al. ^[14] used loss functions to perform Bayesian estimation for the exponential distribution, and Ali et al. ^[15] studied the effect of loss functions on the performance of Bayesian inference under Lindley distribution. If $\vartheta$ is the parameter to be estimated using an estimator $\tilde\vartheta$, the SEL function (say${{\Im }_{s}}(\cdot)$) is defined as

Using (3.1), the Bayes estimator ${{\tilde{\vartheta }}_{S}}$ of $\vartheta$ is acquired as

where $\pi \left(\left. \vartheta \right|\mathbf{g} \right)$ is the posterior PDF of $\vartheta$.

Besides the traditional SEL function, we also use two well-known asymmetric loss functions called linear-exponential loss (LL) and general-entropy loss (GEL) functions because the SEL is often inappropriate in reliability estimation, especially when overestimation is more harmful than underestimation. It is well known that the Bayes estimator in the case of the SEL is the posterior mean where the overestimation and underestimation are treated equally; see Elshahhat et al. ^[16].

First, the LL (say ${{\Im }_{L}}(\cdot)$) is defined as

Using (3.2), the Bayes estimator ${{\tilde{\vartheta }}_{L}}$ of $\vartheta$ is given by

where $\upsilon\rightarrow 0$, the LL (3.2) is quite approximately to the SEL (3.1). Another useful asymmetric loss function is the GEL function, ${{\Im }_{G}}(\cdot)$, is defined as

where $\tau$ is the shape parameter. Using (3.3), the Bayes estimator ${{\tilde{\vartheta }}_{G}}$ of $\vartheta$ is given by

where $\tau = -1$, both estimators ${{\tilde{\vartheta }}_{G}}$ and ${{\tilde{\vartheta }}_{S}}$ coincided. Further, the minimum error under the GEL function occurs at $\vartheta = {{\tilde{\vartheta }}_{G}}$.

3.2.   Bayes estimators

The prior distribution, which reflects knowledge about an unknown parameter, is an important aspect of Bayesian inference. Thus, based on several reasons to consider gamma prior, such as: (ⅰ) It provides various density shapes; (ⅱ) It is flexible; (ⅲ) It is fairly straightforward, concise, and may not lead to a result with a complex estimation issue, we assumed that ${{\xi }_{1}}$ and ${{\xi }_{2}}$ are independently distributed with gamma random variables with PDFs $Gamma({{a}_{1}}, {{b}_{1}})$ and $Gamma({{a}_{2}}, {{b}_{2}})$, respectively. On the other hand, one can easily consider other prior information based on the Rayleigh parameter domain, e.g., Weibull, generalized-exponential, or others.

Hence, the joint prior PDF (say $\pi\left(\cdot\right)$) of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ is

where $\Gamma (\cdot)$ is the complete gamma function.

From (2.3) and (3.4), the joint posterior PDF (say $\Pi\left(\cdot\right)$) of ${{\xi }_{1}}$ and ${{\xi }_{2}}$, becomes

where $C_{D}^{*} = \frac{{{\left({{b}_{1}}+{{q}_{1}} \right)}^{{{a}_{1}}-{m^{*}}}}{{\left({{b}_{2}}+{{q}_{2}} \right)}^{{{a}_{2}}-{n^{*}}}}}{\Gamma \left({{a}_{1}}-{m^{*}} \right)\Gamma \left({{a}_{2}}-{n^{*}} \right)}\cdot$

From (3.5), the joint posterior PDF of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ is obtained as a product of two independent gamma probability densities as ${{\xi }_{1}}\sim Gamma({{a}_{1}}-{m^{*}}, {{b}_{1}}+{{q}_{1}})$ and ${{\xi }_{2}}\sim Gamma({{a}_{2}}-{n^{*}}, {{b}_{2}}+{{q}_{2}})$. As a result, if the improper prior knowledge of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ is available, i.e., ${{a}_{i}}, {{b}_{i}} = 0, \ i = 1, 2,$ Eq (3.5) reduced in proportion to Eq (2.3).

However, using the SEL function (3.1), the Bayes estimators ${{\tilde{\xi }}_{1S}}$ and ${{\tilde{\xi }}_{2S}}$ of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ are given, respectively, by

Similarly, using the LL function (3.2), the Bayes estimators ${{\tilde{\xi }}_{1L}}$ and ${{\tilde{\xi }}_{2L}}$ of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ are given, respectively, by

Again, using the GEL function (3.3), the Bayes estimators ${{\tilde{\xi }}_{1G}}$ and ${{\tilde{\xi }}_{2G}}$ of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ are given, respectively, by

3.3.   Credible intervals

To construct the two bounds $100\, (1-\omega)\%$ Bayes credible intervals (BCIs) of ${{\xi }_{1}}$ and ${{\xi }_{2}}$, the posterior density function (3.5) is used. First, let ${{\delta }_{1}} = 2{{\xi }_{1}}({{q}_{1}}\, +{{b}_{1}})$ and ${{\delta }_{2}} = 2\, {{\xi }_{2}}({{q}_{2}}\, +{{b}_{2}})$, where ${{\delta }_{1}}$ and ${{\delta }_{2}}$ are positive integers, be two pivots follow ${{\chi }^{2}}$ distributions with $2({m^{*}}+{{a}_{1}})$ and $2({n^{*}}+{{a}_{2}})$ degrees of freedom, respectively, see Kundu and Joarder ^[17].

Hence, $100(1-\omega)\%$ BCIs of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ are offered by

respectively, where $\chi _{v, \omega}^{2}$ is the $\omega$th percentile the $\chi _{v}^{2}$ distribution.

Remark 4. If the levels of freedom $2(m^{*}+a_1$ and $2(n^{*}+a_2$ cannot be obtained in integers, then the gamma PDF will be utilized instead of the ${{\chi }^{2}}$ distribution to create the BCIs of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ as

and

respectively, where $F_{G}^{-1}(\omega, \nu_1, \nu_2)$ is the $\omega$th percentile of the gamma distribution with shape $\nu_1$ and scale $\nu_2$ parameters.

4.   Monte-Carlo study

Comprehensive Monte-Carlo simulations are run to examine the behavior of the suggested estimation procedures developed in the preceding sections. For fixed $T = (1, 2)$, when ${{\xi }_{1}}$ and ${{\xi }_{2}}$ are taken as $({{\xi }_{1}}, {{\xi }_{2}}) = (0.5, 0.5)$ and (0.75, 1.5), we simulate 5,000 JHC-T1 samples based on various tests of $n$, $m$, and $r$ such as $n = m = 10$, 30, and 50. We assign the level of $r$ as a failure percentage (of each $N$) such as $(\frac{r}{N})100\% = 30$, 50 and 80%. We also assigned $({{a}_{i}}, {{b}_{i}}) = (0.5, 1)$ for $i = 1, 2,$ when $({{\xi }_{1}}, {{\xi }_{2}}) = (0.5, 0.5)$ and $({a}_{1}, {a}_{2}) = (1.5, 3)$ and ${b}_{i} = 2$ for $i = 1, 2,$ when $({{\xi }_{1}}, {{\xi }_{2}}) = (0.75, 1.5)$. Under SEL, LL($\upsilon( = -3, -0.03, +3)$), and GEL($\tau( = -2, -0.02, +2)$), all offered Bayes estimates are calculated.

For each scenario, the average estimates (AEs), root mean squared-errors (RMSEs), and average relative absolute biases (ARABs) of the unknown parameter ${\xi}_{1}$ (for instance) are calculated using the next formulae, respectively, as

and

where $\hat{\xi}_{1}^{(j)}$ is the obtained estimate at the $j^{th}$ sample of ${\xi}_{1}$.

Further, the corresponding average confidence lengths (ACLs) and coverage probabilities (CPs) related to the $95\%$ ACI/BCI estimates of ${\xi}_{1}$ are acquired using the following formulae, respectively, as

and

where ${\boldsymbol{1}}(\cdot)$ is the indicator operator and ($\textsf{L}(\cdot), \textsf{U}(\cdot)$) denotes the lower and upper sides, respectively, of the ACI (or BCI) of ${\xi}_{1}$. Clearly, in a similar fashion, the AEs, RMSEs, ARABs, ACLs, and CPs of $\xi_2$ can be easily offered.

A Heat-Map is a common tool for data visualization. It is known as a data visualization technique in which each value in a matrix is portrayed by specific colors. So, using this tool via the $\textsf{R}$ programming language, Figures 2–5 display the simulated results (including RMSEs, ARABs, ACLs, and CPs) of ${\xi}_{i}, \ i = 1, 2$. All simulated findings of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ are reported in the Supplementary File. Briefly, some numerical results of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ are presented in Tables 1–3. In these tables, the AEs, RMSEs, and ARABs for each parameter are tabulated in the first, second, and third rows, respectively.

From Figures 2–5, we present the following statements:

● The Bayes' estimation method is better than the frequentist (maximum likelihood) estimation method, and both are applicable and feasible. It can also be seen that the proposed estimation methodologies of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ work satisfactorily.

● As $n$ and $m$ increase, the RMSEs and ARABs of all point estimates decrease as expected. Obviously, to improve numerical results, try increasing the $r$ size.

● As $T$ increases, it is noted that

– The RMSEs and ARABs of all estimates of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ increase.

– The ACLs of 95% ACIs (or BCIs) of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ narrowed down while their CPs grew.

● As $(\frac{r}{N})100\%$ increases, the RMSEs and ARABs of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ decrease at $T = 1$ while these increase at $T = 2$.

● Bayes estimates, due to additional information provided by gamma prior, performed better than others.

● When the true value of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ decreases, the proposed point estimates become even better.

● It is also clear that the proposed Bayes estimates using the LL (or GEL) function of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ are overestimates when $\upsilon(\mbox{or}\ \tau) < 0$ while these are underestimates when $\upsilon(\mbox{or}\ \tau) > 0$.

● This result is due to fact that the Bayes estimates under asymmetric type loss have additional flexibility, due to the shape parameter loss, compared to those obtained based on symmetric type loss.

● When $n$ and $m$ increase, the ACLs of 95% ACIs/BCIs tend to decrease.

● When ${{\xi }_{1}}$ and ${{\xi }_{2}}$ increase, the ACLs of ACIs/BCIs increase while the corresponding CPs decrease. Similar behavior is observed when $T$ increases.

● As the failure percentage increases, for fixed $n$ and $m$, the ACLs and CPs decrease for ACIs/BCIs of ${{\xi }_{1}}$ and ${{\xi }_{2}}$.

● Finally, employing the Bayes paradigm to offer point (or interval) estimates of the Rayleigh population parameters is recommended.

5.   Engineering applications

Two separate engineering scenarios are analyzed to demonstrate how to make use of the suggested inferential methodologies.

5.1.   Insulating fluid

In this application, we shall consider two samples (groups 3 and 6) of size $m = n = 10$ each from Nelson ^[18]. These samples indicate the time required for an insulating liquid exposed to extreme voltage stresses to break down (in minutes). The failure times of groups 3 (say $\mathbf{X}$) are 1.99, 0.64, 2.15, 1.08, 2.57, 0.93, 4.75, 0.82, 2.06, 0.49, as well as the failure times of groups 6 (say $\mathbf{Y}$) are: 2.12, 3.97, 1.56, 1.34, 1.49, 8.71, 2.10, 7.21, 3.83, 5.13.

To highlight the validity of the Rayleigh population for given datasets, the Kolmogorov-Smirnov (K-S) statistic (with its $P$-value) is computed. First, the MLEs with their standard errors (in parentheses) of the Rayleigh parameters $\xi_{i}, \ i = 1, 2$ using $\mathbf{X}$ and $\mathbf{Y}$ datasets are 0.2205(0.0697) and 0.0500(0.0158), respectively. Thus, the K-S distances with their $P$-values (in parentheses) from $\mathbf{X}$ and $\mathbf{Y}$ datasets become 0.2732(0.375) and 0.2987(0.275), respectively. Hence, we can conclude that the real times of $\mathbf{X}$ and $\mathbf{Y}$ come from the Rayleigh populations. For more illustration, using $\mathbf{X}$ and $\mathbf{Y}$ datasets, the fitted/empirical distribution functions of Rayleigh populations, probability-probability (PP), and quantile-quantile (QQ) are displayed in Figure 6. It further confirms the same numerical findings of K-S statistics.

Moreover, before proceeding, to highlight the superiority of the proposed Rayleigh population among others, we reexamine it with two well-known models in the literature called Maxwell-Boltzmann (MB$(\xi)$) and Muth$(\xi)$ lifetime models. Recently, these competitive models have been analyzed by Elshahhat et al. ^[19] and Alotaibi et al. ^[20], respectively. As a result, Table 4 indicates that the Rayleigh population is the best from the insulating liquid data compared to others.

Using various choices of $r$, at $T = 2.5$ (in minutes), three different JHC-T1 samples are generated and reported in Table 5. Since we lack any prior knowledge about ${{\xi }_{1}}$ and ${{\xi }_{2}}$, the Bayes estimates using improper gamma priors, i.e., ${{a}_{i}} = {{b}_{i}} = 0,$ for $i = 1, 2,$ are developed. The Bayes estimates of ${{\xi }_{1}}$ or ${{\xi }_{2}}$ are acquired based on the SEL, LL($\upsilon( = -2, -0.02, +2)$), and GEL($\tau( = -1, -0.01, +1)$) functions. For calculation convenience, we have set all hyperparameter values to 0.001.

From Table 5, the suggested point and 95% interval estimates of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ are computed; see Tables 6 and 7, respectively. From Table 6, we can decide that the offered classical and Bayes estimates of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ are quite similar, as expected. In addition, from Table 7, the 95% credible intervals of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ are also close to those acquired by asymptotic intervals.

To estimate the mean lifetime of the Rayleigh population, the corresponding MLEs of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ based on samples 1–3 are used. Using the invariance property ${\hat{\xi }_{1}}$ and ${\hat{\xi }_{2}}$, the estimated mean lifetimes $\sqrt{\frac{\pi }{4\hat\xi_{1} }}$ and $\sqrt{\frac{\pi }{4\hat\xi_{2} }}$ relative to groups $\mathbf{X}$ and $\mathbf{Y}$ become (1.5068, 1.3434, 1.2476) and (2.6410, 2.7886, 2.2156) from samples (1, 2, 3) respectively. To sum up, the group $\mathbf{X}$ has a higher lifetime compared to the others. Using the offered Bayes' estimates of ${{\xi }_{1}}$ and ${{\xi }_{2}}$, the same result may likewise be reached for the mean lifetime of each group.

5.2.   Mechanical equipment

In this application, from Murthy et al. ^[21], we consider the time between thirty failures for repairable mechanical equipment. To illustrate the findings of the paper, we partitioned (randomly) the complete data into two samples of equal sizes denoted here as groups $\mathbf{X}$ and $\mathbf{Y}$, see Table 8. Using $\mathbf{X}$ and $\mathbf{Y}$ datasets, the MLEs of ${{\xi}_{1}}$ and ${{\xi}_{2}}$ with their (standard errors) are 0.3487(0.0900) and 0.2299(0.0593), respectively.

To see the validity of the proposed model, the K-S($P$-value) for both $\mathbf{X}$ and $\mathbf{Y}$ samples are 0.2132(0.442) and 0.2003(0.520), respectively. These results showed that the Rayleigh populations fit the repairable mechanical equipment datasets quite well. In Figure 7, the fitted/empirical distribution functions of Rayleigh populations as well as PP and QQ plots are displayed based on $\mathbf{X}$ and $\mathbf{Y}$. It also supports the findings of the goodness statistics.

Again, before continuing, we compare the utility of the proposed Rayleigh ensemble with its competitors Maxwell-Boltzmann (MB$(\xi)$) and Muth$(\xi)$ lifetime models. Consequently, Table 9 points out that the suggested Rayleigh population is the best compared to others based on the mechanical equipment data.

Using both $\mathbf{X}$ and $\mathbf{Y}$ datasets, when $T = 1.5$(hours) and $r( = 10, 18, 25)$, three different JHC-T1 samples are created, see Table 10. Using each JHC-T1 sample, the maximum likelihood and Bayes estimates of ${{\xi }_{1}}$ and ${{\xi }_{2}}$, for ${{a}_{i}} = {{b}_{i}} = 0.001, \ i = 1, 2$, are computed and listed in Table 11. Under SEL, LL (for $\upsilon ( = -2, -0.02, +2)$), and GEL (for $\tau ( = -3, -0.03, +3)$), the Bayes estimates of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ are developed. Further, the 95% ACIs/BCIs of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ are calculated and provided in Table 12.

As expected, Tables 11 and 12 showed that the acquired point and 95% interval estimates of ${{\xi }_{1}}$ and ${{\xi }_{2}}$ are similar to each other, as expected. Using the fitted values of ${{\hat{\xi }}_{1}}$ and ${{\hat{\xi }}_{2}}$, the estimated mean lifetimes of the Rayleigh populations from the groups $\mathbf{X}$ and $\mathbf{Y}$ become (0.9174, 1.1446, 1.1646) and (1.1749, 1.2655, 1.2219) under the samples (1, 2, 3), respectively. It can also be seen that the mean lifetimes of the group $\mathbf{Y}$ have a larger lifetime than the others. Using the Bayesian estimates, the same conclusion about the mean lifetimes can be easily drawn. Since the jointly Type-Ⅰ hybrid censoring made a balance between reducing duration tests and observing extreme failures, we therefore recommend terminating the experiment as soon as the first of ${{g}_{(r)}}$ (or $T$) reaches.

6.   Conclusions

A new type of Type-Ⅱ hybrid censoring, which enables the investigation of the lifespan of two (or more) competing goods in the same facility, has been investigated in this study. Various estimation procedures in the presence of such censoring have been proposed when the lifetimes of experimental items from two populations have been assumed to follow Rayleigh distributions with different scale parameters. Besides the maximum likelihood approach, Bayes' inferential approach has also been considered. Monte Carlo computations have been run to assess the efficiency of the suggested methods. The computational results indicate that the acquired estimators of the Rayleigh population parameters derived from the Bayes method perform quite satisfactorily compared to those derived by the classical method. Two actual data sets from engineering-related fields have been shown to test the usefulness of the suggested procedures. To sum up, the use of the asymmetric Bayes paradigm to estimate the Rayleigh population parameters under jointly Type-Ⅰ hybrid censoring is recommended. The conclusions and approach described here may easily be extended to other lifetime groups as a future study, e.g., gamma, Weibull, or generalized exponential distribution, among others. It is better to extend the generalized Rayleigh results in Maiti and Kayal ^[22] to a joint progressive Type-Ⅱ strategy. We also hope that the methodologies discussed here will be beneficial to data analysts and reliability managers.

Use of AI tools declaration

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

Acknowledgments

This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [GRANT No. 5422].

Conflict of interest

The authors declare no conflicts of interest.