On the dissipative solutions for the inviscid Boussinesq equations

1.   Introduction

Statistical models can be used to describe and forecast real-world occurrences. Several extended distributions have been widely employed in data modeling throughout the last few decades. Recent advances have focused on establishing new families that expand well-known distributions while providing tremendous flexibility in modeling data in practice ^[62,63]. A large field of statistics aims at developing distributions with innovative characteristics to create flexible models for data interpretation. In reality, a new distribution can provide a new modeling perspective and a deeper description of the underlying mechanisms establishing the data. A more robust family of distributions is produced by these phenomena of parameter addition, which is effectively used to model data sets from the fields of engineering, economics, biological research, and environmental sciences. Consequently, several well-known generating families of distributions in this respect include the generalized odd Burr III-G ^[31], truncated Cauchy power Weibull-G ^[8], the generalized transmuted-G ^[50], generalized inverted Kumaraswamy-G ^[35], truncated Burr X-G ^[16], odd generalized N-H-G ^[3], sine extended odd Fréchet-G ^[36], generalized odd log-logistic ^[26], arcsine exponentiated-X family ^[33], generalized truncated Fréchet ^[61], tan-G ^[56], extended cosine-G ^[46], type II exponentiated half logistic-G ^[4], logistic-G ^[59], sine-G ^[40], cosine-G ^[57], alpha power transformed family of distributions ^[41], and for more detail see ^{[4,5,17,43,51]}. In 2020, Ijaz ^[34] presented a new family of generalized distributions called the GAP family of distributions and they defined its cumulative distribution function (CDF), probability density function (PDF) as

and

Many authors used the CDF (1.1) and PDF (1.2) to get new generalizations and new sub-models of the Gull alpha power (GAP) family of distributions as the exponentiated generalized Gull alpha power family of distributions ^[39], Gull alpha power Ampadu family of distributions ^[12], Kumaraswamy-Gull alpha power Rayleigh distribution ^[42], and exponentiated Gull alpha power exponential distribution ^[38].

The exponentiated exponential (EE) distribution has been demonstrated to be useful in a variety of applications such as life testing, survival analysis, and dependability. This distribution, which is a particular case of the exponentiated Weibull distribution ^[44,45], was studied in ^[29]. The CDF and PDF of the EE distribution with scale parameter $a$ and shape parameter $b$ are provided via

and

According to the flexibility of the EE distribution, many statisticians utilized it to create new generalizations of the EE distribution, like the beta EE distribution ^[19], Marshall-Olkin EE distribution ^[52], half-Cauchy EE distribution ^[22], odd Lomax EE distribution ^[54], and modified slashed EE distribution ^[14].

When investigations, including the lifespan of test units, must end before full observation, censored data emerges in real-world testing trials. For a number of reasons, including time constraints and financial considerations, censoring is a frequent and necessary routine action. The many forms of censorship have been well studied; types I and II censorship are the most common. A generalized censorship technique known as progressive censored schemes has lately garnered significant attention in the literature compared to standard censorship designs because of its effective use of available resources. The (PTIC) is one of several progressive censored Type-II systems. When a specific number of lifetime test units are consistently removed from the test at the end of each post-test interval, this pattern is seen. According to a study by Balakrishnan et al. ^[15], it can realistically predict the termination time and provides additional design freedom by permitting test units to be terminated during non-terminal time periods.

We now talk about accelerated life tests (ALTs), which are ways to get more data in a shorter amount of time by stressing out items more than they would under normal operating settings. Time and money can be significantly saved with such testing. Hakamipour ^[30] describes the step-stress accelerated life test (SSALT) as one form of ALT. Typically, the researcher starts with a stress level that is slightly over normal condition and gradually increases it at pre-specified time intervals during the test. The test goes on until the time limit is achieved and censoring takes place, or until the full sample of things fails. For more information about SSALT under PTIC, see ^[9,10].

The major goal of this paper is to add to the literature by introducing the Gull alpha power exponentiated exponential distribution (GAPEED) as a novel three-parameter model based on the GAP family of distributions. The subsequent points give adequate cause for examining it:

$(1)$ The GAPEED is a very flexible model whose PDF can be asymmetric (decreasing, unimodal, and right-skewed).

$(2)$ The hazard function (hrf) shape of the GAPEED includes increasing, upside-down and decreasing shapes.

$(3)$ The GAPEED has a closed-form quantile function; it is easy to compute numerous properties and generate random numbers using it.

$(4)$ The parameters of the GAPEED can be estimated utilizing eight different methods of estimation: The maximum likelihood (ML), Anderson-Darling (AD), right-tail Anderson-Darling (RTAD), left-tailed Anderson-Darling (LTAD), Cramér-von Mises (CVM), least-squares (LS), weighted least-squares (WLS), and maximum product of spacing (MPS).

$(5)$ The importance and the flexibility of the GAPEED is discussed using three real datasets, and the GAPEED gives a better fit than well-known distributions such as the Topp-Leone modified Weibull (TLMW), Type II exponentiated half logistic power Lomax (TIIEHLPL), exponential Lomax (EL), Kumaraswamy Weibull (KW), generalized modified Weibull (GMW), Marshall- Olkin alpha power extended Weibull (MOAPEW), exponential Weibull (EW), exponentiated generalized alpha power exponential (EGAPEx), Kavya-Manoharan generalized exponential (KMGEx), exponentiated half logistic inverted Nadarajah- Haghighi (EHLINH), exponentiated exponential (ExEx), and odd Weibull inverse Topp-Leone (OWITL).

$(6)$ We suggest utilizing the GAPEED model to create bivariate SSALTs under PTIC. The optimal test strategy for our suggested bivariate SSALT under PTIC is found by minimizing the asymptotic variance of the MLEs of the scale parameter's.

The remainder of this article is structured as follows: In Section 2, a new three-parameter model utilizing the EE distribution as the parent distribution in the GAP family is presented and discussed. Some important statistical features of the GAPEED are demonstrated in Section 3. Eight different estimation methods, ML, AD, CVM, MPS, LS, RTAD, WLS, and LTAD for the distribution parameters, are proposed in Section 4. In Section 5, we use a Monte Carlo technique to evaluate the quality of different estimators. To illustrate the importance of the GAPEED, we employed three real datasets in Section 6. In Section 7, the bivariate SSALT under the progressive type-I censoring (PTIC) model is discussed. Finally, the paper with concluding remarks.

2.   Model formulation

The GAPEED can be formulated by inserting (1.3) and (1.4) into (1.1) and (1.2), and then the CDF of the new suggested model is defined as

and its PDF is defined as follows

The survival function, hazard rate function (hrf), reversed hrf, and cumulative hrf are provided as

and

Figure 1 shows the plots of the PDF and hrf for the GAPEED for different values of parameters. From Figure 1, we can note that the PDF of the GAPEED can be decreasing, unimodal, and right skewed but the hrf can be decreasing, increasing, and up-side-down.

3.   Statistical properties

3.1.   Quantiles

The quantile function, defined as $Q\left(p; a, b, \tau \right){} = F^{-1}\left(p; a, b, \tau \right), p \in \left({0, {\text{ }}1} \right)$, is computed by inverting Eq (1.1) as

Then, we can rewrite the above equation as

As a result, through the use of the negative Lambert W function, represented by ${W_{ - 1}}\left(. \right)$, we obtain the quantile function of the GAPEED as

Specifically, by inserting $p = 0.25$, $0.5$, and $0.75$, we obtain the first, second (median), and third quantiles. Furthermore, relying on the quantiles, Bowley's skewness ($\alpha_1$) and Moor's kurtosis ($\alpha_2$) are provided via

and

respectively. These metrics provide helpful details about the GAPEED skewness and kurtosis modeling capabilities and have the benefit of being specified for all parameter values. The plots of $\alpha_1$ and $\alpha_2$ for the GAPEED are given in Figure 2.

3.2.   Ordinary moments

The $r$th ordinary moments are essential statistics for determining the measures of dispersion for any distribution. Assume that $X \sim$ GAPEED ($a, b, \tau$) for $x > 0$, then the $r$th ordinary moments of $X$ can computed via

by using the power series

Inserting (3.2) into (3.1), then

Employing the binomial expansion

and inserting (3.4) into (3.3), we get

We can rewrite the above Eq (3.5) as

where ${\pi _{i, j}} = ab{\left({ - 1} \right)^j}\left({\begin{array}{*{20}{c}} i \\ j \end{array}} \right)\frac{{{{\left({\log \left(\tau \right)} \right)}^i}}}{{i!}}.$ Again, using the binomial expansion (3.4) in (3.6), then the $r$th ordinary moments of the GAPEED are given by

where ${\pi _{i, j, k}} = {{\left({ - 1} \right)}^k}{\pi _{i, j}}\left[{\left({\begin{array}{*{20}{c}} {b\left({j + 1} \right) - 1} \\ k \end{array}} \right) - \left({\begin{array}{*{20}{c}} {b\left({j + 2} \right) - 1} \\ k \end{array}} \right)}\log \left(\tau \right) \right].$

Table 1 shows some numerical values of $\mu^\prime_1$, $\mu^\prime_2$, $\mu^\prime_3$, $\mu^\prime_4$, variance ($\sigma^2$), coefficient of variation (CV), skewness, and kurtosis. Also, some 3D plots of moments are provided in Figure 3.

3.3.   Moment generating function and incomplete moments

The moment-generating function for the GAPEED can be computed from (3.7) as

The $s$th lower and upper incomplete moments of the GAPEED are computed as

and

where $\gamma(., .)$ and $\Gamma(., .)$ are the lower and upper incomplete gamma functions.

4.   Estimation of the GAPEED parameters

This section introduces traditional estimation methods specifically designed for estimating the parameters of the GAPEED. These methods are applied in a simulation setting to assess their effectiveness and performance. A total of eight estimation methods are considered for this purpose. Each method involves deriving an estimate by optimizing an objective function to either maximize or minimize a specific value. The estimation setting and the definitions of the functions to be optimized are provided in detail below.

Suppose we have a random sample of values, denoted as $x_1, x_2, \ldots, x_n$, drawn from a random variable that follows the GAPEED. In order to estimate the parameters of the GAPEED, we employ various estimation methods. The first method is maximum likelihood estimation (MLE), where the estimators are obtained by maximizing a specific function which is defined as

Next, we utilize Anderson-Darling estimation (ADE) ^[13] technique for estimating the GAPEED parameters. By considering an ordered sample of values, denoted as $x_{(1)}, x_{(2)}, \ldots, x_{(n)}$, we minimize a certain function to derive the estimators, which is defined as

Similarly, we employ the right-tail Anderson-Darling estimation (RADE) ^[13] by minimizing a specific function, defined as

Additionally, the left-tailed Anderson-Darling estimation (LTADE) ^[47] is utilized to estimate the GAPEED parameters. This estimation method involves minimizing a particular function to obtain the estimators and is defined as

Furthermore, we consider Cramér-von Mises estimation (CVME) ^[23], where the estimators are obtained by minimizing a specific function defined as

Another estimation method employed is least-squares estimation (LSE) ^[58], which involves minimizing a certain function to derive the estimators. This function is defined as follows

Additionally, we employ weighted least-squares estimation (WLSE) ^[58] by minimizing a particular function, and it is defined as

Lastly, the maximum product of spacing estimation (MPSE) ^[37] method is utilized, where the estimators are obtained by maximizing a specific function. This function is defined as

where

4.1.   Confidence intervals

One frequently employed technique for creating confidence intervals (CIs) for parameters relies on the asymptotic normality of MLE and MPS. This method employs the Fisher information matrix, represented as $I(\pmb\theta)$, where $\pmb\theta = (\tau, a, b)$, which is obtained from the second derivatives of the natural logarithm of the likelihood function or product spacing function, calculated at the estimated parameter values $\hat{\pmb\theta} = (\hat{\tau}, \hat{a}, \hat{b})$. The asymptotic variance-covariance matrix of the parameter vector $\pmb\theta$ can be articulated as follows:

The matrix representing the variances and covariances of the estimated parameters, identified as $V(\hat{\pmb\theta})$, is determined by taking the inverse of the Fisher information matrix, denoted as $I^{-1}(\hat{\pmb\theta})$. To create CIs for the parameter vector $\pmb\theta$ based on the MLE's asymptotic normality, one can calculate a $100(1-\alpha)\%$ confidence interval for each parameter using the following procedure:

To calculate the CI for $a$ use this formula: $\hat{a} \pm Z_{0.025} \sqrt{V_{\hat{a}\hat{a}}}$.

To calculate the CI for $b$ use this formula: $\hat{b} \pm Z_{0.025} \sqrt{V_{\hat{b}\hat{b}}}$.

To calculate the CI for $\eta$ use this formula: $\hat{\eta} \pm Z_{0.025} \sqrt{V_{\hat{a}\hat{a}}}$.

In this context, $Z_{0.025}$ denotes the critical value from the standard normal distribution's right tail, with a probability of $\frac{\alpha}{2}$. The values $V_{\hat{a}\hat{a}}$, $V_{\hat{b}\hat{b}}$, and $V_{\hat{\eta}\hat{\eta}}$ correspond to the diagonal components of the variance-covariance matrix $V(\hat{\pmb\theta})$.

5.   Numerical simulation

In our comprehensive simulation study, we investigate the performance of our proposed model using various sample sizes: $n = 35$, $70$, $150$, $300$ and $600$. To generate representative datasets, we employ the inversion of the CDF of our proposed model. For each sample size, we randomly generate datasets based on the following parameter values: $\pmb\theta = (\tau, a, b) = \{(\tau = 0.5, \; a = 0.25, \; b = 0.75), (\tau = 1.5, \; a = 0.75, \; b = 0.5)$, $(\tau = 2, \; a = 0.5, \; b = 1.5), (\tau = 2, \; a = 1.5, \; b = 2), (\tau = 0.75, \; a = 2, \; b = 3), (\tau = 0.25, \; a = 3, \; b = 0.25)\}$. This process is repeated five thousand of times. By varying the sample sizes and incorporating diverse parameter combinations, our simulation study aims to comprehensively evaluate the performance of the proposed model across different data scenarios.

To thoroughly examine the effectiveness of the considered estimation methods, we employ a range of measures that comprehensively evaluate their performance. These measures serve as valuable benchmarks in assessing the quality of the estimators and provide insights into their accuracy, efficiency, and robustness. The following measures are employed to assess the effectiveness of the estimation methods ^[20,53,60]:

● Average of bias:

where $L$ represents the number of iterations and $\widehat{\pmb \theta}_i$ is the considered estimate for $\pmb \theta$ at the $m$-th iteration sample.

● Mean squared error:

● Mean relative error:

● Average absolute difference:

where $F(x; \pmb \theta) = F(x)$ and $x_{ij}$ denotes the values obtained at the $m$-th iteration sample and $j$-th component of this sample.

● Maximum absolute difference

● Average squared absolute error:

where $x_{(i)}$ are the ascending ordered observations. The results of simulating the proposed model parameters using different estimation techniques are presented in Tables 2–7. A graphical representation for some numerical values is presented in Figures 4 and 5. A comprehensive analysis of these tables reveals the following key observations:

First, it is important to note that all the parameter estimation methods for the proposed model demonstrate a high level of reliability, with estimated values that are very close to the actual values. This indicates the precision and accuracy of the estimation techniques employed in capturing the underlying characteristics of the proposed model.

Second, as the sample size $n$ increases, each scenario's calculated measures exhibit a decreasing trend. This observation highlights the influence of sample size on the performance of the estimation methods. Larger sample sizes tend to lead to more precise and accurate parameter estimates. In CIs, Asymptotic CI (ACI) approaches are used for MLE and MPS. The length of ACIs can be denoted as LACI. The confidence level is 95%. Also, the coverage probability (CP) are obtained for MLE and MPS methods. See Tables 9 and 10.

Considering the results derived from the simulation analysis and the subsequent evaluations of rankings in Tables 2–8, we can identify several significant conclusions:

● The property of consistency among the estimators was observed in this investigation. This property signifies that, as the sample size, denoted as $n$, expands, the estimators tend to approach the true parameter values. This convergence not only reaffirms the robustness of these estimators but also underscores their appropriateness for various statistical inference purposes.

● As the sample size "n" increased, a noticeable trend in bias reduction was evident across all the estimating techniques under investigation. This observation highlights larger sample sizes' positive influence on the parameter estimation's precision and impartiality. This phenomenon can be attributed to the decreasing impact of random variations within more extensive datasets.

● With the expansion of the sample size "n", another significant observation pertaining to the consistent reduction is the MSE across all the estimators. The MSE is a comprehensive estimation performance indicator, encompassing bias and variance. The evident decline in the MSE underscores the enhancement in overall estimation accuracy with larger sample sizes.

● For the other measures ($MRE, D_{abs}, D_{max}, ASAE$), we can see that, as the sample size increases, all of these methods' values decrease.

● Based on the overall evaluation of the estimation strategies, the MLE technique emerges as the most effective method for estimating the parameters of the proposed model. As shown in Table 8, which presents the overall ranks for all estimation strategies, the MLE achieves the lowest total score of 67.0. This result further emphasizes the superiority of the MLE technique in accurately estimating the parameters of the proposed model in the context of this study.

6.   Application

To prove that the proposed model is better than previous distributions, a comparison must be made using some data from previous studies. From the previous studies, we note that the basic distribution under study focused on the data of medical field. The three datasets under study have a basic relationship with the medical field in various ways. The GAPEED was applied to the various three datasets, and the outcomes of this comparison were produced with the TLMW ^[11], TIIEHLPL ^[32], EL ^[27], KW ^[25], GMW ^[21], MOAPEW ^[6], EW ^[24], EGAPEx ^[28], KMGEx ^[1], EHLINH ^[1], ExEx ^[48], and OWITL ^[7].

Furthermore, in order to assess the EGAPE model's validity in comparison to other competing models, we utilized various goodness-of-fit metrics, including the Kolmogorov-Smirnov (K-S) statistic with its p-value, Cramer-von Mises (CVM), and Anderson-Darling (AD), as well as other criteria measures like Bayesian information (BI), Akaike information (AI), corrected AI (CAI), and Hannan-Quinn information (HQI). All goodness-of-fit metrics are all taken into account when comparing the fits of all models. We utilize R software and the Maximum Likelihood Estimation (MLE) method to estimate the parameters of the specified distributions and to assess the goodness-of-fit metrics.

Data I: This data set was utilized as "1.1, 1.4, 1.3, 1.7, 1.9, 1.8, 1.6, 2.2, 1.7, 2.7, 4.1, 1.8, 1.5, 1.2, 1.4, 3.0, 1.7, 2.3, 1.6, 2.0" by Barco et al. ^[18], and the data shows how quickly 20 people felt better after taking an analgesic.

Data II: The most recent data cited by ^[2,11], showing the number of daily confirmed death cases linked to COVID-19. The data consists of 89 observed values with an average daily death rate of 18.72. The data set is given as follows: "1, 1, 2, 4, 5, 1, 1, 3, 6, 6, 4, 1, 5, 6, 6, 8, 5, 7, 7, 9, 9, 15, 17, 11, 13, 5, 14, 5, 13, 9, 19, 15, 11, 14, 12, 11, 7, 13, 10, 20, 22, 21, 12, 14, 9, 14, 7, 16, 17, 13, 21, 11, 11, 8, 11, 12, 15, 21, 20, 18, 15, 14, 21, 16, 11, 28, 29, 19, 14, 19, 29, 34, 34, 46, 46, 47, 36, 38, 40, 32, 39, 34, 35, 36, 35, 45, 62." Recently, papers ^[2,11] used this data, for which the Topp-Leone modified Weibull (TLMW) ^[11] has the best results where the KSPV reached 0.7280, while in this paper the KSPV reached 0.7453, and this is better than the another comparative models.

Data III: Survival rates for Guinea pigs infected with virulent tubercle bacilli are shown in the set of data ^[55]. Guinea pigs were chosen for this experiment for a number of reasons, one of which is that it is believed that they are extremely vulnerable to human tuberculosis. The information set is as follows: 0.10, 0.33, 0.44, 0.56, 0.59, 0.72, 0.74, 0.77, 0.92, 0.93, 0.96, 1, 1, 1.02, 1.05, 1.07, 1.07, 1.08, 1.08, 1.08, 1.09, 1.12, 1.13, 1.15, 1.16, 1.2, 1.21, 1.22, 1.22, 1.24 1.30, 1.34, 1.36, 1.39, 1.44, 1.46, 1.53, 1.59, 1.6, 1.63, 1.63, 1.68, 1.71, 1.72 1.76, 1.83, 1.95, 1.96, 1.97, 2.02, 2.13, 2.15, 2.16, 2.22, 2.3, 2.31, 2.4, 2.45, 2.51, 2.53, 2.54, 2.54, 2.78, 2.93, 3.27, 3.42, 3.47, 3.61, 4.02, 4.32, 4.58, 5.55.

Tables 11–Tables 16 present the MLE of the parameters for the GAPEED as well as other distributions, together with standard error (SE) values of the parameters and goodness of fit metrics for each distribution. In order to obtain the likelihood with SE, we first used the "maxLik" package, which implements the Newton Raphson (NR) method of maximization, and the variance covariance matrix. Second, we compare the fits with other distributions to determine if the relevant datasets genuinely fit the GAPEED or not using the goodness-of-fit test. Among all the models that have been fitted to these datasets, the GAPEED provides the lowest values for the KSD, AI, BI, CAI, HAI, CVM, and AD statistics. It also provides the highest value for the P-value when compared to other distributions. Figures 6–14 have been discussed for GAPEED.

Profile likelihood and uniqueness proof of GAPEED parameters have been discussed in Figures 7, 8, 10, 11, 13, and 14. The results in Tables 12, 14, and 16 show that the GAPEED is the most effective model to fit these datasets when compared to the other distributions indicated in Tables 12, 14, and 16. Graphical representations in Figures 6, 9, and 12 reflect these findings.

7.   SSALT based on PTIC

The majority of research employs just one accelerating stress variable. There are situations when increasing one stress variable does not produce enough failure data. For further acceleration, two stress factors might be required. Two stress variables are examined in this paper. It will be possible to better comprehend the impact of two stress variables functioning simultaneously if two stress variables are included in a test design. Furthermore, the test units' failure time is assumed by the author to follow a GAPEED model. The bivariate SSALT under PTIC is discussed in this section. The MLE of the model parameters is also examined.

7.1.   Model assumptions

The bivariate SSALT under PTIC is as follows: Each stress variable (SV) has two levels when using the bivariate SSALT. Let $H_s$ represent the variable l's stress level (SL) $s$, where $l = 1, 2$ and $s = 0, 1, 2$. $H_{10}$ and $H_{20}$ illustrate typical operational scenarios. Allow the experiment to run for $T_1$, during which $n_1$ failures will be logged, with all $n$ units starting at the $1^{st}$ step with SLs $(K_{11}, K_{21})$.

The $1^{st}$ SV is raised from $H_{11}$ to $H_{12}$ at time $T_1$, $c_1$ units are removed at random from the remaining $N-n_1$ units, and the $1^{st}$ SV is raised from $H_{11}$ to $H_{12}$. Until the predetermined time $tau_2$ is calculated at time $tau_2$ from the remaining $N-n_1- c_1- n_2$ units, the $second$ phase is repeated.

The other SV is increased from $H_{21}$ to $H_{22}$ at the conclusion of the $2^{nd}$ step. Up until when $T$ is reached, at which point $n_2$ units fail this stage, the test is repeated. All of the surviving units $c_3 = N-n_1- c_1- n_2- c_2-n_3$ are taken out of the test at time $T$.

In the first phase, GAPEED with cdf in Eq (2.1) is used to calculate the life of test units. A log-linear function of SLs exists for the scale parameter $\alpha_i$ at test step $i$ for $i =$1, 2, and 3.

Step 1.  $\ln(\alpha_1) = B_0+B_1H_{11}+B_2H_{21}$;

Step 2.  $\ln(\alpha_2) = B_0+B_1H_{12}+B_2H_{21}$;

Step 3.  $\ln(\alpha_3) = B_0+B_1H_{12}+B_2H_{22}$,

where $B_0$, $B_1$, and $B_2$ are unidentified parameters that vary based on the test technique and the product. The two pressures are thought to be unrelated to one another.

The model of cumulative exposure is also considered. Regardless of how the chance is calculated, the remaining life in this model is completely based on the current cumulative failure probability and the current SL ^[49].

The shape parameter $\beta$ is constant for all SLs. The cumulative distribution function (cdf) of the test unit lifespan for the bivariate SSALT and cumulative exposure models is then:

where $i = 1, 2, 3$. The pdf of bivariate SSALT for this can be written as

and

7.2.   Likelihood function of bivariate SSALT model

Assume that in a bivariate SSALT, $x_{ij}$ represents the observations produced from a PTIC sample with random deletions, where $i = 1, 2, 3, j = 1, 2, ..., n_i$. Each unit is excluded from the test with the same probability $p$, and the number of items removed from the test at any one time is distributed binomially. In other words,

The joint log-LLF of the bivariate SSALT model under the PTIC sample is as follows if $C_i$ is independent of $x_{ij}$ for all $i$.

where

where in (7.1)–(7.4), $F_i(x_{ij})$ and $f_i(x_{ij})$ will be replaced for $F_i(x_{ij})$ and $f_i(x_{ij})$, respectively.

The ML estimators of the model under bivariate SSALT based on the PTIC sample are shown in Tables 17 and 18, respectively, when p = 0 and p = 0.2. The results in Tables 17 and 18 indicate that the model's effectiveness rises as the probability of binomial elimination rises and the AINC and BINC values fall.

8.   Conclusions

In this article, we derived and studied a new three-parameter lifetime distribution called the GAPEED. Some important statistical and mathematical features (quantile function, ordinary moments, incomplete moments, and moment generating function) were computed. Eight different estimation methods for the distribution parameters, ML, AD, CVM, MPS, LS, RTAD, WLS, and LTAD, were proposed. The Monte Carlo technique was employed to evaluate the quality of different estimators. The importance and flexibility of the GAPEED were demonstrated by utilizing three real datasets. For the GAPEED model, a bivariate SSALT based on PTIC was presented. An optimal test plan under PTIC is expressed by minimizing the asymptotic variance of the MLE of the log of the scale parameter at design stress. Tables 17 and 18 compare the approaches based on various binomial removal values. We conclude from these findings that the effectiveness of this model increases as the value of the binomial removals rises.

Use of AI tools declaration

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

Acknowledgments

This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-RG23142).

Funding statement

This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-RG23142).

Conflicts of interest

The authors declare no conflict of interest.