Constant Stress-Partially Accelerated Life Tests of Vtub-Shaped Lifetime Distribution under Progressive Type II Censoring

Abstract

In lifetime tests, the waiting time for items to fail may be long under usual use conditions, particularly when the products have high reliability. To reduce the cost of testing without sacrificing the quality of the data obtained, the products are exposed to higher stress levels than normal, which quickly causes early failures. Therefore, accelerated life testing is essential since it saves costs and time. This paper considers constant stress-partially accelerated life tests under progressive Type II censored samples. This is realized under the claim that the lifetime of products under usual use conditions follows Vtub-shaped lifetime distribution, which is also known as log-log distribution. The log–log distribution is highly significant and has several real-world applications since it has distinct shapes of its probability density function and hazard rate function. A graphical description of the log–log distribution is exhibited, including plots of the probability density function and hazard rate. The log–log density has different shapes, such as decreasing, unimodal, and approximately symmetric. Several mathematical properties, such as quantiles, probability weighted moments, incomplete moments, moments of residual life, and reversed residual life functions, and entropy of the log–log distribution, are discussed. In addition, the maximum likelihood and maximum product spacing methods are used to obtain the interval and point estimators of the acceleration factor, as well as the model parameters. A simulation study is employed to assess the implementation of the estimation approaches under censoring schemes and different sample sizes. Finally, to demonstrate the viability of the various approaches, two real data sets are investigated.

1. Introduction

To increase the demand and verify trust between the producers and their consumers. Most producers have been doing their best to develop and improve the performance of their products. However, they encounter certain difficulties in the process of improving the product, such as the problem of managing the product’s failure (reliability estimation) within the given test period. To overcome many of the challenges associated with standard reliability, accelerated life testing (ALT) techniques may be used. Enhancing the performance of the item, advancing its development, and identifying the reasons behind its brief lifespan are important. The accelerated testing conditions may include higher levels of temperature, pressure, voltage, load, speed, vibration, etc., depending on the type of product, and they may also include multiple stresses. To estimate the lifespan distribution under typical use conditions, data gathered under such accelerated conditions are then extrapolated using a statistical model that is physically appropriate.

According to [1], there are three forms of ALT, namely constant stress, step stress, and progressive stress. Throughout the duration of the test products’ lives, stress remains at a constant level in constant stress ALT. These types of ALT have been dealt with by several authors, including [1,2,3,4,5,6,7,8].

The mathematical model relating the lifetime of the unit and the stress is known or can be assumed to be the main assumption in ALT. Such life-stress relationships are not known and cannot be assumed in some cases. Thus, another method can be used, which is partially ALT (PALT). In PALT, items are tested at both accelerated and usual use conditions.

In a constant stress-PALT (CS-PALT) study, each test item is subjected to constant stress at standard or accelerated conditions only until the test terminates. CS-PALT analysis has received a lot of attention recently, see [9,10,11,12,13,14,15,16].

In many life-testing and reliability study scenarios, the researcher could not be able to get all the information regarding the failure times of every experimental item. There are also states where the removal of items prior to failure is preplanned to reduce the expense and duration linked with testing. “Censored data” is data gained from such experiments. Type I and Type II censoring are the most popular censoring schemes, although the classical Type I and Type II censoring schemes do not have the flexibility of permitting the removal of items at points other than the endpoint of the experiment. Due to this, this paper considers progressive Type II censoring as a more general censoring technique that generalizes the Type II censoring scheme. For more details about the progressive Type II censoring scheme, see [17,18].

The progressive Type II censoring scheme is flexible as well as efficient than traditional Type I and Type II censoring schemes since it has unique benefits that let the experimenter eliminate survival testing units from the experiment at different testing stages. When the progressive Type II censoring system is utilized, the cost and time for the tests and the efficiency of the experiment will be improved. Due to its many applications in science, engineering, the social sciences, and health, lifetime distributions under the progressive Type II censored scheme have attracted a lot of attention and interest. For more details, see [19,20,21,22,23,24].

Considering the overall importance and advantages of the CS-PALT and progressive Type II censoring scheme, particularly in reliability analysis, as they reduced the required costs and time for experimental tests and improved product performance, it became imperative to merge them to maximize the benefits. Thus, several authors have studied the analysis of CS-PALT based on progressive Type II under different lifetime distributions, such as [11,14,25,26,27,28,29,30].

A variety of software reliability growth models have been obtained over the past thirty years to forecast software reliability methods, such as the number of defects that remain and software reliability. A software reliability model known as the log–log (LL) and also named Pham distribution with a Vtub-shaped hazard rate function (hrf) has been proposed by [31]. In the case of the Vtub-shaped system, the system has a relatively slow rate of increase following the newborn mortality period, but this increase is not continuous and gradually increases with aging-related failures. A Vtub-shaped hrf is a representative model in survival analysis where the hrf initially decreases, then levels off for a period, and finally increases. This shape is often observed in real-life data, specifically in engineering and reliability studies.

The motivation for studying the Vtub-shaped hrf: (1) The initial decreasing hazard rate often represents the early life failure period of a component. This is due to manufacturing defects, material defects, or fabrication errors. (2) The period of relatively stable failure rates, often referred to as the useful life period. During this phase, components operate as designed without significant degradation. (3) The final increasing hrf implies the wear-out period, where components fail due to age, usage, or environmental factors. (4) By understanding the different stages of a component’s life cycle, maintenance strategies can be designed. (5) Preventative maintenance can be scheduled during the early life period to identify and repair possible issues. (6) During the useful life period, condition-based monitoring can be executed to detect abnormal wear-out models. (7) During the wear-out period, replacement planning can be initiated to minimize unexpected failures. (8) The Vtub-shaped hrf helps in estimating component lifetimes and predicting failure probabilities. (9) The Vtub-shaped hrf provides visions into the reliability system, reliability characteristics of helicopter parts, and safety. (10) Based on the Vtub-shape, effective maintenance, replacement strategies, preventing unnecessary maintenance, and avoiding catastrophic failures help to reduce maintenance costs. (11) Recognizing the failure patterns of helicopter parts is necessary for risk estimation and safety management. (12) Discovering the factors that contribute to the different stages of the hrf can help in decreasing risks.

The LL distribution is particularly suitable for modeling data with a Vtub-shaped hrf. This is because the distribution’s parameters can be carefully chosen to obtain the different stages of the hrf curve. The LL distribution could likely achieve a better fit to the helicopter parts data compared to other distributions, leading to more accurate prediction and informed decision-making.

The significance and adaptation of the LL distribution are evident in software reliability modeling. Ref. [31] employed the LL distribution to model helicopter parts data, as the Vtub-shaped hrf of the LL distribution is useful in evaluating the reliability measures of two helicopter components: the main rotor blade and rotor brake assembly. For more details, see [31].

However, there has not been much discussion of the LL distribution in the literature. Ref. [31] used the maximum likelihood (ML) method to get the parameter point estimation. Additionally, Ref. [32] provided an overview of some standard results about the LL distribution. The Bayes estimators for the model parameters were studied by [33]. Ref. [34] used LL distribution to present two new software reliability models that consider the testing coverage subject to operating environment uncertainty and the fault-detection rate.

Ref. [35] introduced some comparisons between the new Pham’s model and other existing models in the field of debugging and test theory. Ref. [36] presented the unit LL distribution. Basic properties and ML estimation of the parameter for unit LL distribution are obtained. Recently, Ref. [37] introduced the exponentiated LL model. They discussed the main properties of the distribution and used ML and Bayesian approaches to estimate the model parameters. Additionally, they studied cancer data to demonstrate the flexibility and potential applications of this model.

Despite the novelty and applicability of the LL distribution in modeling lifetime data, it did not attract attention in the presence of CS-PALT under progressive Type II samples. Furthermore, it appears that there are no available references that address the maximum product spacing (MPS) method based on progressive Type II samples with CS-PALT. Consequently, the main objectives of this article are:

• Highlighting the flexibility and importance of the LL distribution in lifetime applications. Further, studying some of its statistical properties.
• Utilizing two classical procedures, ML and MPS methods based on progressive Type II samples with CS-PALT, to investigate the point and interval estimators for the model parameters and the acceleration factor.
• Evaluating the efficiency of several derived estimators using varying progressive schemes through a simulation study.
• Demonstrating the relevance of the offered study by exploring two real data sets.

The present work is structured as follows: in Section 2, the LL distribution and the basic assumptions of the model are considered. The LL distribution is analyzed in detail to deduce some of its statistical properties and explain its hrf Vtub-shaped behavior. Section 3 revolves around the derivation of both the ML and MPS estimators for the parameters and the acceleration factor of the LL distribution based on CS-PALT under progressive Type II censoring. In the same section, the asymptotic confidence intervals (ACIs) for the parameters and the acceleration factor are derived. In Section 4, a numerical study, involving both a simulation study and two real lifetime data sets, is addressed to illustrate the theoretical results. Finally, general conclusions are introduced in Section 5.

2. The Log–Log Distribution and Basic Assumptions

In this study, the lifetimes of test units are assumed to have the LL distribution. An extensive overview of the LL distribution is provided in this section. Furthermore, the basic assumptions for CS-PALT based on progressive Type II are offered.

2.1. The Log–Log Distribution

Ref. [31] demonstrated that a Vtub-shaped hrf that corresponds to the LL distribution, not only includes distributions with decreasing and increasing failure rates but also provides a wider class of monotone failure rates. The Vtub-shaped hrf can be viewed as: if there exists a change point to such that the hrf is decreasing in $[0, t_0]$ and slowly increasing, as a Vtub-shaped, in $[t_0, \infty )$.

The cumulative distribution function (cdf) and probability density function (pdf) of the LL distribution are given, respectively, by:

$$F_1\left(x;\alpha ,\beta \right)=1-e^{\left(1-{\beta }^{x^{\alpha }}\right)},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\alpha >0,\beta >1,x>0,$$

(1)

and

$$f_1\left(x;\alpha ,\beta \right)=\alpha \ln \left(\beta \right)x^{\alpha -1}{\beta }^{x^{\alpha }}{e}^{1-{\beta }^{x^{\alpha }}},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\alpha >0,\beta >1,x>0,$$

(2)

where $\alpha$ and $\beta$ are shape parameters. The corresponding reliability function (rf) and hrf are, respectively, given by:

$$\overline{F_1}\left(x;\alpha ,\beta \right)=e^{1-{\beta }^{x^{\alpha }}}, \alpha >0,\beta >1,x>0,$$

(3)

and

$$h_1\left(x;\alpha ,\beta \right)=\alpha \ln \left(\beta \right) x^{\alpha -1}{\beta }^{x^{\alpha }}, \alpha >0,\beta >1,x>0.$$

(4)

Figure 1 displays the different plots of the pdf and hrf, respectively, of the LL distribution for some selected values of the parameters. Figure 1 clearly displays the Vtub-shaped hrf of the LL distribution. Based on Figure 1, one can deduce that the LL distribution is flexible regarding the shapes of its pdf and hrf. Additionally, Figure 1 shows that the pdf of the LL distribution can be decreasing, unimodal, and approximately symmetric. Furthermore, at large values of $\beta$, the hrf of LL distribution is sensitive to changes in $\alpha$. Varying $\alpha$ resulted in decreasing, increasing, Vtub-shaped hrf. Due to this, the LL distribution is extremely adaptable and capable of fitting a large number of real-world data sets.

A few of the distribution properties have been investigated. Therefore, in this study, some of the important statistical properties of the LL distribution are covered, including quantiles, probability weighted moments (PWM), moments, moments of residual life, reversed residual life functions, incomplete moments, and entropy. This involves the utilization of both the generalized integro-exponential function, see [38] and the incomplete generalized integro-exponential function, see [39]. They are defined, respectively, by the following integrals:

$$E_s^d\left(z\right)={\displaystyle \frac{1}{d!}}{\int }_1^{\mathrm{\infty }}{\left[\ln \left(u\right)\right]}^d{u}^{-s}{e}^{-zu}du,$$

(5)

and

$$E_s^d\left(z,h\right)={\displaystyle \frac{1}{d!}}{\int }_1^h{\left[\ln \left(u\right)\right]}^d{u}^{-s}{e}^{-zu}du,$$

(6)

where $s\in \left(-\mathrm{\infty },\mathrm{\infty }\right),d>-1,z\in \left(-\mathrm{\infty },\mathrm{\infty }\right)$ and $\underset{h\to \mathrm{\infty }}{\lim }{E}_s^d\left(z,h\right)=E_s^d\left(z\right).$

2.1.1. Quantiles

By inverting the cdf of the LL distribution, given by (1), the quantile function is determined as follows:

$$Q\left(q\right)={\left\{{\displaystyle \frac{\ln \left[1-\ln \left(1-q\right)\right]}{\ln \left(\beta \right)}}\right\}}^{{\displaystyle \frac{1}{\alpha }}},\hspace{1em}\hspace{1em}0<q<1.$$

(7)

The median of the LL distribution can be defined at $q=0.5$. Further, considering $q$ as a uniform random number, a random sample can easily be generated from (7).

Moreover, $Q\left(q\right)$ may be used to calculate the Bowley skewness ($\mathrm{B}\mathrm{S}$) and Moors kurtosis ($\mathrm{M}\mathrm{K}$), see [40,41] which are defined, respectively, by:

$$\mathrm{B}\mathrm{S}={\displaystyle \frac{Q\left(\frac{3}{4}\right)+Q\left(\frac{1}{4}\right)-2Q\left(\frac{1}{2}\right)}{Q\left(\frac{3}{4}\right)-Q\left(\frac{1}{4}\right)}} \mathrm{and}\mathrm{M}\mathrm{K}={\displaystyle \frac{Q\left(\frac{7}{8}\right)-Q\left(\frac{5}{8}\right)+Q\left(\frac{3}{8}\right)-Q\left(\frac{1}{8}\right)}{Q\left(\frac{6}{8}\right)-Q\left(\frac{2}{8}\right)}}.$$

(8)

The $\mathrm{B}\mathrm{S}$ and $\mathrm{M}\mathrm{K}$ measures are less responsive to outliers, and they can be obtained for any distribution even if does not have moments.

2.1.2. Probability Weighted Moments

Ref. [42] proposed the PWM, which is known as power moments. The PWM of a random variable$X$ can be defined in terms of its cdf as:

$$M_{k,p,g}=E\left[X^k{\left(F\left(X\right)\right)}^p{\left(1-F\left(X\right)\right)}^g\right],$$

(9)

where $k,p,$ and $g$ are any real numbers and${ M}_{k,\mathrm{0,0}}$ is the noncentral moment of order$k$. For the LL distribution, the PWM can be expressed using (1) and (9) as:

$$M_{k,p,g}=\alpha \ln \left(\beta \right){\int }_0^{\mathrm{\infty }}{x}^{k+\alpha -1}{\beta }^{x^{\alpha }}{e}^{(g+1)\left(1-{\beta }^{x^{\alpha }}\right)}{\left[1-e^{\left(1-{\beta }^{x^{\alpha }}\right)}\right]}^{p}dx.$$

Appling the binomial expansion to expand ${\left[1-e^{\left(1-{\beta }^{x^{\alpha }}\right)}\right]}^p,$ thus

$$M_{k,p,g}=\alpha \ln \left(\beta \right){\sum }_{\tau =0}^{\mathrm{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{p}{\tau }}\right){\left(-1\right)}^{\tau }{\int }_0^{\mathrm{\infty }}{x}^{k+\alpha -1}{\beta }^{x^{\alpha }}{e}^{(g+1+\tau )\left(1-{\beta }^{x^{\alpha }}\right)}dx.$$

Assuming that ${\beta }^{x^{\alpha }}=u$in the previous equation, it yields

$$M_{k,p,g}={\sum }_{\tau =0}^{\mathrm{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{p}{\tau }}\right){\left(-1\right)}^{\tau }{e}^{(g+1+\tau )}{\left[\ln \left(\beta \right)\right]}^{-{\displaystyle \frac{k}{\alpha }}}{\int }_1^{\mathrm{\infty }}{\left[\ln \left(u\right)\right]}^{{\displaystyle \frac{k}{\alpha }}}{e}^{-(g+1+\tau )u}du,$$

which can be represented using (5) as:

$$M_{k,p,g}={\sum }_{\tau =0}^{\mathrm{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{p}{\tau }}\right){\left(-1\right)}^{\tau }{e}^{\left(g+1+\tau \right)}{\left[\ln (\beta )\right]}^{-{\displaystyle \frac{k}{\alpha }}}\left({\displaystyle \frac{k}{\alpha }}!\right)E_0^{{\displaystyle \frac{k}{\alpha }}}\left(g+\tau +1\right).$$

The derivation of PWM of LL distribution is given in Appendix A.

When $p=0, g=0,$ the $k^{th}$ noncentral moment${ M}_{k,\mathrm{0,0}}$ of the LL distribution is derived to be

$$\mathrm{E}\left(X^k\right)=e\left({\displaystyle \frac{k}{\alpha }}!\right){\left[\ln (\beta )\right]}^{-{\displaystyle \frac{k}{\alpha }}}{E}_0^{{\displaystyle \frac{k}{\alpha }}}\left(1\right).$$

(10)

Consequently, the mean ($E\left(X\right)$), variance ($Var\left(X\right)$), skewness $\left(Sk\right(X\left)\right)$, kurtosis $\left(Ku\left(X\right)\right),$ and the coefficient of variation $\left(\gamma ={\displaystyle \frac{\sqrt{Var\left(X\right)}}{E\left(X\right)}}\right)$ of the LL distribution can be calculated from (10).

Table 1. Numerical values of $\mathit{E}\left(\mathit{X}\right)$, $\mathit{V}\mathit{a}\mathit{r}\left(\mathit{X}\right)$, $\mathit{S}\mathit{k}\left(\mathit{X}\right), \mathit{K}\mathit{u}\left(\mathit{X}\right),$ and $\mathit{\gamma }$ of the LL distribution for some values of $\mathit{\alpha }$ and $\mathit{\beta }$.

$$\mathit{\beta }$$	$$\mathit{\alpha }$$	$$\mathit{E}\left(\mathit{X}\right)$$	$$\mathit{V}\mathit{a}\mathit{r}\left(\mathit{X}\right)$$	$$\mathit{S}\mathit{k}\left(\mathit{X}\right)$$	$$\mathit{K}\mathit{u}\left(\mathit{X}\right)$$	$$\mathit{\gamma }$$
1.5	0.7	1.997	3.507	1.299	4.628	0.938
	1.5	1.214	0.384	0.215	2.367	0.510
	3.5	1.043	0.071	−0.553	2.907	0.255
	5.5	1.019	0.031	−0.853	3.655	0.172
2.5	0.7	0.623	0.341	1.299	4.628	0.938
	1.5	0.705	0.129	0.215	2.367	0.510
	3.5	0.827	0.044	−0.553	2.907	0.255
	5.5	0.878	0.023	−0.853	3.655	0.172
4	0.7	0.345	0.105	1.299	4.628	0.938
	1.5	0.535	0.074	0.215	2.367	0.510
	3.5	0.734	0.035	−0.553	2.907	0.255
	5.5	0.815	0.020	−0.853	3.655	0.172
6	0.7	0.239	0.050	1.299	4.628	0.938
	1.5	0.451	0.053	0.215	2.367	0.510
	3.5	0.683	0.030	−0.553	2.907	0.255
	5.5	0.777	0.018	−0.853	3.655	0.172

displays the numerical values of these measurements for a variety of parameter values. Several shapes of the mean, variance, skewness, kurtosis, and the coefficient of variation of the LL distribution are shown in Figure 2.

From the results provided in Table 1, one can observe that

• For fixed values of β and α increases, then $E\left(X\right)$ increases in most cases, whereas for fixed α and increasing values of β, then $E\left(X\right)$ decreases.
• $Var\left(X\right)$ decreases as either $\alpha$ or $\beta$ increases.
• Furthermore, in most cases, as $\alpha$ increases $Sk\left(X\right), Ku\left(X\right),$ and $\gamma$ decreases, whereas the value of $\beta$ does not effect on the values of $Sk\left(X\right), Ku\left(X\right),$ and $\gamma$.
• For different values of $\alpha$, one can note that; the distribution is right skewed $\left(Sk\right(X)>0)$ for small values of $\alpha$. While, for large values of $\alpha ,$ the distribution is left skewed $\left(Sk\right(X)<0)$.
• Given that $Ku\left(X\right)$ represents the distribution’s peak and $Sk\left(X\right)$ its degree of symmetry, it is evident that the LL distribution included a range of pdf shapes.

2.1.3. Incomplete Moments

In lifetime research, the incomplete moment associated with the Lorenz and Bonferroni curves is significant. For any continuous random variable $X$ with pdf, $f\left(x\right)$, the $k^{th}$ incomplete moment is defined as:

$${\phi }_k\left(t\right)={\int }_{-\mathrm{\infty }}^t{x}^{k}f\left(x\right)dx.$$

Thus, for the LL distribution, the $k^{th}$ incomplete moment is given by:

$${\phi }_k\left(t\right)=\alpha \ln \left(\beta \right)e{\int }_0^t{x}^{k+\alpha -1}{\beta }^{x^{\alpha }}{e}^{-{\beta }^{x^{\alpha }}}dx.$$

Using the transformation ${\beta }^{x^{\alpha }}=u$ and the relation defined by (6), therefore,

$${\phi }_k\left(t\right)=e{\left[\ln \left(\beta \right)\right]}^{-{\displaystyle \frac{k}{\alpha }}}\left({\displaystyle \frac{k}{\alpha }}!\right)E_0^{\left({\displaystyle \frac{k}{\alpha }}\right)}\left(1,{\beta }^{t^{\alpha }}\right).$$

(11)

2.1.4. Moments of Residual Life and Reversed Residual Life Functions

The residual life is a main aging measure that has an important role in reliability theory and life-testing applications. The $k^{th}$ moment of the residual life, say $m_k\left(t\right)=\mathrm{E}\left[\left.{\left(X-t\right)}^k\right|X>t\right], k=\mathrm{1,2},\dots$ is defined as:

$$m_k\left(t\right)={\displaystyle \frac{1}{\overline{F}\left(x\right)}}{\int }_t^{\mathrm{\infty }}{\left(x-t\right)}^{k}f\left(x\right)dx={\displaystyle \frac{1}{\overline{F}\left(t\right)}}{\sum }_{w=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{w}}\right){\left(-t\right)}^{k-w}{\int }_t^{\mathrm{\infty }}{x}^{w}f\left(x\right)dx.$$

(12)

Therefore, the $k^{th}$ moment of the residual life for the LL distribution can be expressed in terms of the $w^{th}$ moment and the ${ w}^{th}$ incomplete moment as:

$$m_k\left(t\right)={\displaystyle \frac{1}{\overline{F_1}\left(x;\alpha ,\beta \right)}}{\sum }_{w=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{w}}\right){\left(-t\right)}^{k-w}\left[E\left(X^w\right)-{\phi }_w\left(t\right)\right],$$

(13)

where $\overline{F_1}\left(x;\alpha ,\beta \right),$ is the rf of the LL distribution given by (3), $E\left(X^w\right),$ and ${\phi }_w\left(t\right)$can be obtained by replacing $k$ by $w$ in (10) and (11), respectively.

The mean residual life function at a given time $t$ measures the expected remaining lifetime of an individual aged $t$. It can be accessed by setting $k=1$ in $m_k\left(t\right)$ outlined by (13).

In parallel with residual life, the reversed residual life also plays an important role in reliability theory and life-testing applications. The $k^{th}$ moment of the reversed residual life, say ${\Phi }_k\left(t\right)=\mathrm{E}\left[\left.{\left(t-X\right)}^k\right|X\le t\right]$ for $t\ge 0$ and $k=\mathrm{1,2},\dots$ is defined by:

$${\Phi }_k\left(t\right)={\displaystyle \frac{1}{F\left(t\right)}}{\sum }_{w=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{w}}\right){\left(t\right)}^{k-w}{\left(-1\right)}^w{\int }_0^t{x}^{w}f\left(x\right)dx.$$

(14)

One more time, using the incomplete moment given in (11), the $k^{th}$ moment of the reversed residual life of the LL distribution can be expressed as:

$${\mathsf{\Phi }}_k\left(t\right)={\displaystyle \frac{1}{\mathrm{F}\left(\mathrm{t}\right)}}{\sum }_{w=0}^{\mathrm{k}}\left({\displaystyle \genfrac{}{}{0pt}{}{\mathrm{k}}{w}}\right){\left(t\right)}^{k-w}{\left(-1\right)}^w{\phi }_w\left(t\right).$$

(15)

The mean inactivity time or the mean waiting time demonstrates the waiting time that passed since the failure of an item on the condition that this failure had occurred in $(0,t)$. For the LL distribution, it can be obtained easily by setting $k=1$ in$\left(15\right)$.

2.1.5. Entropy

In information theory, entropy is fundamental as it measures the degree of uncertainty and disorder in a system or distribution. Rényi entropy, introduced by [43], is a widely used measure of entropy. For a random variable $X$ with pdf, $f\left(x\right),$ the Rényi entropy is defined by:

$$I_{\delta }\left(X\right)={\displaystyle \frac{1}{1-\delta }}\ln \left\{{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left[f\left(x\right)\right]}^{\delta }dx\right\},\hspace{1em}\delta >0,\delta \ne 1.$$

(16)

From the pdf of the LL distribution given in (2), one can obtain

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left[f\left(x\right)\right]}^{\delta }dx={\int }_0^{\mathrm{\infty }}{\left[\alpha \ln \left(\beta \right)x^{\alpha -1}{\beta }^{x^{\alpha }}{e}^{\left(1-{\beta }^{x^{\alpha }}\right)}\right]}^{\delta }dx,$$

which can be rewritten using the transformation ${\beta }^{x^{\alpha }}=u$ and after some simplification as

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left[f\left(x\right)\right]}^{\delta }dx={\alpha }^{\delta -1}{\left[\ln \left(\beta \right)\right]}^{-{\displaystyle \frac{1}{\alpha }}\left(1-\delta \right)}{e}^{\delta }{\int }_1^{\mathrm{\infty }}{\left[\ln (u\right)]}^{\left(\delta -{\displaystyle \frac{\delta }{\alpha }}+{\displaystyle \frac{1}{\alpha }}-1\right)}{u}^{\delta -1}{e}^{-\delta u}du.$$

From (5), it yields that

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left[f\left(x\right)\right]}^{\delta }dx={\alpha }^{\delta -1}{\left[\ln \left(\beta \right)\right]}^{-{\displaystyle \frac{1}{\alpha }}\left(1-\delta \right)}{e}^{\delta }\left[\left(\delta -{\displaystyle \frac{\delta }{\alpha }}+{\displaystyle \frac{1}{\alpha }}-1\right)!\right]\left[E_{\left(1-\delta \right)}^{\left(\delta -{\displaystyle \frac{\delta }{\alpha }}+{\displaystyle \frac{1}{\alpha }}-1\right)}\left(\delta \right)\right].$$

(17)

Substituting (17) in (16), the Rényi entropy for the LL distribution is obtained.

2.2. Basic Assumptions

In CS-PALT, supposing that there are $n$ identical test units, $n_1$ units are randomly selected from $n$ test units for testing in the usual use conditions, and the remaining $n_2=n-n_1$ units are tested under accelerated stress. This is how progressive Type II censoring is implemented: For $s=1, 2,$ when the first failure, say $X_{s1:m_s:n_s},$ simply noted by $X_{s1}$, has happened, $R_{s1}$ units are at random eliminated from the remaining $n_s-1$surviving units. When the second failure, say $X_{s2}$, has happened, $R_{s2}$ units from the remaining $n_s-2-R_{s1}$units at random are eliminated. The test continues at several stress stages until the$m_{s}th$ failure, say $X_{s{m}_s}$, has happened, at which time all remaining $R_{s{m}_s}=n_s-m_s-{\sum }_{v=1}^{m_s-1}{R}_{sv}$ units are withdrawn and the CS-PALT is ended. In this debate, the values of the censoring numbers $R_{si}, s=\mathrm{1,2}, i=\mathrm{1,2},\dots ,m_s$ were preset with $m_s<n_s$.

The experimental methodology to CS-PALT under progressive Type II is brief in Figure 3.

In this study, the progressively Type II censored samples $X_{s1}<X_{s2}<\cdots <X_{s{m}_s}, s=\mathrm{1,2}$, from two testing conditions, of which the cdfs and the pdfs are $F_s\left(x\right)$ and ${ f}_s\left(x\right)$, respectively, with censoring schemes ${ R}_s=(R_{s1}, R_{s2}, . . . , R_{s{m}_s})$.

Denote $x_{s1}<x_{s2}<\cdots <x_{s{m}_s}$ as the observed values of ${ X }_{s1}<X_{s2}<\cdots <X_{s{m}_s}$; the likelihood function (LF) based on progressive Type II censored samples under usual use and accelerated conditions in CS-PALT, can be expressed as:

$$L\left(\underset{\_}{\theta };\underset{\_}{x}\right)=\prod _{s=1}^2\left\{A_S\prod _{i=1}^{m_s}{f}_s\left(x_{si}\right){\left[1-F_s\left(x_{si}\right)\right]}^{R_{si}}\right\},$$

(18)

where $\underset{\_}{\theta }$ is the parameter vector of the lifetime distribution, $\underset{\_}{x}=\left(x_1,x_2\right)$, $s=\mathrm{1,2},$ $x_{si}=\left(x_{s1},\dots ,x_{s{m}_s}\right)$ and $A_S=n_s\left(n_s-1-R_{s1}\right)\left(n_s-2-R_{s1}-R_{s2}\right)\dots \left(n_s-m_s-{\sum }_{v=1}^{m_s-1}{R}_{sv}\right).$

Ref. [44] introduced the MPS method based on a progressive Type II sample. The MPS method chooses the parameter values that make the observed data as uniform as possible, according to a specific quantitative measure of uniformity. Based on CS-PALT under a progressive Type II sample, the product spacing function (PSF) can be written as follows:

$$G\left(\underset{\_}{\theta };\underset{\_}{x}\right)\propto {\prod }_{s=1}^2\left\{{\prod }_{i=1}^{m_s+1}{D}_{si}\left(\underset{\_}{\theta }\right){\prod }_{i=1}^{m_s}{\left[1-F_s\left(x_{si}\right)\right]}^{R_{si}}\right\},$$

(19)

where $\underset{\_}{\theta }$ is the parameter vector of the lifetime distribution, $s=\mathrm{1,2},$

$$D_{si}\left(\underset{\_}{\theta }\right)=\left\{\begin{array}{c}{F}_s\left(x_{s1}\right) \mathrm{if} i=1\\ F_s\left(x_{si}\right)-F_s\left(x_{s(i-1)}\right), \mathrm{if} i=\mathrm{2,3},\dots ,m_s,\\ 1-F_s\left(x_{s{m}_s}\right) \mathrm{if} i=m_s+1,\end{array}\right.$$

(20)

and $\sum D_{si}=1.$

A few basic assumptions regarding CS-PALT under progressive Type II from the LL distribution must be considered as:

(1)

A unit tested under usual use conditions has a lifetime distribution that follows the LL distribution, with the cdf and pdf shown in (1) and (2), respectively.

The distribution of a unit tested under accelerated conditions is derived by using the transformation $X_2={\lambda }^{-1}{X}_1,$ where the equation $X_2={\lambda }^{-1}{X}_1$ essentially scales the time axis. A higher acceleration factor ($\lambda$) implies that a shorter duration under accelerated conditions ($X_2$) is equivalent to a longer duration under normal conditions ($X_1$). The electronics industry frequently employs accelerated aging tests to predict product lifespan under normal conditions. For instance, components like capacitors are subjected to raised temperatures to simulate years of use in a matter of weeks or months. Some other physical mechanism applications are oxidation, where increased temperature accelerates the rate of oxidation, leading to material degradation, and corrosion, where high conditions can increase corrosion processes, impacting material reliability. Additionally, fatigue, where accelerated loading or stress can induce fatigue failure more rapidly, polymer and degradation at higher temperatures speed up the breakdown of polymer chains. Example: If the rate of oxidation doubles for every 10 °C increase in temperature, then exposing a material to 60 °C instead of 30 °C would accelerate the aging process by a factor of 4 (2²). Hence, one day at 60 °C would be equivalent to four days at 30 °C in terms of oxidation damage.

(2)

The distribution of a unit tested under accelerated conditions is derived by using the transformation $X_2={\lambda }^{-1}{X}_1,$ where $\lambda$ is the acceleration factor defined as the ratio of the mean life under usual conditions to that under accelerated conditions and $\lambda >1,$ see [10,45]. Therefore, the pdf, cdf, rf, and hrf under the accelerated conditions are, respectively, given by:

$$f_2\left(x;\alpha ,\beta ,\lambda \right)=\alpha {\lambda }^{\alpha }\ln \left(\beta \right)x^{\alpha -1}{\beta }^{{\left(\lambda x\right)}^{\alpha }}{e}^{1-{\beta }^{{\left(\lambda x\right)}^{\alpha }}},\hspace{1em}\hspace{1em}\alpha >0,\beta ,\lambda >1,x>0,$$

(21)

$$F_2\left(x;\alpha ,\beta ,\lambda \right)=1-e^{1-{\beta }^{{\left(\lambda x\right)}^{\alpha }}},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\alpha >0,\beta ,\lambda >1,x>0,$$

(22)

$$\overline{F_2}\left(x;\alpha ,\beta ,\lambda \right)=e^{1-{\beta }^{{\left(\lambda x\right)}^{\alpha }}},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\alpha >0,\beta ,\lambda >1,x>0,$$

(23)

and

$$h_2\left(x;\alpha ,\beta ,\lambda \right)=\alpha {\lambda }^{\alpha }\ln \left(\beta \right)x^{\alpha -1}{\beta }^{{\left(\lambda x\right)}^{\alpha }},\hspace{1em}\hspace{1em}\hspace{1em}\alpha >0,\beta ,\lambda >1,x>0.$$

(24)

(3)

In this study, there are $n_s, s=\mathrm{1,2}$ units allocated under the usual use conditions with $s=1$ and the accelerated conditions with $s=2$, of which the lifetimes $X_{si}, s=\mathrm{1,2},i=1, 2, . . . , n_s$are mutually independent. Figure 4 shows the pdfs under usual and accelerated conditions.

3. Estimation the Parameters of the Log–Log Distribution

In this section, the ML and MPS estimators of the parameters of the LL distribution and the acceleration factor are derived based on the observed values, $x_{s1}<x_{s2}<\cdots <x_{s{m}_s}$, of the progressive Type II censored sample under CS-PALT. Additionally, the ACIs for the parameters and the acceleration factor of the LL distribution for CS-PALT under progressive Type II censoring are obtained.

3.1. Maximum Likelihood Estimation

Based on Equations (1), (2), (18), (21), and (22), the natural logarithm of the LF for the LL distribution based on CS-PALT under progressive Type II censoring can be expressed as:

$$\begin{gathered} \mathcal{l}\propto \left({m_1+m}_2\right)\ln \left[\alpha \ln \left(\beta \right)\right]+\ln \left(\beta \right)\left[\sum _{i=1}^{m_1}{x}_{1i}^{\alpha }+{\lambda }^{\alpha }\sum _{i=1}^{m_2}{\left(x_{2i}\right)}^{\alpha }\right]+m_2\alpha \ln \left(\lambda \right)+\sum _{i=1}^{m_1}\left(1+R_{1i}\right)\left(1-{\beta }^{x_{1i}^{\alpha }}\right) \\ +\left(\alpha -1\right)\left[{\sum }_{i=1}^{m_1}\ln \left(x_{1i}\right)+{\sum }_{i=1}^{m_2}\ln \left(x_{2i}\right) \right]+{\sum }_{i=1}^{m_2}\left(1+R_{2i}\right)\left[1-{\beta }^{{\left(\lambda x_{2i}\right)}^{\alpha }}\right]. \end{gathered}$$

(25)

Taking the first partial derivatives of (25) with respect to $\alpha , \beta ,$ and $\lambda$ and equating them to zeros, one can derive the ML estimators for $\alpha , \beta ,\mathrm{a}\mathrm{n}\mathrm{d}\lambda$ as:

$$\begin{gathered} {\displaystyle \frac{\partial \mathcal{l}}{\partial \alpha }}={\displaystyle \frac{m_1+m_2}{\alpha }}+m_2\ln \left(\lambda \right)+{\sum }_{i=1}^{m_1}\ln \left(x_{1i}\right)+{\sum }_{i=1}^{m_2}\ln \left(x_{2i}\right)+\ln \left(\beta \right){\sum }_{i=1}^{m_1}{x}_{1i}^{\alpha } \ln \left(x_{1i}\right)\left[1-\left(1+R_{1i}\right) {\beta }^{x_{1i}^{\alpha }}\right] \\ +\ln (\beta \left){\sum }_{i=1}^{m_2}{\left(\lambda x_{2i}\right)}^{\alpha }\ln \left(\lambda x_{2i}\right)\right[1-{(1+R}_{2i}) {\beta }^{{\left(\lambda x_{2i}\right)}^{\alpha }}], \end{gathered}$$

(26)

$${\displaystyle \frac{\partial \mathcal{l}}{\partial \beta }}={\displaystyle \frac{(m_1+m_2)}{\beta \ln (\beta ) }}+{\displaystyle \frac{1}{\beta }}\left[{\sum }_{i=1}^{m_1}{x}_{1i}^{\alpha }\left[1-\left(1+R_{1i}\right){\beta }^{x_{1i}^{\alpha }}\right]\right]+{\displaystyle \frac{1}{\beta }}{\sum }_{i=1}^{m_2}{\left(\lambda x_{2i}\right)}^{\alpha }\left[1-\left(1+R_{2i}\right){\beta }^{{\left({\lambda x}_{2i}\right)}^{\alpha } }\right],$$

(27)

and

$${\displaystyle \frac{\partial \mathcal{l}}{\partial \lambda }}={\displaystyle \frac{m_2\left(\alpha \right)}{\lambda }}+\alpha {\lambda }^{\alpha -1}\ln \left(\beta \right){\sum }_{i=1}^{m_2}{x}_{2i}^{\alpha } \left[1-\left(1+R_{2i}\right){\beta }^{{\left(\lambda x_{2i}\right)}^{\alpha }}\right].$$

(28)

From (26) to (28), it seems that there are no closed form solutions for the ML estimators. Therefore, the ML estimates for the parameters $\alpha ,\beta ,$ and $\lambda$ can be obtained numerically using the Newton–Raphson method as an iterative procedure via the Newton–Raphson method through Mathematica 11.

3.2. Maximum Product Spacing Estimation

The MPS methodology was presented by [46] as an alternative to the ML method. All the ML method’s large sample properties are included in the MPS method, which also retains most of the ML method’s properties under more general conditions. By choosing parameter values that maximize the product of the distances between the distribution function values at nearby ordered points, the MPS estimators are computed. Many authors, including [47,48], have recently employed the MPS approach to estimate the parameters for various lifespan distributions under progressive Type II censoring. MPS estimation is more robust to outliers. The MPS estimators for the parameters $\alpha ,\beta , and \lambda$ of the LL distribution in CS-PALT under progressive Type II censoring sampling can be obtained using (19). Substituting (1) and (22) in (19), the PSF can be written as:

$$G\left(\underset{\_}{\theta };\underset{\_}{x}\right)\propto \left\{{\prod }_{i=1}^{m_1+1}{D}_{1i}\left(\underset{\_}{\theta }\right) {\prod }_{i=1}^{m_1}{e}^{R_{1i}\left[1-{\beta }^{x_{1i}^{\alpha }}\right]}\right\}\left\{{\prod }_{i=1}^{m_2+1}{D}_{2i}\left(\underset{\_}{\theta }\right) {\prod }_{i=1}^{m_2}{e}^{R_{2i}\left[1-{\beta }^{{\left(\lambda x_{2i}\right)}^{\alpha }}\right]}\right\},$$

where $\underset{\_}{\theta }=\left(\alpha ,\beta ,\lambda \right),$ $\underset{\_}{x}=\left(x_1,x_2\right)$, $s=\mathrm{1,2},x_{si}=\left(x_{s1},\dots ,x_{s{m}_s}\right),$ and $D_{si}\left(\underset{\_}{\theta }\right)$ can be obtained explicitly by applying (1) and (22) in (20). Then,

$$\begin{gathered} G(\underset{\_}{\theta };\underset{\_}{x})\propto \left\{\left[1-e^{\left(1-{\beta }^{x_{11}^{\alpha }}\right)}\right]\left[e^{\left(1-{\beta }^{x_{1m_1}^{\alpha }}\right)}\right]e^{{\sum }_{i=1}^{m_1}{R}_{1i}\left(1-{\beta }^{x_{1i}^{\alpha }}\right)}{\prod }_{i=2}^{m_1}\left[e^{\left(1-{\beta }^{x_{1\left(i-1\right)}^{\alpha }}\right)}-e^{\left(1-{\beta }^{x_{1i}^{\alpha }}\right)}\right] \right\} \\ \times \left\{\left[1-e^{\left(1-{\beta }^{{\left(\lambda x_{21}\right)}^{\alpha }}\right)}\right]\left[e^{1-{\beta }^{{\left(\lambda x_{2m_2}\right)}^{\alpha }}}\right]e^{{\sum }_{i=1}^{m_2}{R}_{2i}\left[1-{\beta }^{{\left(\lambda x_{2i}\right)}^{\alpha }}\right]}\prod _{i=2}^{m_2}\left[e^{\left[1-{\beta }^{{\left(\lambda x_{2\left(i-1\right)}\right)}^{\alpha }}\right]}-e^{\left[1-{\beta }^{{\left(\lambda x_{2i}\right)}^{\alpha }}\right]}\right] \right\}. \end{gathered}$$

(29)

From (29), the natural logarithm of the PSF is

$$\begin{gathered} \ln \left[G(\underset{\_}{\theta };\underset{\_}{x})\right]\propto \ln \left[1-e^{\left(1-{\beta }^{x_{11}^{\alpha }}\right)}\right]-{\beta }^{x_{1m_1}^{\alpha }}+\sum _{i=1}^{m_1}{R}_{1i}\left(1-{\beta }^{x_{1i}^{\alpha }}\right)+\sum _{i=2}^{m_1}\ln \left[e^{\left(1-{\beta }^{x_{1\left(i-1\right)}^{\alpha }}\right)}-e^{\left(1-{\beta }^{x_{1i}^{\alpha }}\right)}\right] \\ +\ln \left[1-e^{\left(1-{\beta }^{{\left(\lambda x_{21}\right)}^{\alpha }}\right)}\right]-{\beta }^{{\left(\lambda x_{2m_2}\right)}^{\alpha }}+\sum _{i=1}^{m_2}{R}_{2i}\left[1-{\beta }^{{\left(\lambda x_{2i}\right)}^{\alpha }}\right]+\sum _{i=2}^{m_2}\ln \left[e^{\left[1-{\beta }^{{\left({\lambda x}_{2\left(i-1\right)}\right)}^{\alpha }}\right]}-e^{\left[1-{\beta }^{(\lambda {x_{2i}) }^{\alpha }}\right]}\right]. \end{gathered}$$

(30)

The first partial derivatives of the logarithm for the PSF with respect to $\alpha , \lambda ,$ and $\beta$ are presented below:

$$\begin{gathered} {\displaystyle \frac{\partial \mathcal{l}}{\partial \alpha }}={\displaystyle \frac{{\beta }^{x_{11}^{\alpha }} x_{11}^{\alpha } \ln \left(x_{11}\right)\ln \left(\beta \right)e^{\left(1-{\beta }^{x_{11}^{\alpha }}\right)}}{\left[1-e^{\left(1-{\beta }^{x_{11}^{\alpha }}\right)}\right]}}-{\beta }^{x_{1m_1}^{\alpha }} x_{1m_1}^{\alpha } \ln \left(x_{1m_1}\right)\ln \left(\beta \right)-{\sum }_{i=1}^{m_1}{R}_{1i}{\beta }^{x_{1i}^{\alpha }} x_{1i}^{\alpha } \ln \left(x_{1i}\right) \\ \times \ln \left(\beta \right)-{\sum }_{i=2}^{m_1}{\displaystyle \frac{{\beta }^{x_{1\left(i-1\right)}^{\alpha }} x_{1\left(i-1\right)}^{\alpha } \ln \left(x_{1\left(i-1\right)}\right)\ln \left(\beta \right)e^{\left(1-{\beta }^{x_{1\left(i-1\right)}^{\alpha }}\right)}-{\beta }^{x_{1i}^{\alpha }} x_{1i}^{\alpha } \ln \left(x_{1i}\right)\ln \left(\beta \right)e^{\left(1-{\beta }^{x_{1i}^{\alpha }}\right)}}{\left[e^{\left(1-{\beta }^{x_{1\left(i-1\right)}^{\alpha }}\right)}-e^{\left(1-{\beta }^{x_{1i}^{\alpha }}\right)}\right]}} \\ +{\displaystyle \frac{{\beta }^{{\left(\lambda x_{21}\right)}^{\alpha }} {\left(\lambda x_{21}\right)}^{\alpha } \ln \left(\lambda x_{21}\right)\ln \left(\beta \right)e^{\left(1-{\beta }^{{\left(\lambda x_{21}\right)}^{\alpha }}\right)}}{\left[1-e^{\left(1-{\beta }^{{\left(\lambda x_{21}\right)}^{\alpha }}\right)}\right]}}-{\beta }^{{\left(\lambda x_{2m_2}\right)}^{\alpha }} {\left(\lambda x_{2m_2}\right)}^{\alpha } \ln \left(\lambda x_{2m_2}\right)\ln \left(\beta \right) \\ -{\sum }_{i=1}^{m_2}{R}_{2i}{\beta }^{{\left(\lambda x_{2i}\right)}^{\alpha }} {\left(\lambda x_{2i}\right)}^{\alpha } \ln \left(\lambda x_{2i}\right)\ln \left(\beta \right) \\ -{\sum }_{i=2}^{m_2}{\displaystyle \frac{{\beta }^{{\left(\lambda x_{2\left(i-1\right)}\right)}^{\alpha }} {\left(\lambda x_{2\left(i-1\right)}\right)}^{\alpha } \ln \left(\lambda x_{2\left(i-1\right)}\right)\ln \left(\beta \right)e^{\left[1-{\beta }^{{\left(\lambda x_{2\left(i-1\right)}\right)}^{\alpha }}\right]}-{\beta }^{{\left(\lambda x_{2i}\right)}^{\alpha }} {\left(\lambda x_{2i}\right)}^{\alpha } \ln \left(\lambda x_{2i}\right)\ln \left(\beta \right)e^{\left[1-{\beta }^{{\left(\lambda x_{2i}\right)}^{\alpha }}\right]}}{\left\{e^{\left[1-{\beta }^{{\left(\lambda x_{2\left(i-1\right)}\right)}^{\alpha }}\right]}-e^{\left[1-{\beta }^{{\left(\lambda x_{2i}\right)}^{\alpha }}\right]}\right\}}} \end{gathered}$$

(31)

$$\begin{gathered} {\displaystyle \frac{\partial \mathcal{l}}{\partial \beta }}={\displaystyle \frac{ x_{11}^{\alpha }{\beta }^{\left(x_{11}^{\alpha }-1\right)} e^{\left(1-{\beta }^{x_{11}^{\alpha }}\right)}}{\left[1-e^{\left(1-{\beta }^{x_{11}^{\alpha }}\right)}\right]}}-x_{1m_1}^{\alpha } {\beta }^{\left(x_{1m_1}^{\alpha }-1\right)}-{\sum }_{i=1}^{m_1}{R}_{1i}{x}_{1i}^{\alpha } {\beta }^{\left(x_{1i}^{\alpha }-1\right)} \\ -{\sum }_{i=2}^{m_1}{\displaystyle \frac{ x_{1\left(i-1\right)}^{\alpha }{\beta }^{\left(x_{1\left(i-1\right)}^{\alpha }-1\right)}{e}^{\left(1-{\beta }^{x_{1\left(i-1\right)}^{\alpha }}\right)}-x_{1i}^{\alpha }{\beta }^{\left(x_{1i}^{\alpha }-1\right)} e^{\left(1-{\beta }^{x_{1i}^{\alpha }}\right)}}{\left[e^{\left(1-{\beta }^{x_{1\left(i-1\right)}^{\alpha }}\right)}-e^{\left(1-{\beta }^{x_{1i}^{\alpha }}\right)}\right]}}+{\displaystyle \frac{ {\left(\lambda x_{21}\right)}^{\alpha }{\beta }^{\left[{\left(\lambda x_{21}\right)}^{\alpha }-1\right]} e^{\left[1-{\beta }^{{\left(\lambda x_{21}\right)}^{\alpha }}\right]}}{\left\{1-e^{\left[1-{\beta }^{{\left(\lambda x_{21}\right)}^{\alpha }}\right]}\right\}}} \\ -{\left(\lambda x_{2m_2}\right)}^{\alpha }{\beta }^{\left[{\left(\lambda x_{2m_2}\right)}^{\alpha }-1\right]}-{\sum }_{i=1}^{m_2}{R}_{2i} {\left(\lambda x_{2i}\right)}^{\alpha } {\beta }^{\left[{\left(\lambda x_{2i}\right)}^{\alpha }-1\right]} \\ -{\sum }_{i=2}^{m_2}{\displaystyle \frac{ {\left(\lambda x_{2\left(i-1\right)}\right)}^{\alpha }{\beta }^{\left[{\left(\lambda x_{2\left(i-1\right)}\right)}^{\alpha }-1\right]} e^{\left[1-{\beta }^{{\left(\lambda x_{2\left(i-1\right)}\right)}^{\alpha }}\right]}-{\left(\lambda x_{2i}\right)}^{\alpha } {\beta }^{\left[{\left(\lambda x_{2i}\right)}^{\alpha }-1\right]}{e}^{\left[1-{\beta }^{{\left(\lambda x_{2i}\right)}^{\alpha }}\right]}}{\left\{e^{\left[1-{\beta }^{{\left(\lambda x_{2\left(i-1\right)}\right)}^{\alpha }}\right]}-e^{\left[1-{\beta }^{{\left(\lambda x_{2i}\right)}^{\alpha }}\right]}\right\}}}, \end{gathered}$$

(32)

and

$$\begin{gathered} {\displaystyle \frac{\partial \mathcal{l}}{\partial \lambda }}={\displaystyle \frac{{\alpha {\left(\lambda x_{21}\right)}^{\alpha -1}\left(x_{21}\right)\beta }^{{\left(\lambda x_{21}\right)}^{\alpha }} \ln \left(\beta \right)e^{\left[1-{\beta }^{{\left(\lambda x_{21}\right)}^{\alpha }}\right]}}{\left\{1-e^{\left[1-{\beta }^{{\left(\lambda x_{21}\right)}^{\alpha }}\right]}\right\}}}-{\alpha \left(\lambda x_{2m_2}\right)}^{\alpha -1} \left(x_{2m_2}\right){\beta }^{{\left(\lambda x_{2m_2}\right)}^{\alpha }}\ln \left(\beta \right) \\ -\alpha \ln \left(\beta \right){\sum }_{i=1}^{m_2}{R}_{2i}{\beta }^{{\left(\lambda x_{2i}\right)}^{\alpha }} {\left(\lambda x_{2i}\right)}^{\alpha -1} \left(x_{2i}\right) \\ -{\sum }_{i=2}^{m_2}{\displaystyle \frac{{\alpha \mathrm{l}\mathrm{n}\left(\beta \right)\beta }^{{\left(\lambda x_{2\left(i-1\right)}\right)}^{\alpha }} {\left(\lambda x_{2\left(i-1\right)}\right)}^{\alpha -1} \left(x_{2\left(i-1\right)}\right)e^{\left[1-{\beta }^{{\left(\lambda x_{2\left(i-1\right)}\right)}^{\alpha }}\right]}-\alpha \mathrm{l}\mathrm{n}\left(\beta \right){\beta }^{{\left(\lambda x_{2i}\right)}^{\alpha }} {\left(\lambda x_{2i}\right)}^{\alpha -1} \left(x_{2i}\right)e^{\left[1-{\beta }^{{\left(\lambda x_{2i}\right)}^{\alpha }}\right]}}{\left\{e^{\left[1-{\beta }^{{\left(\lambda x_{2\left(i-1\right)}\right)}^{\alpha }}\right]}-e^{\left[1-{\beta }^{{\left(\lambda x_{2i}\right)}^{\alpha }}\right]}\right\}}}. \end{gathered}$$

(33)

Equating the first partial derivatives of (31)–(33) with respect to $\alpha , \beta ,$ and $\lambda ,$ to zeros, solving simultaneously, thus numerical technique can be applied to obtain the MPS estimates for the parameters$\alpha ,\beta ,$and $\lambda$.

3.3. Asymptotic Confidence Intervals

The asymptotic normality of the ML and MPS estimators under large samples is highly helpful in building the ACIs for the parameters. Therefore, $100(1 - \tau )\%$ ACIs for $\underset{\_}{\theta }=\left(\alpha ,\beta ,\lambda \right)$ based on ML and MPS estimators are given, respectively, by:

$${\left({\widehat{\theta }}_{LL},{\widehat{\theta }}_{UL}\right)}_{ML}={\widehat{\theta }}_{ML}\mp z_{1-{\displaystyle \frac{\tau }{2}}}\sqrt{V\left({\widehat{\theta }}_{ML}\right)},$$

and

$${\left({\widehat{\theta }}_{LL},{\widehat{\theta }}_{UL}\right)}_{MPS}={\widehat{\theta }}_{MPS}\mp z_{1-{\displaystyle \frac{\tau }{2}}}\sqrt{V\left({\widehat{\theta }}_{MPS}\right)},$$

(34)

where $\mathrm{L}\mathrm{L}$ and $\mathrm{U}\mathrm{L}$ denote the lower limit ($\mathrm{L}\mathrm{L})$ and upper limit ($\mathrm{U}\mathrm{L})$ for ACIs, ${\widehat{\theta }}_{ML}$ and ${\widehat{\theta }}_{MPS}$ are, respectively, the ML and the MPS estimators of $\underset{\_}{\theta }=\left(\alpha , \beta ,\lambda \right)$, $z_{{\displaystyle \frac{\tau }{2}}}$is the percentile of the standard normal distribution, and$V\left({\widehat{\theta }}_{ML}\right)$ and $V\left({\widehat{\theta }}_{MPS}\right)$ are the asymptotic variances of the ML and the MPS estimators calculated using the inverse of the observed Fisher information matrix. Where the observed Fisher information matrix in the two methods was obtained based on the second-order partial derivatives of (25) or (30), respectively. The second partial derivative of the parameters for ML and MPS estimators are given in Appendix B.

4. Numerical Illustration

This section aims to investigate the precision of the estimation methods discussed in the previous sections. A simulation study is conducted to assess the performance of the proposed estimation methodologies in CS-PALT under progressive Type II censoring. In addition, real data sets are used to ensure that the proposed model is flexible and applicable in a lot of real-world situations.

4.1. Simulation Study

In this subsection, a simulation study is conducted to illustrate the performance of the presented ML and MPS estimates based on generated data from the LL distribution. ML and MPS estimates of the parameters and the acceleration factor of the LL distribution based on CS-PALT under progressive Type II censoring are computed. All computations were performed using (Mathematica 11).

Ref. [49] proposed an algorithm to generate progressive Type II censored samples. By applying this algorithm, one can generate the progressive Type II censored samples under usual and accelerated conditions from LL distribution as follows:

• Based on the values of $n_s$ and $m_s$($1{\le m}_s\le n_s)$, generate two independent random samples of sizes $m_1$ and $m_2$ from uniform (0,1) distribution, ($U_{s1},U_{s2},\dots , U_{s{m}_s}), s=\mathrm{1,2}.$
• For given values of the progressive censoring scheme $R_{si},$we set $Y_{si}=U_{si}^{1/(i+{\sum }_{w=m_s-i+1}^{m_s}{R}_{sw})},$ where $s=\mathrm{1,2}, i=\mathrm{1,2},\dots ,m_s$.
• Construct the two progressive Type II censored samples, ($U_{s1}^{*}, U_{s2}^{*}, \dots ,U_{s{m}_s}^{*}),$ from the uniform (0,1) distribution, where $U_{si}^{*}=1-{\prod }_{w=m_s-i+1}^{m_s}{Y}_{sw}$.
• Using $U_{si}^{*}$ defined in Step 3 and the cdfs, $F_1\left(x\right)$ and $F_2\left(x\right),$ given in (1) and (22), respectively, one can generate two random samples under usual and accelerated conditions from LL distribution based on progressive Type II censoring. Repeat all the previous steps N = 1000 times.
• Calculate the averages for both the ML and MPS estimates such that
  $\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e}\left(\widehat{\theta }\right)={\displaystyle \frac{{\sum }_{i=1}^N{\widehat{\theta }}_i}{N}},$ where ${\widehat{\theta }}_i$denotes either the ML or the MPS estimate in the $ith$ itration.

The efficiency of various estimates is determined using the following measurements:

(a)

The relative error $\left(\mathrm{R}\mathrm{E}\right)$ of the point estimate, which is calculated using the relation

$$\mathrm{R}\mathrm{E}={\displaystyle \frac{{\sum }_{i=1}^N{\left(\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}-\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{e} \mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}\right)}^2}{N}}.$$

(b)

For each estimation method, the ACIs were measured and compared using the average lengths (AL). For the generated sample, we computed 100(1 − τ) % ACIs of the parameters and the acceleration factor, recorded AL by the following formula:

$$\mathrm{A}\mathrm{L}={\displaystyle \frac{{\sum }_{i=1}^N{(\mathrm{U}\mathrm{L}}_i-{\mathrm{L}\mathrm{L}}_i)}{N}},$$

where ${\mathrm{L}\mathrm{L}}_i$ and ${\mathrm{U}\mathrm{L}}_i$ are the $ith$ lower and upper bounds, respectively, of asymptotic interval estimates. The significance level is selected to be τ = 0.05, in all cases. The ALs of the confidence interval lengths allow for a quantitative comparison of the parameter estimate precision among different schemes and approaches. Shorter intervals generally imply more precise estimates, serving in model selection. To give an overview of the total uncertainty in our model parameters, ALs of the confidence interval lengths are calculated. This helps assess the model’s reliability and predictive performance. The ALs of the confidence interval lengths confirm our results, whereas as the sample sizes $(n_s,m_s)$ increase, the values of ALs decrease. For example, the shorter intervals for the parameters align with the higher significance of their effects.

For $s=\mathrm{1,2},$ three progressive censoring schemes in both usual and accelerated conditions were considered as follows:

Scheme I: $R_{s1}=n_s-m_s \mathrm{a}\mathrm{n}\mathrm{d} R_{si}=0$ for $i\ne 1$.

Scheme II: $R_{{sm}_s}=n_s-m_s \mathrm{a}\mathrm{n}\mathrm{d} R_{si}=0$ for $i\ne m_s$.

Scheme III: $R_{s\left(m_s-1\right)/2}=n_s-m_s,R_{si}=0$ for $i\ne \left(m_s-1\right)/2$ if $m_s$ is odd and $R_{m_s/2}=n_s-m_s,R_{si}=0$ for $i\ne {\displaystyle \frac{m_s}{2}}$ if $m$ is even.

Remark. 

When the scheme ${ R}_{{sm}_s}=n_s-m_s and R_{si}=0$ for $i\ne m_s$ (Scheme II) is applied, all the results obtained in this paper based on progressive Type II censoring reduce to those of Type II censored samples. Additionally, if $R_{si}=0, s=\mathrm{1,2}, i=\mathrm{1,2},\dots ,m_s$ and $n_s=m_s$ is applied, all the results obtained in this paper based on progressive Type II censoring reduce to the complete sample case.

The best scheme is that which minimizes the REs and ALs of the estimates.

Regarding the population parameters and sample sizes, two cases are considered as follows:

(i)

The population parameter values ($\lambda =9.8,\beta =3.9 \mathrm{a}\mathrm{n}\mathrm{d} \alpha =2.2$), with sample sizes ($n_1=n_2=n$), observed failure times ($m_1=m_2=m$), which represent $60\%$ and $80\%$ of the sample size. The results are shown in

Table 2. Averages of the ML, averages of MPS, REs and ALs for ACIs for the parameters of LL distribution based on CS-PALT under different progressive censoring schemes ($\mathit{\lambda }=9.8,\mathit{\beta }=3.9,\mathit{\alpha }=2.2$ at ${\mathit{n}}_1={\mathit{n}}_2=\mathit{n} \mathrm{a}\mathrm{n}\mathrm{d} {\mathit{m}}_1={\mathit{m}}_2=\mathit{m})$.

$$\mathit{n}$$	$$\mathit{m}$$	Scheme		$$\mathsf{\alpha }$$			$$\mathsf{\beta }$$			$$\mathsf{\lambda }$$
			Methods	Average	RE	$$\mathbf{A}\mathbf{L}$$	Average	$$\mathbf{R}\mathbf{E}$$	$$\mathbf{A}\mathbf{L}$$	Average	$$\mathbf{R}\mathbf{E}$$	$$\mathbf{A}\mathbf{L}$$
30	18	I	ML	1.7545	0.2056	0.3087	3.5061	0.1012	0.0891	9.9372	0.0141	0.0689
			MPS	1.7617	0.2020	0.2899	3.4895	0.1053	0.0530	9.9279	0.0132	0.0793
		II	ML	1.4089	0.3672	0.6443	3.6766	0.0583	0.1641	10.0555	0.0266	0.2038
			MPS	1.4090	0.3670	0.6329	3.6740	0.0594	0.2023	10.0547	0.0266	0.2140
		III	ML	1.5803	0.2902	0.6027	3.3423	0.1436	0.1952	9.6165	0.0189	0.1022
			MPS	1.5855	0.2881	0.6073	3.3311	0.1463	0.1799	9.6097	0.0196	0.1059
	24	I	ML	1.8042	0.1827	0.2754	3.6260	0.0704	0.0722	9.9206	0.0124	0.0616
			MPS	1.8085	0.1810	0.2834	3.6099	0.0745	0.0578	9.9119	0.0116	0.0738
		II	ML	1.5949	0.2768	0.2672	3.7214	0.0468	0.1511	9.9958	0.0201	0.0794
			MPS	1.5979	0.2751	0.2448	3.7104	0.0497	0.1608	9.9913	0.0196	0.0748
		III	ML	1.6519	0.2515	0.2946	3.4856	0.1065	0.1073	9.8957	0.0099	0.0685
			MPS	1.6565	0.2497	0.3153	3.4750	0.1091	0.0893	9.8896	0.0094	0.0796
60	36	I	ML	2.1018	0.0452	0.0618	3.8593	0.0105	0.0161	9.8686	0.0070	0.0196
			MPS	2.1092	0.0419	0.0615	3.8248	0.0194	0.0311	9.8391	0.0041	0.0337
		II	ML	1.8583	0.1557	0.0929	3.7191	0.0468	0.1001	9.9155	0.0118	0.0149
			MPS	1.8607	0.1546	0.0960	3.7090	0.0494	0.1022	9.9110	0.0113	0.0205
		III	ML	2.0780	0.0563	0.0859	3.8600	0.0111	0.0639	9.8725	0.0074	0.0225
			MPS	2.0843	0.0536	0.0879	3.8305	0.0181	0.0486	9.8472	0.0049	0.0393
	48	I	ML	2.1426	0.0263	0.0335	3.8673	0.0084	0.0094	9.8502	0.0051	0.0151
			MPS	2.1488	0.0236	0.0361	3.8321	0.0174	0.0080	9.8286	0.0030	0.0290
		II	ML	2.0801	0.0547	0.0450	3.8420	0.0157	0.0779	9.8750	0.0077	0.0119
			MPS	2.0862	0.0520	0.0463	3.8125	0.0230	0.0786	9.8517	0.0053	0.0186
		III	ML	2.1012	0.0453	0.0487	3.8691	0.0079	0.0117	9.8657	0.0067	0.0190
			MPS	2.1085	0.0420	0.0511	3.8327	0.0173	0.0081	9.8345	0.0036	0.0351
100	60	I	ML	2.1783	0.0100	0.0165	3.8672	0.0084	0.0048	9.8266	0.0027	0.0072
			MPS	2.1838	0.0077	0.0180	3.8363	0.0163	0.0064	9.8076	0.0009	0.0145
		II	ML	2.1515	0.0223	0.0290	3.8455	0.0145	0.0634	9.8428	0.0044	0.0089
			MPS	2.1541	0.0211	0.0314	3.8319	0.0181	0.0768	9.8346	0.0036	0.0180
		III	ML	2.1497	0.0231	0.0306	3.8720	0.0072	0.0076	9.8511	0.0052	0.0141
			MPS	2.1556	0.0205	0.0326	3.8384	0.0158	0.0080	9.8309	0.0032	0.0239
	80	I	ML	2.2054	0.0027	0.0103	3.8699	0.0077	0.0036	9.8171	0.0017	0.0042
			MPS	2.2102	0.0048	0.0113	3.8431	0.0146	0.0049	9.8002	0.0002	0.0097
		II	ML	2.2337	0.0153	0.0086	3.8541	0.0123	0.0552	9.8208	0.0021	0.0047
			MPS	2.2374	0.0170	0.0094	3.8332	0.0174	0.0523	9.8087	0.0009	0.0068
		III	ML	2.1539	0.0211	0.0194	3.8757	0.0062	0.0054	9.8457	0.0046	0.0087
			MPS	2.1595	0.0186	0.0209	3.8440	0.0143	0.0054	9.8264	0.0027	0.0156

.

(ii)

The population parameter values ($\lambda =6.8,\beta =5.1 \mathrm{a}\mathrm{n}\mathrm{d} \alpha =2.5$), with sample sizes ($n_1\ne n_2$), observed failure times ($m_1\ne m_2$), with different levels of censoring sample sizes. The results are given in

Table 3. Averages of the ML, averages of MPS, REs and ALs for ACIs for the parameters of LL distribution based on CS-PALT under different progressive censoring schemes ($\mathit{\lambda }=6.8,\mathit{\beta }=5.1,\mathit{\alpha }=2.5$ at ${\mathit{n}}_1\ne {\mathit{n}}_2 \mathrm{a}\mathrm{n}\mathrm{d} {\mathit{m}}_1\ne {\mathit{m}}_2)$.

n1 n2	m1 m2	Scheme		$$\mathsf{\alpha }$$			$$\mathsf{\beta }$$			$$\mathsf{\lambda }$$
			Methods	Average	RE	$$\mathbf{A}\mathbf{L}$$	Average	$$\mathbf{R}\mathbf{E}$$	$$\mathbf{A}\mathbf{L}$$	Average	$$\mathbf{R}\mathbf{E}$$	$$\mathbf{A}\mathbf{L}$$
40	22	I	ML	2.2929	0.0841	0.1465	5.0770	0.0045	0.0116	6.8724	0.0108	0.0437
30	16		MPS	2.1987	0.1232	0.2498	5.0582	0.0082	0.0122	6.8842	0.0130	0.1065
		II	ML	2.1750	0.1311	0.1686	5.0684	0.0063	0.0195	6.9034	0.0153	0.0470
			MPS	2.1773	0.1302	0.1692	5.0643	0.0070	0.0171	6.8966	0.0143	0.0521
		III	ML	2.0467	0.1903	0.5657	5.0611	0.0081	0.0556	6.9539	0.0236	0.1754
			MPS	2.0512	0.1887	0.5698	5.0527	0.0095	0.0434	6.9383	0.0217	0.2048
30	30	I	ML	2.3549	0.0584	0.0599	5.0846	0.0030	0.0049	6.8490	0.0072	0.0182
40	22		MPS	2.3604	0.0562	0.0642	5.0748	0.0049	0.0080	6.8275	0.0043	0.0403
		II	ML	2.2562	0.0981	0.1058	5.0769	0.0046	0.0116	6.8799	0.0118	0.0305
			MPS	2.2593	0.0969	0.1069	5.0711	0.0057	0.0084	6.8695	0.0103	0.0373
		III	ML	2.1992	0.1237	0.2833	5.0743	0.0052	0.0230	6.9067	0.0160	0.0894
			MPS	2.2043	0.1223	0.3075	5.0615	0.0076	0.0146	6.8807	0.0128	0.1281
60	33	I	ML	2.3791	0.0486	0.0482	5.0857	0.0028	0.0047	6.8431	0.0064	0.0144
80	44		MPS	2.3848	0.0464	0.0521	5.0759	0.0047	0.0076	6.8208	0.0033	0.0344
		II	ML	2.3003	0.0802	0.0687	5.0782	0.0043	0.0085	6.8644	0.0095	0.0181
			MPS	2.3023	0.0794	0.0692	5.0746	0.0050	0.0071	6.8582	0.0086	0.0211
		III	ML	2.3479	0.0625	0.1410	5.0837	0.0032	0.0152	6.8520	0.0078	0.0349
			MPS	2.3521	0.0610	0.1473	5.0759	0.0047	0.0113	6.8360	0.0057	0.0602
	45	I	ML	2.4214	0.0316	0.0300	5.0879	0.0024	0.0043	6.8302	0.0045	0.0081
	60		MPS	2.4267	0.0295	0.0334	5.0793	0.0041	0.0070	6.8095	0.0017	0.0251
		II	ML	2.3099	0.0763	0.0619	5.0807	0.0038	0.0070	6.8643	0.0095	0.0175
			MPS	2.3435	0.0628	0.0522	5.0782	0.0043	0.0043	6.8449	0.0066	0.0182
		III	ML	2.3785	0.0488	0.0442	5.0856	0.0028	0.0048	6.8433	0.0064	0.0129
			MPS	2.3842	0.0465	0.0477	5.0759	0.0047	0.0078	6.8212	0.0033	0.0325
100	55	I	ML	2.4453	0.0220	0.0249	5.0892	0.0021	0.0040	6.8255	0.0037	0.0065
120	66		MPS	2.4504	0.0200	0.0281	5.0806	0.0038	0.0063	6.8049	0.0011	0.0221
		II	ML	2.3689	0.0526	0.0376	5.0825	0.0034	0.0051	6.8466	0.0068	0.0093
			MPS	2.4479	0.0209	0.0182	5.0865	0.0026	0.0026	6.8170	0.0025	0.0049
		III	ML	2.4398	0.0244	0.0420	5.0893	0.0020	0.0028	6.8273	0.0040	0.0124
			MPS	2.4444	0.0227	0.0462	5.0812	0.0037	0.0073	6.8088	0.0018	0.0318
	75	I	ML	2.4537	0.0187	0.0219	5.0904	0.0019	0.0036	6.8254	0.0037	0.0056
	90		MPS	2.4594	0.0164	0.0238	5.0809	0.0037	0.0005	6.8026	0.0007	0.0159
		II	ML	2.4095	0.0363	0.0203	5.0880	0.0023	0.0026	6.8381	0.0056	0.0053
			MPS	2.4705	0.0118	0.0102	5.0896	0.0020	0.0010	6.8124	0.0018	0.0036
		III	ML	2.4510	0.0197	0.0192	5.0904	0.0018	0.0027	6.8262	0.0038	0.0053
			MPS	2.4563	0.0176	0.0216	5.0814	0.0037	0.0052	6.8050	0.0010	0.0176

.

Under several choices of the parameter values, sample sizes, and censoring schemes, the results of criteria quantities for both ML and MPS estimates are presented in Table 2 and Table 3.

4.2. Real Data Analysis

In this subsection, two real lifetime CS-PALT data sets are investigated to demonstrate the flexibility and adaptability of the LL distribution as well as to present the actual use of the estimation approaches covered in the earlier sections.

The first data (DATA I) set shows the oil breakdown time of insulating liquids in real time. Electrical insulation has a lifespan that affects a lot of industrial components. The life test may be made more effective by accelerating conditions, like raising the voltage. Based on this background, Ref. [1] investigated accelerated testing by subjecting various experimental groups to varying voltage magnitudes during the tests. The empirically analyzed data in this paper are derived from this study. In the empirical analysis, the sample at 32 kilovolt (KV) was used as data under the usual condition, while the sample at 38 KV was used as data under the accelerated condition.

The second data set (DATA II) presented by [50], describes the ordered times to failure in rolling contact fatigue of 10 hardened steel specimens tested at each of the four values of contact stress. In the empirical analysis, the sample at $0.990\times {10}^6$psi was used as data under the usual condition, and the sample at $1.18\times {10}^6$psi was used as data under the accelerated condition.

The Kolmogorov–Smirnov (KS) goodness of fit test was applied to check if the LL distribution can be used as a proper model to fit the two real data sets.

Table 4. KS distances and p-values of the LL distribution for the two real data sets.

DATA		KS	p-Values
	Usual condition
DATA I	0.27, 0.4, 0.69, 0.79, 2.75, 3.91, 9.88, 13.95, 15.93, 27.8, 53.24, 82.85, 89.29, 100.58, 215.1	$0.1227$	$0.9776$
	Accelerated condition
	0.09, 0.39, 0.47, 0.73, 0.74, 1.13, 1.40, 2.38	$0.1704$	$0.9743$
DATA II	Usual condition
	0.80, 1.00, 1.37, 2.25, 2.95, 3.70, 6.07, 6.65, 7.05, 7.37	$0.2040$	$0.7996$
	Accelerated condition
	0.073, 0.098, 0.117, 0.135, 0.175, 0.262, 0.27, 0.35, 0.386, 0.456	$0.1697$	$0.9355$

displays the two real data sets, KS distances, and the corresponding p-values in both usual and accelerated conditions.

The p-values indicated that the LL distribution is an adequate model for the two real data sets. In addition, one can use the information about the data that is represented by the hrf to select a proper model. A graphical technique known as the total time on test (TTT) transform can be considered for this purpose, see [51]. The TTT plots and box plots for the two real data sets are presented in Figure 5 and Figure 6, respectively. Figure 5 and Figure 6 indicated convex, straight diagonal, and concave TTT plots, which implied that the data sets have decreasing, constant, and increasing hrfs, respectively. Note that all these shapes are contained in the LL hrf.

Moreover, some criteria quantities are utilized to demonstrate the superiority of the LL distribution over other competitive distributions in modeling lifetime data sets. The compared models are the inverted Kumaraswamy (IKUM), extended odd Weibull exponential (EOWE), Nadarajah–Haghighi (NH), flexible Weibull (FW), and Lomax distributions. The used criteria quantities are Akaike’s information criteria (AIC), consistent Akaike’s information criteria (CAIC), Bayesian information criteria (BIC), and Hannan–Quinn information criteria (HQIC). These quantities are calculated and presented for the two real data sets in

Table 5. $\mathbf{A}\mathbf{I}\mathbf{C},\mathbf{B}\mathbf{I}\mathbf{C},\mathbf{C}\mathbf{A}\mathbf{I}\mathbf{C},$ and $\mathbf{H}\mathbf{Q}\mathbf{I}\mathbf{C}$ for DATA I.

Results under Usual Conditions
Model	$\mathbf{A}\mathbf{I}\mathbf{C}$	$\mathbf{B}\mathbf{I}\mathbf{C}$	$\mathbf{C}\mathbf{A}\mathbf{I}\mathbf{C}$	$\mathbf{H}\mathbf{Q}\mathbf{I}\mathbf{C}$
$\mathrm{L}\mathrm{L}$	$135.4324$	$136.8485$	$136.4324$	$135.4174$
$\mathrm{I}\mathrm{K}\mathrm{U}\mathrm{M}$	$138.9307$	$140.3468$	$139.9307$	$138.9156$
$\mathrm{B}\mathrm{I}\mathrm{I}\mathrm{I}$	$137.7816$	$139.1977$	$138.7816$	$137.7665$
$\mathrm{E}\mathrm{O}\mathrm{W}\mathrm{E}$	$136.4687$	$138.5928$	$138.6505$	$136.4460$
$\mathrm{N}\mathrm{H}$	$167.5239$	$168.9400$	$168.5239$	$167.5088$
$\mathrm{F}\mathrm{W}$	$147.1618$	$148.5779$	$148.1618$	$147.1467$
$\mathrm{L}\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{x}$	$138.6048$	$140.0209$	$139.6048$	$138.5897$
Results under accelerated conditions
Model	$\mathbf{A}\mathbf{I}\mathbf{C}$	$\mathbf{B}\mathbf{I}\mathbf{C}$	$\mathbf{C}\mathbf{A}\mathbf{I}\mathbf{C}$	$\mathbf{H}\mathbf{Q}\mathbf{I}\mathbf{C}$
$\mathrm{L}\mathrm{L}$	$17.7381$	$17.8970$	$20.1381$	$16.6665$
$\mathrm{I}\mathrm{K}\mathrm{U}\mathrm{M}$	$18.5689$	$18.7278$	$20.9689$	$17.4973$
$\mathrm{B}\mathrm{I}\mathrm{I}\mathrm{I}$	$17.9328$	$18.0917$	$20.3328$	$16.8612$
$\mathrm{E}\mathrm{O}\mathrm{W}\mathrm{E}$	$19.5711$	$19.8094$	$25.5711$	$17.9637$
$\mathrm{N}\mathrm{H}$	$35.9909$	$36.1498$	$38.3909$	$34.9193$
$\mathrm{F}\mathrm{W}$	$49.8442$	$50.0031$	$52.2442$	$48.7726$
$\mathrm{L}\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{x}$	$18.6006$	$18.7595$	$21.0006$	$17.5290$

and

Table 6. $\mathbf{A}\mathbf{I}\mathbf{C},\mathbf{B}\mathbf{I}\mathbf{C},\mathbf{C}\mathbf{A}\mathbf{I}\mathbf{C},$ and $\mathbf{H}\mathbf{Q}\mathbf{I}\mathbf{C}$ for DATA II.

Results under Usual Conditions
Model	$\mathbf{A}\mathbf{I}\mathbf{C}$	$\mathbf{B}\mathbf{I}\mathbf{C}$	$\mathbf{C}\mathbf{A}\mathbf{I}\mathbf{C}$	$\mathbf{H}\mathbf{Q}\mathbf{I}\mathbf{C}$
$\mathrm{L}\mathrm{L}$	$47.9943$	$48.5995$	$49.7086$	$47.3305$
$\mathrm{I}\mathrm{K}\mathrm{U}\mathrm{M}$	$50.8126$	$51.4177$	$52.5269$	$50.1487$
$\mathrm{B}\mathrm{I}\mathrm{I}\mathrm{I}$	$51.1503$	$51.7555$	$52.8646$	$50.4865$
$\mathrm{E}\mathrm{O}\mathrm{W}\mathrm{E}$	$51.9160$	$52.8238$	$55.9160$	$50.9202$
$\mathrm{N}\mathrm{H}$	$78.8345$	$79.4396$	$80.5488$	$78.1706$
$\mathrm{F}\mathrm{W}$	$58.0596$	$58.6648$	$59.7739$	$57.3957$
$\mathrm{L}\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{x}$	$51.3270$	$51.9322$	$53.0413$	$50.6632$
Results under accelerated conditions
Model	$\mathbf{A}\mathbf{I}\mathbf{C}$	$\mathbf{B}\mathbf{I}\mathbf{C}$	$\mathbf{C}\mathbf{A}\mathbf{I}\mathbf{C}$	$\mathbf{H}\mathbf{Q}\mathbf{I}\mathbf{C}$
$\mathrm{L}\mathrm{L}$	$-10.6153$	$-10.0102$	$-8.9010$	$-11.2792$
$\mathrm{I}\mathrm{K}\mathrm{U}\mathrm{M}$	$-10.3542$	$-9.7490$	$-8.6399$	$-11.0181$
$\mathrm{B}\mathrm{I}\mathrm{I}\mathrm{I}$	$1.1972$	$1.8023$	$2.9115$	$0.5333$
$\mathrm{E}\mathrm{O}\mathrm{W}\mathrm{E}$	$-8.5160$	$-7.6082$	$-4.5160$	$-9.5117$
$\mathrm{N}\mathrm{H}$	$13.4815$	$14.0867$	$15.1958$	$12.8177$
$\mathrm{F}\mathrm{W}$	$47.0419$	$47.6471$	$48.7562$	$46.3780$
$\mathrm{L}\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{x}$	$-5.2031$	$-4.5979$	$-3.4888$	$-5.8669$

, respectively. The best distribution is the one with the least values of the AIC, CAIC, BIC, and HQIC. Table 5 and Table 6 strongly indicate that the LL distribution is the most adequate among others.

The three progressive censoring schemes considered in the simulation study were applied to the two real data sets to find the ML and MPS estimates of the unknown parameters and the acceleration factor for the LL distribution. The results are displayed in

Table 7. ML and MPS estimates of the unknown parameters for the LL distribution based on CS-PALT under different progressive censoring schemes from the two real data sets.

DATA	n1 n2	m1 m2	Scheme	$\mathsf{\alpha }$		$\mathsf{\beta }$		$\mathsf{\lambda }$
				$\mathbf{M}\mathbf{L}$	$\mathbf{M}\mathbf{P}\mathbf{S}$	$\mathbf{M}\mathbf{L}$	$\mathbf{M}\mathbf{P}\mathbf{S}$	$\mathbf{M}\mathbf{L}$	$\mathbf{M}\mathbf{P}\mathbf{S}$
DATA I	15	2	I	2.8709	3.0911	3.3834	1.0711	3.9384	1.5410
		1	II	0.3399	0.3430	6.7597	6.7712	4.5478	4.5437
			III	1.6081	1.8001	2.5419	1.3995	2.0821	2.0305
	8	6	I	2.1106	2.3006	3.9514	1.2004	2.8364	2.2330
		4	II	3.8681	4.2005	1.5783	1.2005	2.2120	1.5093
			III	3.5735	3.8002	2.5477	1.4002	1.9142	2.3991
DATA II	10	1	I	1.1516	0.6697	7.4517	3.2287	5.3283	3.4303
		1	II	0.5095	0.5119	3.1381	3.2027	4.7263	4.2159
			III	0.7573	0.6800	3.2446	3.2236	3.4268	3.4301
	10	4	I	1.0827	0.4466	6.7757	3.2405	4.7005	4.2655
		4	II	1.4038	1.4017	3.3063	3.3209	3.3263	3.3263
			III	0.7959	0.7922	3.0952	3.1113	3.5145	3.5135

.

5. Conclusions

In recent years, progressive censoring has attracted much attention, mainly because high-speed computer systems are now readily available. This makes progressive censoring a viable topic for researchers to conduct simulation studies on and an efficient method for practitioners to collect lifetime data. This article examines a CS-PALT model with progressively Type II censored data, where the observed failure times originate from the LL distribution. Additionally, the article highlights the statistical inference on an untapped distribution, although it is applicable, known as the LL distribution, and thoroughly examines the idea of a Vtub hrf. A variety of the statistical properties of the LL distribution have been deduced.

Based on CS-PALT under progressive Type II censored data, the ML and MPS estimators for the unknown parameters, and the acceleration factor of the LL distribution are addressed. Furthermore, the ACIs for the unknown parameters and the acceleration factor are identified using both estimate approaches. A simulation study was performed to examine and compare the performance of the proposed estimation methods for a variety of sample sizes, censoring schemes, acceleration factors, and parameter values. The applicability of the used methods and LL distribution is demonstrated through two real-life CS-PALT data sets.

The significant differences in ML and MPS estimates of the model parameters under different censoring schemes can be referred to some factors:

• Information available in the Data and the Data distribution since different censoring schemes (e.g., Type I, Type II, progressive) obtain different amounts of information about the main survival distribution. The shape of the underlying survival distribution interacts with the censoring scheme. Heavy-tailed distributions might be more sensitive to censoring than light-tailed ones.
• Estimation Method Sensitivity: ML estimation can be sensitive to outliers and model misspecification even if it is known for effectiveness under correct model assumptions. Different censoring schemes might lead to different levels of sensitivity. MPS estimation is more robust to outliers but might be less efficient than MLE under ideal conditions. Its performance can vary across different censoring schemes.
• Model Assumptions: The chosen model can affect the impact of censoring. Some models are more robust to censoring than others. If the censoring distribution is not correctly specified, it can bias the estimates.

Considering the empirical results, one may indicate the following:

• Evidently, LL distribution is a highly competitive model that can describe a large amount of lifetime data.
• As the sample sizes $(n_s,m_s)$ increase, the values of REs and ALs decrease. It can be assumed that the estimation methods have good consistency.
• For a fixed number of units$(n_s,m_s)$, the estimates under the censored Schemes I and II are superior to the estimates of Scheme III.
• With small sample sizes, the ML estimates appear to perform better than the MPS estimates; conversely, with large sample sizes, the MPS estimates appear to perform better than the ML estimates.
• Merging all the earlier results, one can suggest using the MPS estimation method to estimate the parameters for large samples and the ML estimation method for small samples of the LL distribution based on CS-PALT under progressive Type II censoring.

Some suggestions for future research can be carried out as follows: studying Bayesian estimation methods based on different types of loss functions for estimating the parameters of the LL distribution. Considering other methods of estimation, such as the modified ML method or modified moments. Additionally, other types of progressive censoring schemes can be considered for estimating the parameters, such as progressive first-failure censoring, adaptive Type II progressive censoring, and Type II stepwise progressive censoring schemes. Studying the ML and Bayesian prediction (point and interval) for future observation of the LL distribution. An optimal accelerated life testing model for the LL distribution may also be derived. Many researchers have applied these methods to many distributions, such as [52,53,54,55].