Engineering Applications with Stress-Strength for a New Flexible Extension of Inverse Lomax Model: Bayesian and Non-Bayesian Inference

Abstract

In this paper, we suggest a brand new extension of the inverse Lomax distribution for fitting engineering time data. The newly developed distribution, termed the transmuted Topp–Leone inverse Lomax (TTLILo) distribution, is characterized by an additional shape and transmuted parameters. It is critical to notice that the skewness, kurtosis, and tail weights of the distribution are strongly influenced by these additional characteristics of the extra parameters. The TTLILo model is capable of producing right-skewed, J-shaped, uni-modal, and reversed-J-shaped densities. The proposed model’s statistical characteristics, including the moments, entropy values, stochastic ordering, stress-strength model, incomplete moments, and quantile function, are examined. Moreover, characterization based on two truncated moments is offered. Using Bayesian and non-Bayesian estimating techniques, we estimate the distribution parameters of the suggested distribution. The bootstrap procedure, approximation, and Bayesian credibility are the three forms of confidence intervals that have been created. A simulation study is used to assess the efficiency of the estimated parameters. The TTLILo model is then put to the test by being applied to actual engineering datasets, demonstrating that it offers a good match when compared to alternative models. Two applications based on real engineering datasets are taken into consideration: one on the failure times of airplane air conditioning systems and the other on the active repair times of airborne communication transceivers. Also, we consider the problem of estimating the stress-strength parameter $R=P(Z_2<Z_1)$ with engineering application.

1. Introduction

The modeling of actual occurrences using probability distributions is one of the most crucial jobs of statistics. In many fields, including economics, industry, health, and agriculture, actual lifetime data have been extensively modeled using traditional statistical distributions. A crucial issue is choosing an effective model from which to draw inferences. Weibull [1] first presented the Weibull distribution, a well-known continuous distribution used in survival and reliability studies. This distribution’s inability to accommodate non-monotone hazard rates is one of its main weaknesses. Due to this, it is necessary to investigate further generalizing this model. The exponentiated Weibull model is the first generalization that permits non-monotone hazard rates (see [2,3]). Stacy [4] created the generalized gamma distribution, which includes unique sub-models, such as the exponential, Weibull, gamma, and Rayleigh distributions, among others. Other semi-parametric models (see [5,6]) have been employed to model different forms of survival data.

Our interest in one significant lifetime model with applications in the actuarial and economic sciences is the ILo (inverse Lomax) distribution, as mentioned by Kleiber and Kotz [7]. Furthermore, the ILo distribution is strongly recommended in both stochastic modeling and life testing. To obtain the Lorenz ordering connection among order statistics, Kleiber [8] took into account the ILo distribution. Using geophysical information, this distribution was put into practice [9]. The ILo distribution is a special case of the generalized beta distribution of the second kind. The ILo distribution can be derived from the Lomax distribution using the transformation $Z=1/Y$, where Y has a Lomax distribution. Rahman and Aslam [10] employed a two-component mixture ILo model to forecast the next-ordered observations using a Bayesian framework. Yadav et al. [11] discussed estimating the ILo distribution’s parameters using hybrid censored samples. An estimation of a two-component mixture ILo model using a Bayesian technique was proposed by Rahman and Aslam [12]. Bayesian estimates of the ILo distribution parameters using several approximation approaches were discussed by Jan and Ahmad [13]. Some extended forms for ILo distribution have been presented (see, for example, [14,15,16,17,18]). The SS (stress-strength) reliability estimator of the ILo model, based on extreme and median ranked set sampling, has been presented, respectively, by Al-Omari et al. [19] and Hassan et al. [20]. The CDF (cumulative distribution function) of a two-parameter ILo distribution, with shape parameter $\omega$ and scale parameter $\rho$, is specified by the following:

$$G\left(z\right)={\left(1+\left(\rho /\phantom{\rho z}z\right)\right)}^{-\omega };z,\rho ,\omega >0.$$

(1)

The PDF (probability density function) is as follows:

$$g\left(z\right)=\frac{\omega \rho }{z^2}{\left(1+\left(\rho /\phantom{\rho z}z\right)\right)}^{-\omega -1};z,\rho ,\omega >0.$$

(2)

The goal of this paper is mainly to suggest a new version of the ILo distribution based on the TTL-G (transmuted Topp–Leone-generated) family [21]. The CDF and PDF of the TTL-G family, for $\left|c\right|\le 1,z\in \mathrm{R}$, are defined as follows:

$$F\left(z\right)=\left(1+c\right){\left\{1-{\left[1-G(z;\varsigma )\right]}^2\right\}}^b-c{\left\{1-{\left[1-G(z;\varsigma )\right]}^2\right\}}^{2b},$$

(3)

$$f\left(z\right)=2bg(z;\varsigma )\left(1-G(z;\varsigma )\right){\left\{1-{\left[1-G(z;\varsigma )\right]}^2\right\}}^{b-1}\left\{1+c-2c{\left\{1-{\left[1-G(z;\varsigma )\right]}^2\right\}}^b\right\},$$

(4)

where b is the shape parameter, c is the transmuted parameter, $G(z;\varsigma )$ and $g(z;\varsigma )$ are the CDF and PDF of the baseline distribution, respectively, with $\varsigma$ as the parameter vector. The main motivations behind the use of the TTL-G family are as follows. The TTL-G family in (3) includes the TL-G (Topp–Leone-generated) family considered by Rezaei et al. [22] and the transmuted generated family proposed by Shaw and Buckley [23]. For c = 0, the TL-G family is provided. The baseline distribution’s beneficial features can be enhanced by the TTL-G family and can be constructed heavy-tailed distributions for modeling various real datasets.

The ILo distribution has a limited application and is the most suitable for modeling data with a high failure rate, which is often unfeasible in practical settings. To accommodate different HF (hazard function) forms, the ILo distribution must be expanded to provide greater flexibility. So, the primary goal of this study is to take advantage of the TTL-G family of distributions, which was created by Yousof et al. [21], to provide an enhanced extension of the ILo distribution that is called the TTLILo (transmuted Topp–Leone inverse Lomax) distribution. This is intended to improve the baseline distribution’s fit by increasing its modeling capability. When compared to the baseline model (ILo distribution), the newly constructed model exhibits diverse shapes of the HF and provides superior fits for various types of datasets. The TTLILo distribution contains the ILo and TLILo distributions as special sub-models. We propose the TL ILo distribution according to the following:

• To produce several forms for the HF and PDF.
• To improve the ILo distribution’s flexibility for modeling various data.
• To enhance the adaptability of the standard ILo distribution’s mean, variance, skewness, and kurtosis properties.
• To estimate the TTLILo distribution parameters using Bayesian and non-Bayesian approaches.
• To construct the ACIs (approximate confidence intervals), Bayesian credible intervals, and BCIs (bootstrap confidence intervals).
• To take the the data analysis of jute fiber-breaking strengths at two different gauge lengths and investigate for SS application purposes and compare to some other models.
• To model skewed data, which are difficult to examine with other traditional models.
• To demonstrate with two engineering datasets that the TTLILo distribution gives a better fit than some other models.

The rest of this paper unfolds as follows. The TTLILo distribution is introduced in Section 2, along with its useful expansion and quantile function. Some vital aspects are covered in Section 3. Section 4 discusses some findings of the characterization. Section 5 discusses how to estimate model parameters using maximum likelihood and Bayesian techniques. The simulation results are used to assess the performance of the estimation techniques, and they are provided in Section 6. In Section 7, we provide examples using real data to emphasize the importance of the proposed distribution. Application of the SS reliability model is discussed in Section 8. Some last remarks are included in Section 9.

2. The Construction of the TTLILo Distribution

In this section, we present the TTLILo distribution. Representations for its PDF and CDF are derived. The reliability and HF formulas are also provided.

Let the random variable Z have the ILo distribution with CDF (1) and PDF (2), and then from (1) to (3), the CDF of the TTLILo distribution is defined, for z >0, as follows:

$$F\left(z;\Delta \right)=\left(1+c\right){\left\{1-{\left[1-{\left(1+\left(\rho /\phantom{\rho z}z\right)\right)}^{-\omega }\right]}^2\right\}}^b-c{\left\{1-{\left[1-{\left(1+\left(\rho /\phantom{\rho z}z\right)\right)}^{-\omega }\right]}^2\right\}}^{2b}.$$

(5)

For the simplified form, let $\mathrm{{\rm Y}}(z,\rho ,\omega )=\left(1-{\left(1+\left(\rho /\phantom{\rho z}z\right)\right)}^{-\omega }\right)$ in (5), and then the CDF of the TTLILo distribution can be written as

$$F\left(z;\Delta \right)=\left(1+c\right){\left\{1-{\mathrm{{\rm Y}}}^2(z,\rho ,\omega )\right\}}^b-c{\left\{1-{\mathrm{{\rm Y}}}^2(z,\rho ,\omega )\right\}}^{2b},z>0.$$

(6)

The PDF of the TTLILo distribution is as follows

$$\begin{array}{cc}\hfill f\left(z;\Delta \right)=& 2b\rho \omega z^{-2}{\left(1+\left(\rho /\phantom{\rho z}z\right)\right)}^{-\omega -1}\mathrm{{\rm Y}}(z,\rho ,\omega ){\left\{1-{\mathrm{{\rm Y}}}^2(z,\rho ,\omega )\right\}}^{b-1}\hfill \\ & \left\{1+c-2c{\left\{1-{\mathrm{{\rm Y}}}^2(z,\rho ,\omega )\right\}}^b\right\},z>0,\hfill \end{array}$$

(7)

where $b,\omega >0,$ are the shape parameters; $\rho >0,$ is the scale parameter; $\left|c\right|\le 1$ is the transmuted parameters; and $\Delta \equiv (b,c,\rho ,\omega )$ is the set of parameters. A random variable Z with PDF (7) will be denoted by Z ∼ TTLILo $(b,c,\rho ,\omega ).$ For c = 0, CDF (6) reduces to the TLILo distribution prepared by Hassan and Ismail [17]. The HF of Z is as below:

$$\begin{array}{cc}\hfill h(z;\Delta )=& 2b\rho \omega z^{-2}{\left(1+\left(\rho /\phantom{\rho z}z\right)\right)}^{-\omega -1}\mathrm{{\rm Y}}(z,\rho ,\omega ){\left\{1-{\mathrm{{\rm Y}}}^2(z,\rho ,\omega )\right\}}^{b-1}\hfill \\ & \frac{\left\{1+c-2c{\left\{1-{\mathrm{{\rm Y}}}^2(z,\rho ,\omega )\right\}}^b\right\}}{1-\left[\left(1+c\right){\left\{1-{\mathrm{{\rm Y}}}^2(z,\rho ,\omega )\right\}}^b-c{\left\{1-{\mathrm{{\rm Y}}}^2(z,\rho ,\omega )\right\}}^{2b}\right]}.\hfill \end{array}$$

(8)

Plots of the PDF in (7) and the HF in (8) can be seen in Figure 1 and Figure 2 for some variant parameter values. Figure 1 shows that the PDF of Z is extremely adaptable and may assume several forms, including asymmetric, uni-modal, J-shaped, reversed-J-shaped, and upside-down forms. Finally, they emphasize the advantages of the proposed model. In addition, Figure 2 shows that the HF has asymmetric, uni-modal, J-shaped, reversed-J-shaped, and upside-down forms. Finally, they emphasize the advantages of the proposed model.

2.1. Usefuel Expansions of the TTLILo Model

In this subsection, a representation of the TTLILo CDF, as well as the TTLILo PDF, is provided. By using the binomial expansion more than once in (6), then the CDF in (6) can be represented as follows:

$$\begin{array}{c}F\left(z;\Delta \right)=\left(1+c\right){\displaystyle {\sum }_{i_1=0}^{\infty }}{\displaystyle {\sum }_{m=0}^{2i_1}}{\left(-1\right)}^{i_1+m}\left(\begin{array}{c}b\\ i_1\end{array}\right)\left(\begin{array}{c}2i_1\\ m\end{array}\right){\left(1+\left(\rho /\phantom{\rho z}z\right)\right)}^{-\omega m}\hfill \\ -c{\displaystyle {\sum }_{i_1=0}^{\infty }}{\displaystyle {\sum }_{m=0}^{2i_1}}{\left(-1\right)}^{i_1+m}\left(\begin{array}{c}2b\\ i_1\end{array}\right)\left(\begin{array}{c}2i_1\\ m\end{array}\right){\left(1+\left(\rho /\phantom{\rho z}z\right)\right)}^{-\omega m}.\hfill \end{array}$$

Hence, the above CDF of the TTLILo distribution can be written as

$$F\left(z;\Delta \right)=\sum _{m=0}^{2i_1}{\eta }_{i_1,m}{\pi }_m\left(z\right),$$

(9)

where ${\eta }_{i_i,m}={\sum }_{i_1=0}^{\infty }{\left(-1\right)}^{i_1+m}\left[(1+c)\left(\begin{array}{c}b\\ i_1\end{array}\right)-c\left(\begin{array}{c}2b\\ i_1\end{array}\right)\right]\left(\begin{array}{c}2i_1\\ m\end{array}\right),$ and ${\pi }_m\left(z\right)={\left(1+\left(\rho /\phantom{\rho z}z\right)\right)}^{-\omega m}$ is the CDF of the ILo distribution with parameters $(\rho ,\omega m).$ By differentiating (9), we obtain the PDF of the TTLILo distribution as follows:

$$f\left(z;\Delta \right)=\sum _{m=0}^{2i_1}{\eta }_{i_1,m}{\pi }^{\ast }{}_{m+1}\left(z\right),$$

$${\pi }^{\ast }{}_{m+1}\left(z\right)=\frac{\omega (m+1)\rho }{z^2}{\left(1+\left(\rho /\phantom{\rho z}z\right)\right)}^{-\omega (m+1)-1}.$$

(10)

2.2. Quantile Function

It is easy to generate the TTLILo distribution by inverting (5) as shown below: If u has a uniform distribution U (0, 1), then the nonlinear equation’s solution for c ≠ 0 is

$$\begin{array}{c}\hfill z_u=\rho {\left\{{\left\{1-{\left[1-{\left(k\right)}^{1/\phantom{1b}b}\right]}^{0.5}\right\}}^{\frac{-1}{\omega }}-1\right\}}^{-1},\\ \hfill k={\left(2c\right)}^{-1}\left[\left(1+c\right)-\sqrt{{\left(1+c\right)}^2-4cu}\right].\end{array}$$

This scheme is helpful for generating TTLILo random variates.

3. Structural Properties

Stochastic ordering (SO), incomplete moments, moments, and some variability measures are the few significant statistical aspects of the TTLILo distribution that are described in this section.

3.1. Stochastic Ordering

SO is a well-studied idea in probability distributions used to evaluate the performance of random variables in reliability theory especially. Let Z${}_i$ have the parameters ${\Delta }_i\equiv (b_i,c_i,{\rho }_i,{\omega }_i)$i = 1, 2 from the TTLILo distribution. Suppose that $F_i(z;{\Delta }_i)$ and $f_i(z;{\Delta }_i)$ indicate, respectively, Z${}_i$’s CDF and PDF.

If $f_1(z;{\Delta }_1)/\phantom{f_1(z;{\Delta }_1)f_2(z;{\Delta }_2)}{f}_2(z;{\Delta }_2)$ is a decreasing function ∀ z, then, in terms of the LRO (likelihood ratio order), Z${}_1$ is said to be stochastically less than Z${}_2$, which is simplified by (Z${}_1$${\le }_{sr}$Z${}_2$).

Let Z${}_1$∼ TTLILo $\left({\Delta }_1\right)$ and Z${}_2$∼ TTLILo $\left({\Delta }_2\right),$ and then the LRO is

$$\begin{array}{cc}\hfill \frac{f_1(z;{\Delta }_1)}{f_2(z;{\Delta }_2)}=& \frac{b_1{\rho }_1{\omega }_1{\left(1+\left({\rho }_1/\phantom{{\rho }_{1}z}z\right)\right)}^{-{\omega }_1-1}{\mathrm{{\rm Y}}}_1(z,{\rho }_1,{\omega }_1){\left[1-{\mathrm{{\rm Y}}}_1^2(z,{\rho }_1,{\omega }_1)\right]}^{b_1-1}}{b_2{\rho }_2{\omega }_2{\left(1+\left({\rho }_2/\phantom{{\rho }_{2}z}z\right)\right)}^{-{\omega }_2-1}{\mathrm{{\rm Y}}}_2(z,{\rho }_2,{\omega }_2){\left[1-{\mathrm{{\rm Y}}}_2^2(z,{\rho }_2,{\omega }_2)\right]}^{b_2-1}}\hfill \\ & \times \frac{\left\{1+c_1-2c_1{\left[1-{\mathrm{{\rm Y}}}_1^2(z,{\rho }_1,{\omega }_1)\right]}^{b_1}\right\}}{\left\{1+c_2-2c_2{\left[1-{\mathrm{{\rm Y}}}_2^2(z,{\rho }_2,{\omega }_2)\right]}^{b_2}\right\}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill \frac{d}{dz}log\left[\frac{f_1(z;{\Delta }_1)}{f_2(z;{\Delta }_2)}\right]=& \frac{({\omega }_1+1){\rho }_1}{{\rho }_1+z^2}-\frac{({\omega }_2+1){\rho }_2}{{\rho }_2+z^2}+\hfill \\ & \left[\frac{1}{{\mathrm{{\rm Y}}}_1(z,{\rho }_1,{\omega }_1)}-\frac{\left(2(b_1-1)-2c_1{b}_1\right){\mathrm{{\rm Y}}}_1(z,{\rho }_1,{\omega }_1)}{1-{\mathrm{{\rm Y}}}_1^2(z,{\rho }_1,{\omega }_1)}\right]\frac{\partial {\mathrm{{\rm Y}}}_1(z,{\rho }_1,{\omega }_1)}{\partial z}-\hfill \\ & \left[\frac{1}{{\mathrm{{\rm Y}}}_2(z,{\rho }_2,{\omega }_2)}-\frac{\left(2(b_2-1)-2c_2{b}_2\right){\mathrm{{\rm Y}}}_2(z,{\rho }_2,{\omega }_2)}{1-{\mathrm{{\rm Y}}}_2^2(z,{\rho }_2,{\omega }_2)}\right]\frac{\partial {\mathrm{{\rm Y}}}_2(z,{\rho }_2,{\omega }_2)}{\partial z},\hfill \end{array}$$

$${\mathrm{{\rm Y}}}_i(z,{\rho }_i,{\omega }_i)=1-{\left(1+\left({\rho }_i/\phantom{{\rho }_{i}z}z\right)\right)}^{-{\omega }_i},\frac{\partial {\mathrm{{\rm Y}}}_i(z,{\rho }_i,{\omega }_i)}{\partial z}=\frac{-{\rho }_i{\omega }_i}{z^2}{\left(1+\left({\rho }_i/\phantom{{\rho }_{i}z}z\right)\right)}^{-{\omega }_i-1},i=1,2$$

For $b_1<b_2,c_1<c_2,{\omega }_1<{\omega }_2,c_1=c_2$, we obtain $\frac{d}{dz}log\left[\frac{f_1(z;{\Delta }_1)}{f_2(z;{\Delta }_2)}\right]<0,$ for all $z\ge 0,$; hence, $\frac{f_1(z;{\Delta }_1)}{f_2(z;{\Delta }_2)}$ is decreasing in z and hence Z${}_1$${\le }_{lr}$Z${}_2$. Moreover, Z${}_1$ is said to be smaller than Z${}_2$ in other different orderings, such as SO (denoted by Z${}_1$${\le }_{sr}$Z${}_2$), the HF order that is denoted by $Z_1{\le }_{HFo}{Z}_2$, and the reversed HF (denoted by $Z_1{\le }_{RHFo}{Z}_2$).

3.2. Moments and Incomplete Moments

Moments are useful in statistical studies, particularly in applications. They may be used to research the most significant distributional aspects and features. For instance, these features are dispersion, skewness, kurtosis, and tendency. The s$th$ moment of the TTLILo distribution that has a PDF in (10) can be calculated as follows:

$${\mu }_s^{\prime }=\sum _{m=0}^{2i_1}{\eta }_{i_1,m}{\int }_0^{\infty }{z}^s\frac{\omega (m+1)\rho }{z^2}{\left(1+\left(\rho /\phantom{\rho z}z\right)\right)}^{-\omega (m+1)-1}dz.$$

After simplification, the s$th$ moment of the TTLILo model is obtained as follows:

$${\mu }_s^{\prime }=\sum _{m=0}^{2i_1}{\eta }_{i_1,m}{\rho }^s\omega (m+1)\mathrm{B}\left(1-s,\omega (m+1)+s\right),1<s,$$

where B(.,.) is the beta function. Now, we arrive at a straightforward expression for the s$th$ inverse moment of Z, say, ${\xi }_s,$ as follows:

$$\begin{array}{cc}\hfill {\xi }_s& =\sum _{m=0}^{2i_1}{\eta }_{i_1,m}{\int }_0^{\infty }{z}^{-s}\frac{\omega (m+1)\rho }{z^2}{\left(1+\left(\rho /\phantom{\rho z}z\right)\right)}^{-\omega (m+1)-1}dz\hfill \\ & =\sum _{m=0}^{2i_1}{\eta }_{i_1,m}{\rho }^s\omega (m+1)\mathrm{B}\left(1+s,\omega (m+1)-s\right).\hfill \end{array}$$

Note that the harmonic mean of the TTLILo distribution can be obtained by using the first inverse moment.

3.3. Variability Measures

The entropy of a random variable is a measure of knowledge variation. The reliability of the data decreases as the entropy increases. A higher entropy value denotes a lower degree of accuracy in the data. As a result, the entropy takes into account several informative measures. One of these metrics is the RE (Rényi entropy), which was identified by Renyi [24]. The RE is given by

$$\mathsf{𝔍}\left(z\right)={(1-\epsilon )}^{-1}log\left[{\int }_{-\infty }^{\infty }{\left(f\left(z\right)\right)}^{\epsilon }dz\right],\epsilon >0,\epsilon \ne 1.$$

The TTLILo distribution’s PDF, ${\left(f(z;\Delta )\right)}^{\epsilon },$ is stated as follows.

$$\begin{array}{cc}\hfill {\left(f(z;\Delta )\right)}^{\epsilon }=& \frac{{\left(2b\rho \omega \right)}^{\epsilon }{(1+c)}^{\epsilon }{\left(1+\left(\rho /\phantom{\rho z}z\right)\right)}^{-\epsilon (\omega +1)}}{z^{2\epsilon }}{\mathrm{{\rm Y}}}^{\epsilon }(\rho ,\omega ){\left\{1-{\mathrm{{\rm Y}}}^2(z,\rho ,\omega )\right\}}^{\epsilon (b-1)}\hfill \\ & \times {\left\{1-\frac{2c}{1+c}{\left\{1-{\mathrm{{\rm Y}}}^2(z,\rho ,\omega )\right\}}^b\right\}}^{\epsilon }.\hfill \end{array}$$

By applying the binomial expansions, under the consideration that $\frac{2c}{1+c}{\left\{1-{\mathrm{{\rm Y}}}^2{(z,\rho ,\omega )}^2\right\}}^b$$<1,$ we obtain

$${\left(f(z;\Delta )\right)}^{\epsilon }=\sum _{j_1,j_2,j_3=0}^{\infty }{\Xi }_{j_1,j_2,j_3}{z}^{-2\epsilon }{\left(1+\left(\rho /\phantom{\rho z}z\right)\right)}^{-\epsilon (\omega +1)-\omega j_3},$$

(11)

$${\Xi }_{j_1,j_2,j_3}={(-1)}^{j_1+j_2+j_3}{\left(2b\rho \omega \right)}^{\epsilon }{(1+c)}^{\epsilon -j_1}{\left(2c\right)}^{j_1}\left(\begin{array}{c}\epsilon \\ j_1\end{array}\right)\left(\begin{array}{c}\epsilon +2j_2\\ j_3\end{array}\right)\left(\begin{array}{c}\epsilon (b-1)+b{j}_1\\ j_2\end{array}\right).$$

Hence, the RE of the TTLILo distribution is as follows:

$$\mathsf{𝔍}\left(z\right)={(1-\epsilon )}^{-1}log\left[\sum _{j_1,j_2,j_3=0}^{\infty }{\Xi }_{j_1,j_2,j_3}{\rho }^{1-2\epsilon }\mathrm{B}\left(2\epsilon -1,\omega (\epsilon +j_3)-\epsilon +1\right)\right].$$

Tsallis [25] generalized Shannon’s entropy and defined the measure as:

$$T_{\epsilon }\left(z\right)=\frac{1}{\epsilon -1}\left[1-{\int }_{-\infty }^{\infty }{\left(f\left(z\right)\right)}^{\epsilon }dz\right],\epsilon \ne 1,\epsilon >0.$$

Based on (11), the Tsallis entropy of the TTLILo distribution is obtained as follows:

$$T_{\epsilon }\left(z\right)=\frac{1}{\epsilon -1}\left[1-\sum _{j_1,j_2,j_3=0}^{\infty }{\Xi }_{j_1,j_2,j_3}{\rho }^{1-2\epsilon }\mathrm{B}\left(2\epsilon -1,\omega (\epsilon +j_3)-\epsilon +1\right)\right].$$

The $\epsilon$-generalized entropy presented by Mathai and Haubold [26] is another generalized form of the Shannon entropy. It is determined by

$${\eta }_{\epsilon }\left(z\right)={(\epsilon -1)}^{-1}\left[{\int }_{-\infty }^{\infty }{\left(f\left(z\right)\right)}^{2-\epsilon }dz-1\right],0<\epsilon <2,\epsilon \ne 1.$$

Using a similar way to (11),

$${\left(f(z;\Delta )\right)}^{2-\epsilon }=\sum _{j_1,j_2,j_3=0}^{\infty }{\Xi }_{j_1,j_2,j_3}^{\ast }{z}^{-2\epsilon }{\left(1+\left(\rho /\phantom{\rho z}z\right)\right)}^{-\epsilon (\omega +1)-\omega j_3},$$

$${\Xi }_{j_1,j_2,j_3}^{\ast }={(-1)}^{j_1+j_2+j_3}{\left(2b\rho \omega \right)}^{\epsilon }{(1+c)}^{\epsilon -j_1}{\left(2c\right)}^{j_1}\left(\begin{array}{c}2-\epsilon \\ j_1\end{array}\right)\left(\begin{array}{c}\epsilon +2j_2\\ j_3\end{array}\right)\left(\begin{array}{c}\epsilon (b-1)+b{j}_1\\ j_2\end{array}\right).$$

Then, ${\eta }_{\epsilon }\left(z\right)$ of the TTLILo distribution is given by:

$${\eta }_{\epsilon }\left(z\right)={(\epsilon -1)}^{-1}\sum _{j_1,j_2,j_3=0}^{\infty }{\Xi }_{j_1,j_2,j_3}^{\ast }{\rho }^{1-2\epsilon }\mathrm{B}\left(2\epsilon -1,\omega (\epsilon +j_3)-\epsilon +1\right).$$

For some of the selected parameter values,

Table 1. Entropy values for the TTLILo model.

	$\mathit{\epsilon }=0.8$			$\mathit{\epsilon }=1.5$
$(\mathit{b},\mathit{c},\mathit{\rho },\mathit{\omega })$	$\mathsf{𝔍}\left(\mathit{z}\right)$	${\mathit{T}}_{\mathit{\epsilon }}\left(\mathit{z}\right)$	${\mathit{\eta }}_{\mathit{\epsilon }}\left(\mathit{z}\right)$	$\mathsf{𝔍}\left(\mathit{z}\right)$	${\mathit{T}}_{\mathit{\epsilon }}\left(\mathit{z}\right)$	${\mathit{\eta }}_{\mathit{\epsilon }}\left(\mathit{z}\right)$
(1.5, −0.5, 1.5, 0.5)	1.176	1.326	0.629	0.453	0.406	3.97
(2.5, −0.5, 1.5, 0.5)	1.588	1.869	1.047	1.013	0.795	5.014
(4, −0.5, 1.5, 0.5)	1.906	2.32	1.313	1.376	0.995	6.048
(1.5, −0.5, 3, 0.5)	1.87	2.267	1.195	1.147	0.873	6.443
(2.5, −0.5, 3, 0.5)	2.281	2.891	1.559	1.706	1.148	7.919
(4, −0.5, 3, 0.5)	2.599	3.408	1.79	2.07	1.289	9.382
(1.5, 0.5, 1.5, 0.5)	0.352	0.365	−0.287	−0.571	−0.661	2.255
(2.5, 0.5, 1.5, 0.5)	0.889	0.973	0.415	0.253	0.237	3.121
(4, 0.5, 1.5, 0.5)	1.273	1.45	0.803	0.724	0.607	3.965
(1.5, 0.5, 3, 0.5)	1.045	1.162	0.398	0.122	0.118	4.018
(2.5, 0.5, 3, 0.5)	1.583	1.862	1.008	0.946	0.754	5.242
(4, 0.5, 3, 0.5)	1.966	2.409	1.347	1.417	1.015	6.436

lists the numerical entropy values.

We draw some conclusions from Table 1:

• The values of $\mathsf{𝔍}\left(z\right)$ and $T_{\epsilon }\left(z\right)$ when $\epsilon =0.8$ are greater than the values of $\mathsf{𝔍}\left(z\right)$ and $T_{\epsilon }\left(z\right)$ at $\epsilon =1.8$. Then, it can be concluded that the values of $\mathsf{𝔍}\left(z\right)$ decrease as the value of $\epsilon$ increases, whereas the ${\eta }_{\epsilon }\left(z\right)$ values increase, for all the values of the selected parameters.
• At $\epsilon =0.8,$ the ${\eta }_{\epsilon }\left(z\right)$ measure’s value is lower than the values for the other two measures, which indicate more information.
• For the same set of parameters, all the measures have values that are lower for the positive transmuted parameter values than for the negative transmuted parameter values. For example, the results of (1.5, −0.5, 1.5, 0.5) and (1.5, 0.5, 1.5, 0.5).
• There is less uncertainty in all the measure values in accordance with their smaller values, as the value of b increases with the same values of the other parameters. For example, see the results of (1.5, −0.5, 1.5, 0.5) and (2.5, −0.5, 1.5, 0.5).
• All the uncertainty measures decrease as the value of $\rho$ increases with the same values of the other parameters, suggesting less uncertainty. For example, see the results of (1.5, −0.5, 1.5, 0.5) and (1.5, −0.5, 3, 0.5).

4. Characterizations

Statistical characterizations of distributions have received increasing attention in recent years. Consequently, it is worth mentioning that the various TTLILo dispersion characterizations are presented. Two abbreviated moments have a straightforward link that supports these classifications. This characterization result employs a Glänzel [27] theorem. In addition, the result is still valid when the interval K and the CDF do not have a closed form. According to Glänzel [28], this characterization is stable in the sense of weak convergence. This section provides characterizations based on two truncated moments.

Theorem 1.

Let $(O,F,P)$ be a given probability space and suppose that K = [h${}_1$, h${}_2$] is an interval for some $h_1<h_2(h_1=-\infty ,h_2=\infty )$. Let $Z:\mathrm{\Omega }\to (0,\infty )$ be a continuous random variable with the CDF F, and let $d_1$ and $d_2$ be two real functions defined on K such that

$$E\left(d_i\left(Z\right)\left|Z\ge z\right.\right)={\int }_z^{\infty }{d}_i\left(t\right)\frac{f\left(t\right)}{1-F\left(z\right)}dti=1,2,$$

and $E\left(d_2\left(Z\right)\left|Z\ge z\right.\right)=E\left(d_1\left(Z\right)\left|Z\ge z\right.\right)\tau \left(z\right),z\in K,$ is defined for some real function $\tau .$ Assume that $d_1,d_2\in C^1\left(K\right),\tau \in C^2\left(K\right),$ and F is a twice continuously differentiable and strictly monotone function on the set. Assume that the equation $\tau d_1=d_2$ has no real solution in the interior of K. Then, F is uniquely determined by the functions d${}_1$(z), d${}_2$(z), and $\tau \left(z\right)$ as below:

$$F\left(z\right)={\int }_{h_1}^{z}C\left[\frac{{\tau }^{{}^{\prime }}\left(u\right)}{\tau \left(u\right)d_1\left(u\right)-d_2\left(u\right)}\right]e^{-\left(s\right(u\left)\right)}du,$$

where the function s is a solution of the differential equation $s^{\prime }={\tau }^{{}^{\prime }}{d}_1/\phantom{{\tau }^{{}^{\prime }}{d}_1\left(\tau d_1-d_2\right)}\left(\tau d_1-d_2\right),$ and C is the normalization constant, such that ${\int }_{\mathrm{K}}dF=1.$

Proposition 1.

Let $Z:\mathrm{\Omega }\to (0,\infty )$ be a continuous random variable and let

$d_1\left(z\right)={\left\{1+c-2c{\left\{1-{\left[\mathrm{{\rm Y}}(z,\rho ,\omega )\right]}^2\right\}}^b\right\}}^{-1},$ $d_2\left(z\right)=d_1\left(z\right){\left\{1-{\left[\mathrm{{\rm Y}}(z,\rho ,\omega )\right]}^2\right\}}^b,$ for z > 0, where $\mathrm{{\rm Y}}(z,\rho ,\omega )=\left(1-{\left(1+\left(\rho /\phantom{\rho z}z\right)\right)}^{-\omega }\right).$

The random variable Z belongs to the TTLILo distribution in (7) if and only if the function $\tau$ defined in Theorem 1 has the form

$$\tau (z)=\frac{1}{2}\left[1+{\left\{1-{\left[\mathrm{{\rm Y}}(z,\rho ,\omega )\right]}^2\right\}}^b\right],z>0.$$

Proof. 

Let Z be a random variable with PDF (7), and then

$$\left(1-F\left(z\right)\right)E\left(d_1\left(Z\right)\left|Z\ge z\right.\right)=1-{\left\{1-{\left[\mathrm{{\rm Y}}(z,\rho ,\omega )\right]}^2\right\}}^b,z>0.$$

And,

$$\left(1-F\left(z\right)\right)E\left(d_2\left(Z\right)\left|Z\ge z\right.\right)=\frac{1}{2}\left[1-{\left\{1-{\left[\mathrm{{\rm Y}}(\rho ,\omega )\right]}^2\right\}}^{2b}\right],z>0.$$

Finally,

$$\tau (z)d_1\left(z\right)-d_2\left(z\right)=\frac{1}{2}{d}_1\left(z\right)\left[1-{\left\{1-{\left[\mathrm{{\rm Y}}(z,\rho ,\omega )\right]}^2\right\}}^b\right],z>0.$$

Conversely, if $\tau$ is given as above, then

$$\begin{array}{c}\hfill s^{\prime }\left(z\right)=\frac{{\tau }^{\prime }\left(z\right)d_1\left(z\right)}{\tau \left(\mathrm{z}\right)d_1\left(z\right)-d_2\left(z\right)}=\frac{2b\rho \omega z^{-2}{\left(1+\left(\rho /\phantom{\rho z}z\right)\right)}^{-\omega -1}\mathrm{{\rm Y}}(z,\rho ,\omega ){\left\{1-{\left[\mathrm{{\rm Y}}(z,\rho ,\omega )\right]}^2\right\}}^{b-1}}{1-{\left\{1-{\left[\mathrm{{\rm Y}}(z,\rho ,\omega )\right]}^2\right\}}^b},\end{array}$$

where $z>0$. Hence,

$$s\left(z\right)=-l\mathrm{og}\left[1-{\left\{1-{\left[\mathrm{{\rm Y}}(z,\rho ,\omega )\right]}^2\right\}}^b\right],z>0.$$

Now, in view of Theorem 1, Z has density (7). □

Corollary 1.

Let $Z:\mathrm{\Omega }\to (0,\infty )$ be a continuous random variable, and let d${}_1$(z) be as in Proposition 1. The PDF of Z is (7) if and only if there exist functions d${}_2$ and τ defined in Theorem 1 satisfying the differential equation

$$\frac{{\tau }^{\prime }\left(z\right)d_1\left(z\right)}{\tau \left(\mathrm{z}\right)d_1\left(z\right)-d_2\left(z\right)}=\frac{2b\rho \omega z^{-2}{\left(1+\left(\rho /\phantom{\rho z}z\right)\right)}^{-\omega -1}\mathrm{{\rm Y}}(z,\rho ,\omega ){\left\{1-{\left[\mathrm{{\rm Y}}(z,\rho ,\omega )\right]}^2\right\}}^{b-1}}{1-{\left\{1-{\left[\mathrm{{\rm Y}}(z,\rho ,\omega )\right]}^2\right\}}^b}.$$

(12)

The general solution of the differential equation in Corollary 1 is

$$\begin{array}{cc}\hfill \tau \left(z\right)=& {\left[1-{\left\{1-{\left[\mathrm{{\rm Y}}(z,\rho ,\omega )\right]}^2\right\}}^b\right]}^{-1}\left[-\int 2b\rho \omega z^{-2}{\left(1+\left(\rho /\phantom{\rho z}z\right)\right)}^{-\omega -1}\mathrm{{\rm Y}}(z,\rho ,\omega )\right.\hfill \\ & \left.\times {\left\{1-{\left[\mathrm{{\rm Y}}(z,\rho ,\omega )\right]}^2\right\}}^{b-1}{d}_2\left(z\right)dz+D{\left(d_1\left(z\right)\right)}^{-1}\right],\hfill \end{array}$$

(13)

where D is a constant. Note that a set of functions satisfying the differential Equation (12) is given in Proposition 1 with $D=0.5.$ However, it should be noted that there are other triplets $(d_1,d_2,\tau )$ satisfying the conditions of Theorem 1.

5. Bayesian and Non-Bayesian Inference

In this section, the MLEs (maximum likelihood estimates) and BEs (Bayesian estimates) of the TTLILo distribution parameters are discussed.

5.1. Maximum Likelihood Estimator

Suppose that Z${}_1$, Z${}_2$,…, Z${}_n$ represent the observed samples from the TTILo distribution with the given set of parameters $\Delta \equiv {(b,c,\rho ,\omega )}^T.$ The log-likelihood function, represented by ${\ell }^{\ast },$ is as follows:

$$\begin{array}{cc}\hfill {\ell }^{\ast }=& nln\left(2b\rho \omega \right)-(\omega +1)\sum _{i=1}^{n}ln\left(1+\left(\rho /\phantom{\rho z_i}{z}_i\right)\right)-2\sum _{i=1}^{n}ln\left(z_i\right)+\sum _{i=1}^{n}ln\mathrm{{\rm Y}}(z_i,\rho ,\omega )\hfill \\ & +(b-1)\sum _{i=1}^{n}ln\left\{1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right\}+\sum _{i=1}^{n}ln\left\{1+c-2c{\left\{1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right\}}^b\right\},\hfill \end{array}$$

(14)

where $\mathrm{{\rm Y}}(z_i,\rho ,\omega )=\left(1-{\left(1+\left(\rho /\phantom{\rho z_i}{z}_i\right)\right)}^{-\omega }\right).$

The MLEs of $b,c,\rho ,$ and $\omega$, denoted by ($\widehat{b},\widehat{c},\widehat{\rho },\widehat{\omega }$), are obtained by maximizing (14) with respect to $b,c,\rho ,$ and $\omega$. Thus, we consider the partial derivatives of ${\ell }^{\ast },$ with respect to $b,c,\rho ,$ and $\omega$; in this regard, the components of the score vector $U_L={(U_b,U_c,U_{\rho },U_{\omega })}^T$ are as follows:

$$U_b=\frac{n}{b}+\sum _{i=1}^{n}ln\left\{1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right\}-\sum _{i=1}^n\frac{2c{\left\{1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right\}}^{b}ln\left\{1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right\}}{\left\{1+c-2c{\left\{1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right\}}^b\right\}},$$

$$U_c=\frac{n}{c}+\sum _{i=1}^n\frac{1-2{\left\{1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right\}}^b}{\left\{1+c-2c{\left\{1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right\}}^b\right\}},$$

$$\begin{array}{cc}\hfill U_{\rho }=& \frac{n}{\rho }-\sum _{i=1}^n\frac{(\omega +1)}{\rho +z_i}+\sum _{i=1}^n\frac{\omega {\left(1+\rho /\phantom{\rho z_i}{z}_i\right)}^{-\omega -1}}{z_i\mathrm{{\rm Y}}(z_i,\rho ,\omega )}-\sum _{i=1}^n\frac{2(b-1)\mathrm{{\rm Y}}(z_i,\rho ,\omega )\omega {(1+\rho /\phantom{\rho z_i}{z}_i)}_{}^{-\omega -1}}{z_i\left(1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right)}\hfill \\ & +\sum _{i=1}^n\frac{4cb\omega {\left\{1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right\}}^{b-1}\mathrm{{\rm Y}}(z_i,\rho ,\omega ){(1+\rho /\phantom{\rho z_i}{z}_i)}_{}^{-\omega -1}}{z_i\left\{1+c-2c{\left\{1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right\}}^b\right\}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill U_{\omega }=& \frac{n}{\omega }+\sum _{i=1}^n\frac{\left(1+\rho /\phantom{\rho z_i}{z}_i\right)ln\left(1+\rho /\phantom{\rho z_i}{z}_i\right)}{\mathrm{{\rm Y}}(z_i,\rho ,\omega )}-\sum _{i=1}^n\frac{2(b-1)\mathrm{{\rm Y}}(z_i,\rho ,\omega )\left(1+\rho /\phantom{\rho z_i}{z}_i\right)ln\left(1+\rho /\phantom{\rho z_i}{z}_i\right)}{\left\{1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right\}}\hfill \\ & -\sum _{i=1}^{n}ln\left(1+\frac{\rho }{z_i}\right)+\sum _{i=1}^n\frac{4cb{\left\{1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right\}}^b{}^{-1}\mathrm{{\rm Y}}(z_i,\rho ,\omega )\left(1+\rho /\phantom{\rho z_i}{z}_i\right)ln\left(1+\rho /\phantom{\rho z_i}{z}_i\right)}{\left\{1+c-2c{\left\{1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right\}}^b\right\}}.\hfill \end{array}$$

The MLEs ($\widehat{b},\widehat{c},\widehat{\rho },\widehat{\omega }$) of $(b,c,\rho ,\omega )$ are produced by putting $U_b=0,U_c=0,U_{\rho }=0$, and $U_{\omega }=0$ and solving simultaneously. These equations can be solved numerically using statistical software.

It is known that the MLE is consistent and asymptotically normally distributed. Therefore, the two-sided approximate $(1-\gamma )$100% confidence limits for a certain population parameter $\Delta$ can be obtained such that:

$$P\left[-z\le \frac{\widehat{\Delta }-\Delta }{\sigma \left(\widehat{\Delta }\right)}\le z\right]\cong \gamma ,$$

where z is the [100(1 −$\gamma$/2)]$th$ standard normal percentile and $\sigma \left(\widehat{\Delta }\right)$ is the standard deviation of the MLE of $\Delta .$ Hence, the two-sided approximate $\gamma$100 percent confidence limits for $\Delta =(b,c,\rho ,\omega ),$ are given, respectively, as follows: $U_{\Delta }=\widehat{\Delta }+z_{\gamma /\phantom{\gamma 2}2}\sigma \left(\widehat{\Delta }\right)$ and $L_{\Delta }=\widehat{\Delta }-z_{\gamma /\phantom{\gamma 2}2}\sigma \left(\widehat{\Delta }\right)$.

5.2. Bayesian Estimator

The parameters are investigated by the Bayesian approach as random variables with a probability distribution. The Bayesian method is extremely beneficial in a reliability analysis because it allows for the incorporation of prior knowledge. We presume that the priors of b, $\rho ,$ and $\omega$ have the following PDFs for their gamma distribution. Also, assume that the prior distribution of parameter c has a uniform distribution on the interval (−1, 1). The joint prior density of b, c, $\rho$, and $\omega$ can be formed as:

$$\pi \left(b,c,\rho ,\omega \right)\propto b^{a_1-1}{e}^{-(b_{1}b+b_3\rho +b_4\omega )}{\rho }^{a_3-1}{\omega }^{a_4-1};a_1,b_1,a_3,b_3,a_4,b_4>0,-1<c<1.$$

(15)

The joint posterior function of the TTLILo distribution is as follows:

$$\begin{array}{cc}\hfill \mathrm{\Pi }(\Delta |z)\propto & b^{n+a_1-1}{\rho }^{n+a_3-1}{\omega }^{n+a_4-1}\prod _{i=1}^n{\left(1+\left(\frac{\rho }{z_i}\right)\right)}^{-\omega -1}\mathrm{{\rm Y}}(z_i,\rho ,\omega ){\left\{1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right\}}^{b-1}\hfill \\ & e^{-(b_{1}b+b_3\rho +b_4\omega )}\prod _{i=1}^n\left\{1+c-2c{\left\{1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right\}}^b\right\}.\hfill \end{array}$$

The BE is considered under the squared error (SE) loss function, which can be defined as $L(\tilde{b},b)={(\tilde{b}-b)}^2,$ $L(\tilde{c},c)={(\tilde{c}-c)}^2,$ $L(\tilde{\rho },\rho )={(\tilde{\rho }-\rho )}^2,$ and $L(\tilde{\omega },\omega )={(\tilde{\omega }-\omega )}^2.$ The MCMC (Markov Chain Monte Carlo) technique can be used to receive the BEs. Important subclasses of the MCMC methods include Gibbs sampling and the more versatile Metropolis inside Gibbs samplers. The MH (Metropolis–Hastings) algorithm and Gibbs sampling algorithm are the two most commonly used MCMC techniques. The MH approach makes the same assumption as acceptance–rejection sampling: a candidate value can be generated from the TTLILo distributions for each iteration of the procedure. Using the MCMC steps and the MH, the following random samples are generated from conditional posterior densities:

$$\mathrm{\Pi }\left(b\right|\Delta )\propto b^{n+a_1-1}{e}^{-b_{1}b}\prod _{i=1}^n{\left\{1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right\}}^{b-1}\left\{1+c-2c{\left\{1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right\}}^b\right\},$$

(16)

$$\mathrm{\Pi }\left(c\right|\Delta )\propto \prod _{i=1}^n\left\{1+c-2c{\left\{1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right\}}^b\right\},$$

(17)

$$\begin{array}{c}\mathrm{\Pi }\left(\rho \right|\Delta )\propto {\rho }^{n+a_3-1}{e}^{-b_3\rho }{\displaystyle {\prod }_{i=1}^n}{\left(1+\left(\frac{\rho }{z_i}\right)\right)}^{-\omega -1}\mathrm{{\rm Y}}(z_i,\rho ,\omega ){\left\{1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right\}}^{b-1}\hfill \\ \times {\displaystyle {\prod }_{i=1}^n}\left\{1+c-2c{\left\{1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right\}}^b\right\},\hfill \end{array}$$

(18)

$$\begin{array}{c}\mathrm{\Pi }\left(\omega \right|\Delta )\propto {\omega }^{n+a_4-1}{e}^{-b_4\omega }{\displaystyle {\prod }_{i=1}^n}{\left(1+\left(\frac{\rho }{z_i}\right)\right)}^{-\omega -1}\mathrm{{\rm Y}}(z_i,\rho ,\omega ){\left\{1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right\}}^{b-1}\hfill \\ \times {\displaystyle {\prod }_{i=1}^n}\left\{1+c-2c{\left\{1-{\mathrm{{\rm Y}}}^2(z_i,\rho ,\omega )\right\}}^b\right\}.\hfill \end{array}$$

(19)

Because the marginal posterior densities in (16)–(19) have unknown distributions, we will utilize the MH sampler to create values for b, c, $\rho$, and $\omega$ together with samples from the normal proposal distribution.

The 95% two-sided highest density region credible interval for the unknown parameters or any function of them is given as $\left[{\Delta }_{0.025N:N},{\Delta }_{0.975N:N}\right]$ by using the method proposed by Chen and Shao [29].

5.3. Bootstrap Confidence Intervals

A well-known inferential technique from the early days of computers is the bootstrap [30,31]. The bootstrap is predicated on the notion that, in some cases, using only the available data can produce better results than relying on unfounded assumptions about the populations we are attempting to estimate. The fundamental working principle of the bootstrap is data-driven simulation, known as sampling with replacement. Thus, learning about the bootstrap is an opportunity to not only learn about an alternative to the traditional parametric approaches of statistical inference but also about simulations and how we choose our methods. The bootstrap method has been used in the study of fatigue life and the reliability of materials [32,33]. Bradley Efron’s key study [30], which originally described the bootstrap approach discussed above as the percentile bootstrap, was published in 1979. The bootstrap-t technique, also known as percentile-t bootstrap, uses the bootstrap to calculate a data-driven T distribution under the null hypothesis. Since 1979, numerous additional bootstrap techniques have been created, but we will only discuss one other variant in a later section. In this subsection, two bootstrap approaches have been derived as the percentile bootstrap and bootstrap-t for TTLILo distribution parameters as follows:

i. 

Algorithm Percentile BCIs

For the PBCIs (percentile BCIs) approach, the following steps were executed:

1.

Compute the estimators of $\Delta \equiv (b,c,\rho ,\omega )$ for the TTILo distribution.

2.

Generate a bootstrap sample using b, c, $\rho$, and $\omega$ to obtain the bootstrap estimate of b say ${\widehat{b}}^B$, c say ${\widehat{c}}^B$, $\rho$ say ${\widehat{\rho }}^B$, and $\omega$ say ${\widehat{\omega }}^B$ using the bootstrap sample.

3.

Repeat step 2 N times to obtain $\left\{b^{B\left(1\right)},b^{B\left(2\right)},\dots ,b^{B\left(N\right)}\right\}$, $\left\{c^{B\left(1\right)},c^{B\left(2\right)},\dots ,c^{B\left(N\right)}\right\}$, $\left\{{\rho }^{B\left(1\right)},{\rho }^{B\left(2\right)},\dots ,{\rho }^{B\left(N\right)}\right\}$, and $\left\{{\omega }^{B\left(1\right)},{\omega }^{B\left(2\right)},\dots ,{\omega }^{B\left(N\right)}\right\}$.

4.

Arrange $\left\{b^{B\left(1\right)},b^{B\left(2\right)},\dots ,b^{B\left(N\right)}\right\}$, $\left\{c^{B\left(1\right)},c^{B\left(2\right)},\dots ,c^{B\left(N\right)}\right\}$, $\left\{{\rho }^{B\left(1\right)},{\rho }^{B\left(2\right)},\dots ,{\rho }^{B\left(N\right)}\right\}$, and $\left\{{\omega }^{B\left(1\right)},{\omega }^{B\left(2\right)},\dots ,{\omega }^{B\left(N\right)}\right\}$ in ascending order as $\left\{b^{B\left[1\right]},b^{B\left[2\right]},\dots ,b^{B\left[N\right]}\right\}$, $\left\{c^{B\left[1\right]},c^{B\left[2\right]},\dots ,c^{B\left[N\right]}\right\}$, $\left\{{\rho }^{B\left[1\right]},{\rho }^{B\left[2\right]},\dots ,{\rho }^{B\left[N\right]}\right\}$, and $\left\{{\omega }^{B\left[1\right]},{\omega }^{B\left[2\right]},\dots ,{\omega }^{B\left[N\right]}\right\}$.

5.

Set a two-sided $100\left(1-\gamma \right)\%$ PBCI for the unknown parameters b, c, $\rho ,$ and $\omega$ by $\left\{{\widehat{b}}^{B\left[N\gamma /2\right]},{\widehat{b}}^{B\left[N\left(1-\gamma /2\right)\right]}\right\}$, $\left\{{\widehat{c}}^{B\left[N\gamma /2\right]},{\widehat{c}}^{B\left[N\left(1-\gamma /2\right)\right]}\right\}$, $\left\{{\widehat{\rho }}^{B\left[N\gamma /2\right]},{\widehat{\rho }}^{B\left[N\left(1-\gamma /2\right)\right]}\right\}$, and $\left\{{\widehat{\omega }}^{B\left[N\gamma /2\right]},{\widehat{\omega }}^{B\left[N\left(1-\gamma /2\right)\right]}\right\}$.

ii. 

Algorithm Bootstrap-t CIs

For the BTCIs (bootstrap-t CIs) approach, we follow the outlined steps below:

1.

Follow the same steps as (1–2) in the previous algorithm.

2.

Compute the t-statistic of $\Delta \equiv (b,c,\rho ,\omega )$ as $T_j=\left({\widehat{\Delta }}_j^B-{\widehat{\Delta }}_j\right)/\sqrt{V\left({\widehat{\Delta }}_j^B\right)}$ where j = 1, 2, 3, 4, and $V\left({\widehat{\Delta }}_j^B\right)$ is the asymptotic variance of ${\widehat{\Delta }}_j^B$ and it can be obtained using the Fisher information matrix.

3.

Repeat steps 2–3 B times and obtain $T_j^{\left(1\right)},T_j^{\left(2\right)},\dots ,T_j^{\left(N\right)}$.

4.

Arrange $T_j^{\left(1\right)},T_j^{\left(2\right)},\dots ,T_j^{\left(N\right)}$ in ascending order as $T_j^{\left[1\right]},T_j^{\left[2\right]},\dots ,T_j^{\left[N\right]}$.

5.

A two-sided $100\left(1-\gamma \right)\%$ BTCI for the unknown parameters b, c, $\rho$, and $\omega$ is given by

$$\left\{\widehat{b}+T_1^{[N\gamma /2]}\sqrt{V\left(\widehat{b}\right)},\widehat{b}+T_1^{\left[N\left(1-\gamma /2\right)\right]}\sqrt{V\left(\widehat{b}\right)}\right\},\left\{\widehat{c}+T_2^{[N\gamma /2]}\sqrt{V\left(\widehat{c}\right)},\widehat{c}+T_2^{\left[N\left(1-\gamma /2\right)\right]}\sqrt{V\left(\widehat{c}\right)}\right\},$$

$$\left\{\widehat{\rho }+T_3^{[N\gamma /2]}\sqrt{V\left(\rho \right)},\widehat{\rho }+T_3^{\left[N\left(1-\gamma /2\right)\right]}\sqrt{V\left(\widehat{\rho }\right)}\right\},\left\{\widehat{\omega }+T_4^{[N\gamma /2]}\sqrt{V\left(\omega \right)},\widehat{\omega }+T_4^{\left[N\left(1-\gamma /2\right)\right]}\sqrt{V\left(\widehat{\omega }\right)}\right\}.$$

6. Simulation Study

The proposed MLEs and BEs for the TTLILo distribution were compared in this section using simulated data. A total of 5000 simulated samples were used to study the comparison between the MLEs and BEs. We obtained samples from the TTLILo distribution with n = 50, 100, and 200. The primary purpose of a simulation is to comprehend the behavior of a model for actual implementation, which necessitates selecting the appropriate parameter values. Using the design of experiments to vary the TTLILo model’s parameters over tenable ranges and then creating a response surface model to evaluate the effects of those changes is a reasonable strategy. Therefore, to examine the behavior of the model under various parameter effects $(b>0,-1<c<1,\rho >0,\omega >0)$, we chose different values of the actual values (growing and decreasing) as follows:

Case 1: b = 0.4, c = −0.6, $\rho$ = 2.5, $\omega$ = 1.8 & b = 0.4, c = 0.6, $\rho$ = 2.5, $\omega$ = 1.8 & b = 0.9, c = −0.6, $\rho$ = 2.5, $\omega$ = 1.8 & b = 0.9, c = 0.6, $\rho$ = 2.5, $\omega$ = 1.8. (See Figure 3).

Case 2: b = 1.6, c = −0.5, $\rho$ = 1.2, $\omega$ = 0.8 & b = 1.6, c = 0.5, $\rho$ = 1.2, $\omega$ = 0.8 & b = 1.6, c = −0.5, $\rho$ = 1.2, $\omega$ = 1.3 & b = 1.6, c = 0.5, $\rho$ = 1.2, $\omega$ = 1.3. (See Figure 4).

Case 3: b = 1.5, c = −0.3, $\rho$ = 0.2, $\omega$ = 0.8 & b = 1.5, c = 0.2, $\rho$ = 0.2, $\omega$ = 0.8 & b = 1.5, c = −0.3, $\rho$ = 0.7, $\omega$ = 0.8 & b = 1.5, c = 0.2, $\rho$ = 0.7, $\omega$ = 0.8. (See Figure 5).

For the TTLILo distribution, the MLEs and BEs are calculated. To obtain posterior samples from (16)–(19) using the ‘coda’ package, we generated 12,000 MCMC samples, discarding the initial 2000 samples as burn-in. Subsequently, with the remaining 10,000 MCMC samples, we calculated the Bayes point and interval estimates for the TTLILo parameters as proposed.

Table A1. Parameter estimates and different CIs of the TTLILo model for Case 1.

$\mathit{b}=1.5,\mathit{\omega }=0.8$				MLE		Bayesian		CI				Bootstraping
$\mathit{\rho }$	$\mathit{c}$	n		Bias	MSE	Bias	MSE	LACI	CP	LCCI	CP	LBP.MLE	LBT.MLE	LBP.Bayes	LBT.Bayes
0.2	−0.3	50	b	0.0987	0.0180	0.1347	0.0142	0.3568	94.0%	0.3456	95.5%	0.0244	0.0249	0.1037	0.1025
			c	0.0911	0.0479	0.0294	0.0411	0.7820	94.6%	0.6818	94.4%	0.0526	0.0552	0.0958	0.0957
			$\rho$	−0.0587	0.0065	−0.0374	0.0058	0.2185	93.5%	0.2098	94.5%	0.0542	0.0554	0.0260	0.0268
			$\omega$	0.1097	0.0389	0.1140	0.0310	0.6443	93.5%	0.6010	96.0%	0.0473	0.0471	0.0776	0.0764
		100	b	0.0896	0.0111	0.0781	0.0106	0.2164	94.5%	0.1932	96.3%	0.0151	0.0149	0.0665	0.0665
			c	0.0813	0.0363	0.0251	0.0312	0.7189	95.0%	0.6176	94.5%	0.0469	0.0419	0.0810	0.0808
			$\rho$	−0.0407	0.0064	−0.0226	0.0056	0.2099	94.5%	0.2018	95.5%	0.0516	0.0511	0.0167	0.0165
			$\omega$	0.0873	0.0247	0.0821	0.0240	0.5136	96.5%	0.5074	98.0%	0.0358	0.0356	0.0535	0.0543
		200	b	0.0784	0.0093	0.0729	0.0084	0.2021	98.5%	0.1917	96.5%	0.0146	0.0137	0.0539	0.0539
			c	0.0804	0.0302	0.0145	0.0277	0.5468	95.3%	0.5100	95.5%	0.0378	0.0377	0.0747	0.0749
			$\rho$	−0.0360	0.0044	−0.0214	0.0041	0.1115	95.0%	0.1048	96.5%	0.0083	0.0083	0.0109	0.0108
			$\omega$	0.0798	0.0131	0.0593	0.0102	0.3214	96.8%	0.3052	99.5%	0.0232	0.0233	0.0352	0.0354
	0.2	50	b	0.1040	0.0830	0.0874	0.0727	0.5438	95.5%	0.5060	95.5%	0.0379	0.0385	0.1404	0.1395
			c	−0.0487	0.1529	0.0421	0.1216	5.3828	96.2%	0.8769	94.5%	0.3498	0.5153	0.0673	0.0678
			$\rho$	−0.0578	0.0041	0.0163	0.0040	1.2462	96.5%	0.2798	95.5%	0.0860	0.1147	0.0204	0.0206
			$\omega$	0.1082	0.0432	0.0894	0.0391	0.6969	94.3%	0.6135	94.5%	0.0504	0.0504	0.0888	0.0886
		100	b	0.0906	0.0137	0.0819	0.0117	0.2919	95.8%	0.2498	96.3%	0.0204	0.0210	0.0667	0.0657
			c	−0.0318	0.1465	−0.0229	0.1054	1.3318	96.5%	0.8691	94.9%	0.0944	0.1096	0.0636	0.0638
			$\rho$	−0.0418	0.0038	0.0135	0.0031	0.1788	97.5%	0.1625	95.8%	0.0122	0.0125	0.0192	0.0190
			$\omega$	0.1035	0.0267	0.0809	0.0246	0.4969	94.5%	0.4720	97.5%	0.0356	0.0358	0.0614	0.0613
		200	b	0.0901	0.0137	0.0781	0.0103	0.2557	95.9%	0.2476	96.6%	0.0185	0.0181	0.0532	0.0529
			c	−0.0212	0.0655	−0.0080	0.0494	0.5662	96.9%	0.5175	95.5%	0.0396	0.0401	0.0627	0.0619
			$\rho$	−0.0406	0.0028	0.0113	0.0024	0.1064	97.9%	0.0998	96.9%	0.0079	0.0079	0.0174	0.0174
			$\omega$	0.0981	0.0181	0.0781	0.0132	0.3616	96.0%	0.3529	99.0%	0.0262	0.0263	0.0380	0.0379
0.7	−0.3	50	b	0.2234	0.0861	0.1473	0.0724	0.7480	93.5%	0.7111	94.5%	0.1316	0.1330	0.0539	0.0533
			c	0.0909	0.2026	0.0891	0.1310	1.7325	95.0%	1.2359	94.5%	0.1193	0.1342	0.0951	0.0951
			$\rho$	−0.1550	0.0943	−0.0116	0.0610	1.0418	97.5%	0.9226	95.0%	0.0756	0.0783	0.0737	0.0737
			$\omega$	0.0467	0.0388	0.0313	0.0374	0.7520	94.0%	0.7107	94.5%	0.0866	0.0881	0.0520	0.0518
		100	b	0.1737	0.0417	0.1061	0.0406	0.4213	93.9%	0.3992	95.5%	0.0678	0.0683	0.0295	0.0304
			c	0.0815	0.0692	0.0450	0.0610	0.8503	96.5%	0.8118	94.9%	0.0860	0.0864	0.0609	0.0612
			$\rho$	−0.1484	0.0533	−0.0097	0.0301	0.5489	97.8%	0.5373	98.0%	0.0403	0.0413	0.0398	0.0399
			$\omega$	0.0453	0.0171	0.0304	0.0144	0.4690	95.0%	0.4567	98.0%	0.0518	0.0566	0.0343	0.0350
		200	b	0.1709	0.0406	0.1003	0.0395	0.4015	95.5%	0.3972	95.7%	0.0512	0.0513	0.0279	0.0282
			c	0.0816	0.0555	0.0426	0.0508	0.6682	98.5%	0.6605	96.5%	0.0779	0.0786	0.0459	0.0454
			$\rho$	−0.1375	0.0423	−0.0081	0.0256	0.4238	98.5%	0.4174	98.5%	0.0317	0.0319	0.0294	0.0292
			$\omega$	0.0371	0.0099	0.0301	0.0090	0.3623	97.5%	0.3467	99.5%	0.0332	0.0327	0.0252	0.0250
	0.2	50	b	0.1274	0.1538	0.1181	0.1490	1.4575	92.0%	1.4219	94.5%	0.1121	0.1117	0.1043	0.1041
			c	0.7316	5.1970	-0.0191	0.0534	7.0476	95.5%	0.8373	94.5%	0.8035	0.8260	0.0664	0.0666
			$\rho$	0.1820	1.1833	0.0576	0.0864	4.2146	92.5%	1.0963	94.5%	0.2836	0.3248	0.0799	0.0793
			$\omega$	0.1665	0.1865	0.1833	0.1077	1.5658	95.5%	1.0288	96.0%	0.1089	0.1095	0.0730	0.0729
		100	b	0.1141	0.0583	0.0979	0.0568	0.7709	95.5%	0.7093	95.7%	0.0565	0.0568	0.0603	0.0607
			c	0.0607	1.0970	0.0030	0.0500	2.9508	95.7%	0.8093	94.9%	0.5583	0.4704	0.0634	0.0644
			$\rho$	−0.0494	0.1924	0.0186	0.0356	1.7128	97.5%	0.6662	97.5%	0.1197	0.1376	0.0522	0.0535
			$\omega$	0.0745	0.0325	0.1395	0.0304	0.6451	97.0%	0.5851	97.5%	0.0440	0.0447	0.0408	0.0404
		200	b	0.0914	0.0338	0.0983	0.0305	0.4517	96.0%	0.4174	96.2%	0.0313	0.0313	0.0539	0.0546
			c	−0.0508	0.1535	0.0029	0.0453	1.1830	96.0%	0.7982	95.1%	0.0854	0.0897	0.0567	0.0561
			$\rho$	−0.0461	0.0939	−0.0091	0.0218	1.0921	97.7%	0.5316	98.0%	0.0747	0.0844	0.0398	0.0400
			$\omega$	0.0513	0.0093	0.1187	0.0030	0.3199	97.6%	0.3059	98.5%	0.0228	0.0225	0.0386	0.0390

,

Table A2. Parameter estimates and different CIs of the TTLILo model for Case 2.

$\mathit{b}=1.6,\mathit{\rho }=1.2$				MLE		Bayesian		CI				Bootstraping
$\mathit{\omega }$	$\mathit{c}$	n		Bias	MSE	Bias	MSE	LACI	CP	LCCI	CP	LBP.MLE	LBP.MLE	LBP.Bayes	LBP.Bayes
0.8	−0.5	50	b	0.2785	0.1057	0.0733	0.0807	0.6597	95.0%	0.6343	95.5%	0.1143	0.1159	0.0455	0.0463
			c	0.2140	0.8928	0.2016	0.1466	3.6167	94.0%	1.1627	94.5%	0.2511	0.3356	0.0862	0.0855
			$\rho$	−0.2215	0.1791	−0.0437	0.0933	1.4172	94.7%	1.1076	95.5%	0.0999	0.0970	0.0889	0.0864
			$\omega$	−0.0135	0.0267	0.0127	0.0196	0.6402	95.0%	0.6178	94.5%	0.0665	0.0666	0.0458	0.0456
		100	b	0.2648	0.0894	0.0557	0.0599	0.5462	95.5%	0.5089	95.8%	0.0649	0.0667	0.0384	0.0388
			c	0.2034	0.1626	0.1921	0.1360	0.8246	96.0%	0.8091	94.9%	0.0849	0.0847	0.0609	0.0601
			$\rho$	−0.2157	0.1434	−0.0381	0.0447	1.0911	95.0%	0.7314	95.6%	0.0788	0.0786	0.0545	0.0547
			$\omega$	0.0058	0.0140	0.0042	0.0125	0.4648	95.5%	0.4574	95.0%	0.0435	0.0435	0.0334	0.0328
		200	b	0.1604	0.0856	0.0381	0.0431	0.4918	96.0%	0.4745	99.5%	0.0523	0.0525	0.0331	0.0334
			c	0.1832	0.1352	0.1621	0.1259	0.7091	98.5%	0.6913	95.5%	0.0792	0.0790	0.0491	0.0492
			$\rho$	−0.1831	0.1353	−0.0213	0.0400	0.7600	96.0%	0.5897	99.0%	0.0554	0.0554	0.0444	0.0444
			$\omega$	0.0041	0.0066	−0.0038	0.0051	0.3170	95.9%	0.3045	95.9%	0.0321	0.0321	0.0225	0.0221
	0.5	50	b	−0.0261	0.3511	0.1189	0.2015	2.3262	88.0%	1.7245	94.5%	0.1599	0.1635	0.1202	0.1213
			c	0.3539	2.9478	−0.2588	0.1927	6.0802	94.0%	1.1817	94.5%	0.5993	0.5021	0.0996	0.0997
			$\rho$	0.6329	2.6104	0.0967	0.1214	5.8418	91.5%	1.1861	94.8%	0.4255	0.4246	0.0875	0.0869
			$\omega$	0.3925	0.6845	0.1867	0.1121	2.8622	91.5%	0.9358	97.0%	0.2001	0.2049	0.0761	0.0789
		100	b	0.0234	0.2686	0.0814	0.0741	2.0322	89.5%	1.0189	95.5%	0.1496	0.1493	0.0725	0.0718
			c	0.0911	1.2316	−0.1835	0.1331	5.8596	96.5%	1.1397	94.6%	0.3931	0.4560	0.0849	0.0851
			$\rho$	0.4104	1.5044	0.0912	0.0691	4.5422	92.0%	0.8662	95.8%	0.3243	0.3327	0.0622	0.0616
			$\omega$	0.2200	0.2939	0.1356	0.0438	1.9471	92.0%	0.5704	97.5%	0.1345	0.1340	0.0461	0.0469
		200	b	0.0225	0.1988	0.0520	0.0480	1.7411	92.0%	0.8353	95.8%	0.1236	0.1224	0.0600	0.0601
			c	−0.0707	1.2063	−0.1723	0.1240	4.3073	97.5%	1.0516	95.5%	0.3071	0.3297	0.0804	0.0805
			$\rho$	0.3335	1.0860	0.0718	0.0466	3.8800	95.0%	0.7355	99.5%	0.2778	0.2920	0.0593	0.0584
			$\omega$	0.1370	0.1788	0.1124	0.0322	1.5720	95.0%	0.4989	98.0%	0.1116	0.1141	0.0393	0.0396
1.3	−0.5	50	b	0.2524	0.1641	0.1804	0.1521	1.2449	95.0%	1.1570	94.8%	0.0884	0.0897	0.1143	0.1137
			c	−0.3919	2.3380	0.1912	0.1628	6.6250	95.7%	1.2487	94.5%	0.5645	0.5381	0.1110	0.1097
			$\rho$	−0.2872	0.3443	−0.1272	0.0822	2.0107	94.6%	1.0862	95.5%	0.1467	0.1458	0.0829	0.0827
			$\omega$	0.0929	0.1613	0.0331	0.0789	1.5356	92.0%	1.0154	94.5%	0.1105	0.1097	0.0770	0.0764
		100	b	0.2483	0.1176	0.0821	0.0536	0.9294	96.0%	0.8459	95.9%	0.0652	0.0683	0.0596	0.0594
			c	0.1503	1.7181	0.1479	0.1563	5.1474	96.5%	1.2020	94.6%	0.3521	0.4778	0.1003	0.1003
			$\rho$	−0.2359	0.2577	−0.1198	0.0462	1.4106	95.9%	0.6865	97.9%	0.1006	0.1074	0.0486	0.0483
			$\omega$	0.0911	0.0749	0.0240	0.0455	0.9781	93.5%	0.8103	97.9%	0.0717	0.0723	0.0597	0.0598
		200	b	0.2393	0.1110	0.0711	0.0444	0.6195	96.5%	0.6079	96.5%	0.0456	0.0474	0.0527	0.0524
			c	0.1472	0.1915	0.1485	0.1420	0.6256	97.0%	0.6110	96.5%	0.0451	0.0451	0.0901	0.0881
			$\rho$	−0.2240	0.1903	−0.0915	0.0445	0.6672	96.3%	0.5760	98.9%	0.0457	0.0455	0.0398	0.0399
			$\omega$	0.0813	0.0442	−0.0081	0.0263	0.6525	96.5%	0.6021	98.5%	0.0474	0.0472	0.0449	0.0439
	0.5	50	b	0.1251	0.3617	0.1713	0.1775	2.3613	92.5%	1.4204	93.9%	0.1648	0.1612	0.1088	0.1083
			c	0.5983	5.1007	−0.2273	0.1846	7.9351	94.5%	1.1812	94.5%	0.7939	0.8247	0.1004	0.1011
			$\rho$	0.5473	2.9872	0.1371	0.1294	6.4427	92.5%	1.2650	98.0%	0.4600	0.4776	0.0922	0.0910
			$\omega$	0.5010	0.8039	0.2085	0.1681	2.9219	93.0%	1.2700	94.2%	0.2204	0.2224	0.0980	0.0989
		100	b	−0.0446	0.2835	0.0945	0.0458	2.0850	93.5%	0.7416	94.5%	0.1466	0.1492	0.0561	0.0571
			c	0.1286	3.6066	−0.1031	0.1721	7.4623	96.5%	1.1750	94.7%	0.5298	0.5228	0.0935	0.0930
			$\rho$	0.5306	2.3538	0.0791	0.0543	5.6571	93.5%	0.8181	99.5%	0.4259	0.4255	0.0580	0.0580
			$\omega$	0.3782	0.5966	0.1372	0.0584	2.6468	93.5%	0.7643	94.5%	0.1903	0.1910	0.0502	0.0497
		200	b	0.0381	0.1578	0.0861	0.0325	1.5540	94.2%	0.6116	94.8%	0.1110	0.1122	0.0420	0.0419
			c	−0.1006	0.4073	−0.0925	0.1565	2.3810	97.5%	1.0812	95.5%	0.1619	0.1602	0.0836	0.0840
			$\rho$	0.1848	0.9123	0.0634	0.0452	3.6827	94.0%	0.7640	99.6%	0.2573	0.2591	0.0572	0.0582
			$\omega$	0.2342	0.1536	0.1270	0.0346	1.2350	95.5%	0.5240	95.9%	0.0888	0.0896	0.0372	0.0368

and

Table A3. Parameter estimates and different CIs of the TTLILo model for Case 3.

$\mathit{\rho }=2.5,\mathit{\omega }=1.8$				MLE		Bayesian		CI				Bootstraping
$\mathit{b}$	$\mathit{c}$	n		Bias	MSE	Bias	MSE	LACI	CP	LCCI	CP	LBP.MLE	LBP.MLE	LBP.Bayes	LBP.Bayes
0.4	−0.6	50	b	0.1596	0.0552	0.0534	0.0399	0.8929	94.7%	0.7023	94.5%	0.0645	0.0634	0.0543	0.0553
			c	−0.7587	4.7153	0.3274	0.2938	8.6923	95.5%	1.1968	94.3%	0.9750	0.9979	0.0987	0.0986
			$\rho$	−0.2257	0.3981	−0.0296	0.0591	2.3154	95.7%	0.9447	94.3%	0.1544	0.1527	0.0664	0.0662
			$\omega$	−0.2204	0.2213	−0.0933	0.1645	1.6331	95.5%	1.6324	93.6%	0.1157	0.1156	0.1186	0.1173
		100	b	0.1483	0.0348	0.0272	0.0119	0.6554	95.5%	0.3919	97.1%	0.0457	0.0459	0.0294	0.0298
			c	−0.4509	3.4587	0.3041	0.2316	7.3058	96.0%	1.1679	94.5%	0.4991	0.5519	0.0977	0.0984
			$\rho$	−0.2135	0.2755	−0.0187	0.0323	1.5268	96.5%	0.6005	94.5%	0.1156	0.1142	0.0427	0.0420
			$\omega$	−0.2070	0.1356	−0.0746	0.0664	0.9842	97.5%	0.9768	94.5%	0.0671	0.0667	0.0684	0.0682
		200	b	0.1411	0.0335	0.0234	0.0116	0.6468	96.2%	0.3888	97.5%	0.0438	0.0437	0.0278	0.0278
			c	0.2555	1.1127	0.2658	0.2258	4.0219	96.4%	1.0907	95.5%	0.2880	0.3111	0.0908	0.0905
			$\rho$	−0.2099	0.1273	−0.0113	0.0313	0.9972	96.6%	0.5724	94.8%	0.0729	0.0732	0.0394	0.0396
			$\omega$	−0.1833	0.1263	−0.0510	0.0506	0.9047	97.7%	0.7390	94.8%	0.0640	0.0632	0.0566	0.0563
	0.6	50	b	0.0658	0.0259	0.0650	0.0236	0.5766	93.5%	0.5387	94.6%	0.0408	0.0408	0.0450	0.0449
			c	−0.3979	2.1328	−0.3706	0.2493	5.7371	95.0%	1.1482	94.5%	0.3900	0.4161	0.0961	0.0957
			$\rho$	0.3901	1.7463	0.1207	0.0703	4.9618	94.0%	0.9433	94.5%	0.3700	0.3715	0.0669	0.0659
			$\omega$	0.2069	0.6551	0.1561	0.2030	3.0751	94.5%	1.5362	94.3%	0.2110	0.2109	0.1150	0.1157
		100	b	0.0582	0.0209	0.0482	0.0202	0.4686	94.0%	0.4488	96.0%	0.0315	0.0319	0.0342	0.0342
			c	−0.3192	0.3495	−0.3041	0.2377	1.9554	95.4%	1.0745	94.6%	0.1398	0.1413	0.0897	0.0908
			$\rho$	0.1867	0.6496	0.0604	0.0311	3.0812	94.5%	0.6434	94.7%	0.2182	0.2202	0.0464	0.0463
			$\omega$	0.0815	0.2393	0.0597	0.0665	1.9214	95.0%	0.9357	94.5%	0.1367	0.1360	0.0708	0.0703
		200	b	0.0410	0.0206	0.0378	0.0140	0.4061	94.6%	0.3436	98.5%	0.0289	0.0289	0.0241	0.0241
			c	−0.2857	0.3102	−0.2406	0.2040	1.5790	95.6%	0.9900	95.5%	0.1064	0.1065	0.0732	0.0722
			$\rho$	0.0252	0.2286	0.0239	0.0213	1.8765	95.5%	0.5135	96.1%	0.1264	0.1289	0.0385	0.0387
			$\omega$	−0.0786	0.1024	0.0279	0.0448	1.2191	95.5%	0.8127	99.0%	0.0902	0.0901	0.0603	0.0611
0.9	0.2	50	b	0.1503	0.1753	0.1426	0.1292	1.5358	95.1%	1.2499	95.5%	0.1119	0.1116	0.0949	0.0991
			c	0.2925	2.6366	0.0032	0.0480	6.2768	95.0%	0.8870	94.5%	0.4647	0.4918	0.0671	0.0642
			$\rho$	0.1919	2.1393	0.0524	0.0634	5.6982	92.5%	0.9463	96.1%	0.3972	0.3998	0.0661	0.0667
			$\omega$	0.3002	0.8372	0.0694	0.1031	3.3967	92.5%	1.1605	94.9%	0.2489	0.2531	0.0870	0.0888
		100	b	0.1500	0.1444	0.1406	0.0662	1.3720	95.3%	0.7227	98.9%	0.0985	0.0983	0.0576	0.0577
			c	−0.0300	1.1120	−0.0018	0.0465	4.1424	95.5%	0.8220	94.7%	0.2943	0.3082	0.0597	0.0598
			$\rho$	0.1012	1.1802	0.0108	0.0306	4.2506	94.5%	0.7183	96.7%	0.2953	0.2950	0.0466	0.0470
			$\omega$	0.1769	0.3890	0.0527	0.0764	2.3504	93.5%	0.9299	95.5%	0.1685	0.1675	0.0728	0.0730
		200	b	0.1422	0.1141	0.1363	0.0433	1.0162	95.5%	0.5090	99.0%	0.0749	0.0751	0.0381	0.0385
			c	−0.0267	0.3299	−0.0010	0.0387	2.2017	96.5%	0.7541	94.9%	0.1601	0.1655	0.0533	0.0531
			$\rho$	−0.0914	0.6118	−0.0098	0.0223	3.0202	96.2%	0.5550	96.9%	0.2140	0.2153	0.0422	0.0421
			$\omega$	−0.0515	0.1718	0.0485	0.0364	1.6277	93.9%	0.7128	96.3%	0.1126	0.1147	0.0507	0.0509
	0.6	50	b	0.1124	0.1803	0.1023	0.1504	1.6092	96.5%	1.1377	97.5%	0.1174	0.1169	0.0852	0.0852
			c	0.1933	2.9809	−0.1907	0.2650	6.7845	95.2%	1.1380	94.4%	0.5019	0.5148	0.0938	0.0962
			$\rho$	0.3246	2.9453	0.0816	0.0768	6.6226	93.5%	1.0038	94.5%	0.4767	0.4759	0.0713	0.0714
			$\omega$	0.4664	1.2229	0.1084	0.1337	3.9404	93.5%	1.3896	98.3%	0.2839	0.2847	0.0967	0.0980
		100	b	0.1026	0.1051	0.0911	0.0491	1.1742	96.7%	0.7275	99.0%	0.0849	0.0840	0.0520	0.0515
			c	−0.1876	0.3729	−0.1743	0.2580	1.8901	95.4%	1.1119	94.5%	0.1377	0.1391	0.0807	0.0809
			$\rho$	0.0544	0.8320	0.0536	0.0280	3.5783	93.7%	0.6422	94.6%	0.2536	0.2539	0.0432	0.0430
			$\omega$	0.1637	0.2831	0.0987	0.0539	1.9894	93.7%	0.8245	98.5%	0.1441	0.1445	0.0582	0.0581
		200	b	0.0916	0.0912	0.0851	0.0426	1.1237	97.5%	0.5620	99.5%	0.0838	0.0828	0.0445	0.0453
			c	−0.1721	0.3255	−0.1706	0.2112	1.0657	99.5%	0.9882	95.0%	0.1259	0.1243	0.0714	0.0713
			$\rho$	0.0507	0.7463	0.0315	0.0220	2.7135	97.5%	0.5842	94.7%	0.2401	0.2336	0.0397	0.0399
			$\omega$	0.1384	0.2556	0.0726	0.0446	1.8537	97.5%	0.7275	99.5%	0.1369	0.1309	0.0526	0.0533

provide the details of the bias of all the estimates and associated MSEs (mean squared errors). In the CIs, the ACI, CCI (credible CI), and bootstrap approaches were used. The length of the ACI and CCI can be denoted as LACI and LCCI, respectively. The bootstrap methods were executed by the length of the simulation results as the LPB (length of percentile bootstrap) and LBT (length of bootstrap-t). Also, the coverage probability (CP) was obtained based on the Bayesian and non-Bayesian methods. Hence, we can denote LBP.MLE and LBT.MLE for the MLE, and LBP.Bayes and LBT.Bayes for the BE.

We performed numerical assessments for the maximum likelihood, Bayesian, and bootstrap estimators using the ‘maxLik’, ‘coda’, and ‘boot’ packages, respectively, within the R 4.3. software environment.

Visually, the simulated values for the TTLILo parameters, along with their MSE, are presented in heat map plots for cases 2 and 3, as depicted in Figure 6, Figure 7 and Figure 8, respectively.

From Table A1, Table A2 and Table A3 and Figure 6, Figure 7 and Figure 8, we can infer the following:

• The suggested point or interval estimates for $b,c,\rho$, and $\omega$ have demonstrated strong performance across the provided parameter sets.
• As the value of n grows, all the recommended estimates perform effectively, thus confirming the consistency property of the obtained estimates.
• The Bayes estimating approach performs better than the maximum likelihood method.
• The Bayes estimation approach is the most effective for the TTLILo distribution.
• The BCIs have the shortest CIs.
• The CCIs are better than ACIs in terms of their shortest length.
• In roughly most scenarios, the CP of the CI estimates increases as the sample sizes rise.
• The CP of the estimates at a negative true value of c has larger values compared to the corresponding at a positive true value.
• For all choices of n, the CP is quite near the desired level of significance.

7. Real Data Analysis

The different datasets were used to apply the TTLILo distribution, and the results of this comparison were made with the TTLW (TTL Weibull) [34], IELoP (inverse exponentiated Lomax–Poisson) [35], PL (Poisson–Lomax) [36], PLP (Power Lomax–Poisson) [37], OBWP (odd Burr–Weibull–Poisson) [38], and KW (Kumaraswamy–Weibull) [39] distributions. The AIC (Akaike information criteria) and BIC (Bayesian IC) and the KS (Kolmogorov–Smirnov), AD (Anderson–Darling), and CVM (Cramér–von Mises) tests and their statistics were employed as goodness-of-fit criteria to choose the distribution with the best performance. In the meantime, the best model was chosen based on the distribution that matches the lowest value of these criteria.

Dataset I: This dataset has been discussed by Gupta and Kundu [40], which shows the failure times of the air conditioning system of an airplane. Utilizing these metrics, the optimal distribution is identified by the lowest values of the AIC, BIC, AD, CVM, and KS statistics, in addition to the highest p-value via R software with the ‘AdequacyModel’ package. All unknown parameters are estimated using the maximum likelihood method, and their standard errors (SE) are calculated and presented in

Table 2. Estimates for all models with different goodness-of-fit measures for dataset I.

		b	c	$\mathit{\rho }$	$\mathit{\omega }$	$\mathit{\delta }$	KS	p-Value	CVM	AD	AIC	BIC
TTLILo	estimates	0.1703	−0.0439	20.5164	8.8115		0.1072	0.8807	0.0795	0.4333	311.4148	317.0196
	SE	0.0582	0.5545	0.3608	0.3567
KW	estimates	1.2367	0.6078	1.4584	0.0804		0.1941	0.2083	0.0862	0.4687	313.9641	319.5689
	SE	0.5945	0.0020	0.6861	0.0149
TTLW	estimates	3.4231	0.0955	0.0210	0.4761		0.1473	0.5334	0.0810	0.4358	311.7251	317.3299
	SE	4.6060	0.7433	0.0349	0.2882
IELoP	estimates	86.7457	1.1845	0.4980	2.5505		0.1258	0.7296	0.0905	0.4791	311.5726	317.1774
	SE	21.3210	0.3504	1.8010	2.1276
PL	estimates	0.7387	15.2763	54.7040			0.1570	0.4503	0.1010	0.7367	315.9373	320.1409
	SE	0.0942	2.5666	3.6562
PLP	estimates	0.0104	2.1150	39.1073	230.6120		0.1210	0.7720	0.0904	0.5829	315.5517	321.1565
	SE	0.0047	0.8415	17.1134	51.1562
OBWP	estimates	0.0025	0.6161	1.2651	10.7397	8.0621	0.1354	0.6409	0.0950	0.5201	321.8135	328.8195
	SE	0.0005	0.0765	0.1629	5.3759	4.9321

. Furthermore, Table 2 displays the estimated goodness-of-fit measures for the failure times of the airplane air conditioning dataset. Table 2 presents the findings of the analysis of dataset I. The TTLILo distribution fits dataset I better than the other distributions, according to the Table 2 results, where the KS distance is 0.1072, CVM statistic is 0.0795, AD statistic is 0.4333, AIC is 311.4148, and BIC is 317.0196. It is clear that the TTLILo distribution outperforms the other distributions when applied to the failure times of the airplane air conditioning dataset.

To verify that these data are appropriate for the hazard model, Figure 9 presents the TTT (total time on test) and HF of the TTLILo for dataset I. The comparison between the histogram plot of the data used and the plot of the TTLILo distribution, as shown in Figure 10, can further confirm this, where the p-value of the TTLILo is 0.8807 (more than 0.05 and the highest value of the p-value of the KS test). Additionally, Figure 10 provides a visual representation, showing the histograms, fitted density curves, and comparison of the fitted/empirical CDF functions along with a PP plot for the TTLILo distribution. These graphical presentations in Figure 10 reinforce the findings in Table 2, highlighting that the TTLILo distribution is the most suitable model for fitting the failure times of the airplane air conditioning dataset when compared to all the listed distributions in Table 2.

To verify that the results provided in Table 2 are unique and have the maximum number of points, Figure 11 and Figure 12 display the convex, non-convex, and existence plots of the parameter for the TTLILo for data I. To theoretically prove that the log-likelihood is convex, the Hessian matrix, eigenvalues, and eigenvector should be obtained.

Table 3. Hessian matrix, eigenvalues, and eigenvectors of TTLILo model: dataset I.

	Hessian Matrix					Eigenvectors
	$\mathit{b}$	$\mathit{c}$	$\mathit{\rho }$	$\mathit{\omega }$	Eigenvalues	$\mathit{b}$	$\mathit{c}$	$\mathit{\rho }$	$\mathit{\omega }$
b	1011.2481				1019.3303	0.9960	0.0206	0.0107	−0.0862
c	−87.4628	10.6091			17.8560	−0.0864	0.0037	0.0153	−0.9961
$\rho$	5.5160	−0.6141	12.6414		12.4330	0.0055	0.1822	−0.9831	−0.0149
$\omega$	21.4747	−1.9702	−0.8537	18.1396	3.0188	0.0215	−0.9830	−0.1820	−0.0083

shows the Hessian matrix, eigenvalues, and eigenvector of the log-likelihood of the TTLILo model for dataset I. From the results in Table 3, we can denote that the estimates of the log-likelihood of the TTLILo model are convex because eigenvalues are non-negative. Convex optimization is a concept that relates eigenvalues with convexity. Eigenvalues are essential for assessing a function’s or problem’s convexity in convex optimization. If the second-order partial derivatives of a log-likelihood function make up its Hessian matrix and the matrix is positive semi-definite, the function is said to be convex. All of the Hessian matrix’s eigenvalues must be non-negative for the matrix to be positive semi-definite. According to stringent convexity, the function has all positive eigenvalues; for more information about these measures, see [41].

Dataset II: This dataset contains information about 40 airborne communication transceivers’ active repair times (measured in hours), which was covered by Jorgense [42]. The TTT and HF of the TTLILo model for dataset II are shown in Figure 13 to demonstrate that these data are appropriate for the hazard model. We have employed the maximum likelihood method to estimate all undisclosed parameters and their SEs are computed and showcased in

Table 4. Estimates for all models with different goodness-of-fit measures for dataset II.

		b	c	$\mathit{\rho }$	$\mathit{\omega }$	$\mathit{\delta }$	KS	p-Value	CVM	AD	AIC	BIC
TTLILo	Estimates	0.1600	0.1317	0.0699	167.4198		0.1202	0.6104	0.0543	0.3589	186.6263	193.3818
	SE	0.1535	0.5072	0.0716	3.8966
TTLW	Estimates	58.1011	0.1803	8.0034	0.2664		0.1533	0.3038	0.0606	0.4152	188.0099	194.7654
	SE	48.8839	0.4997	9.2094	0.0450
IELoP	Estimates	0.1669	12.8608	0.0827	1.5100		0.122	0.603	0.0638	0.4102	187.5061	194.2616
	SE	0.1185	7.9279	1.3748	0.4414
PLP	Estimates	0.0822	2.4468	6.6152	1.9044		0.1278	0.5844	0.1024	0.6467	192.1852	198.9407
	SE	0.1923	2.3213	5.2638	4.9810
OBWP	Estimates	0.0026	1.0358	0.9683	9.4829	9.2072	0.1279	0.5297	0.1226	0.8713	202.9075	211.3519
	SE	0.0006	0.2114	0.1557	6.7102	6.7098
KW	Estimates	3.8258	0.6738	13.9079	0.1657		0.1277	0.6014	0.0553	0.3958	187.5262	194.2817
	SE	0.0073	0.0078	1.0124	0.0269

. Additionally, Table 4 illustrates the estimated goodness-of-fit metrics for the dataset concerning the active repair times of airborne communication transceivers. It is evident that the TTLILo distribution surpasses other distribution models when applied to the dataset involving the active repair times of airborne communication transceivers. The TTLILo distribution fits dataset II better than the other distributions, according to the Table 4 results, where the KS distance is 0.1202, CVM statistic is 0.0543, AD statistic is 0.3589, AIC is 186.6263, and BIC is 193.3818.

Figure 14 illustrates the comparison between the plots of the TTLILo distribution and the histogram plot of dataset II under examination. The p-value of the TTLILo is 0.6104, indicating a significant difference (greater than 0.05 and the highest p-value of the KS test). This confirms earlier findings. Furthermore, Figure 14 offers a visual representation, featuring histograms, fitted density curves, and a comparison between the fitted and empirical CDF functions, along with a PP plot specifically for the TTLILo distribution. These graphical presentations within Figure 14 corroborate the findings in Table 4, underscoring that the TTLILo distribution stands as the most appropriate model for fitting the dataset related to the active repair times of airborne communication transceivers in comparison to all the distributions listed in Table 4.

Figure 15 shows the contour plots of the parameters for the TTLILo for dataset II to confirm that the results are unique and have the maximum points. Figure 15 and Figure 16 display the convex, non-convex, and existence plots of the parameter for the TTLILo for data II.

In

Table 5. Hessian matrix, eigenvalues, and eigenvectors of TTLILo model: dataset II.

	Hessian Matrix					Eigenvectors
	$\mathit{b}$	$\mathit{c}$	$\mathit{\rho }$	$\mathit{\omega }$	Eigenvalues	$\mathit{b}$	$\mathit{c}$	$\mathit{\rho }$	$\mathit{\omega }$
b	1675.1444				11,569.5888	0.3759	0.9142	0.1515	−0.0005
c	−118.8257	14.8123			47.7700	−0.0289	0.1750	−0.9841	0.0008
$\rho$	4012.4206	−312.9006	9931.1742		3.7739	0.9262	−0.3656	−0.0922	−0.0002
$\omega$	1.7366	−0.1332	4.2407	0.0677	0.0659	0.0004	0.0003	0.0008	1.0000

, a log-likelihood function of the TTLILo distribution is said to be convex where the Hessian matrix, which consists of the second-order partial derivatives of the function, is positive semi-definite. The positive semi-definiteness of the Hessian matrix is equivalent to all the eigenvalues of the matrix being non-negative. If all eigenvalues are positive, the function is strictly convex.

8. Stress-Strength Application

The concept of the SS reliability model, indicated by $R=P\left[Z_2<Z_1\right],$ represented by Z${}_1$ for the component strength and Z${}_2$ for stress, is demonstrated. In the SS reliability model, the system is a mail function if Z${}_2$ > Z${}_1$. The determination of Z${}_1$∼ TTLILo$(b_1,c_1,{\omega }_1,\rho ),$ and Z${}_2$∼ TTLILo$(b_2,c_2,{\omega }_2,\rho ),$ is considered as follows:

$$R={\int }_0^{\infty }{f}_1(z;b_1,c_1,{\omega }_1,\rho )F_2(z;b_2,c_2,{\omega }_2,\rho )dz.$$

By using the binomial expansions several times, we obtain

$$R=\sum _{m=0}^{2i_1}\sum _{t=0}^{2i_2}{\Delta }_{m,t}{\int }_0^{\infty }\frac{{\omega }_1\rho }{z^2}{\left(1+\left(\rho /\phantom{\rho z}z\right)\right)}^{-({\omega }_{1}m+{\omega }_{2}t+1)-1}dz,$$

$$\begin{array}{cc}\hfill {\Delta }_{m,t}=& \sum _{i_1=i_2=0}^{\infty }{\left(-1\right)}^{i_1+i_2+m+t}(m+1)\left(\begin{array}{c}2i_1\\ m\end{array}\right)\left(\begin{array}{c}2i_2\\ t\end{array}\right)\left[(1+c_1)\left(\begin{array}{c}{b}_1\\ i_1\end{array}\right)-c_1\left(\begin{array}{c}2b_1\\ i_1\end{array}\right)\right]\hfill \\ & \left[(1+c_2)\left(\begin{array}{c}{b}_2\\ i_2\end{array}\right)-c_2\left(\begin{array}{c}2b_2\\ i_2\end{array}\right)\right]\hfill \end{array}$$

Then, the SS reliability model is given by:

$$R=\sum _{m=0}^{2i_1}\sum _{t=0}^{2i_2}\frac{{\Delta }_{m,t}{\omega }_1}{({\omega }_{1}m+{\omega }_{2}t+1)}.$$

This section describes the analysis of two real datasets for SS application purposes. Xia et al. [43] employed two datasets that represent the breaking strengths of jute fiber at two distinct gauge lengths. Also, more papers discussed these data to estimate the SS reliability, such as [44,45].

First, it was determined whether or not these datasets could be analyzed using the TTLILo distribution. In

Table 6. MLE with SE and different values of goodness-of-fit measures of jute fiber at two distinct gauge lengths.

		b	c	$\mathit{\rho }$	$\mathit{\omega }$	KS	p-Value	CVM	AD	AIC	BIC
Z${}_1$	Estimates	7.4166	−0.5586	172.9725	0.6006	0.2147	0.1081	0.0897	0.6820	419.5942	425.1990
	SE	44.1519	0.5949	323.2337	2.8588
Z${}_2$	Estimates	31.8271	−0.2593	342.4166	0.1634	0.1676	0.3307	0.1214	0.8252	418.6250	424.2298
	SE	8.1252	0.6559	14.1515	0.0915

, the MLEs and goodness-of-fit metrics are presented. The p-values for the KS are more than 0.05. The TTLILo distribution as the fitting of the data cannot be ruled out based on the p-values. The notations listed below have been applied: Jute fiber with a diameter of 10 mm has a breaking strength of $Z_1$, while fiber with a diameter of 20 mm has a breaking strength of $Z_2$. The estimate of $R=P(Z_2<Z_1)$ using both methods is listed in

Table 7. Estimates of SS model.

	Maximum Likelihood		Bayesian
	Estimate	SE	Estimate	SE
$b_1$	13.6892	96.5180	5.9063	4.6592
$c_1$	−0.5568	0.6069	−0.0853	0.2912
${\omega }_1$	0.3726	1.8839	0.2542	0.1439
$b_2$	3.6497	17.5410	4.5396	3.6980
$c_2$	−0.2449	0.6666	−0.3828	0.3578
${\omega }_2$	0.7876	3.2043	0.6023	0.5364
$\rho$	207.0288	340.0919	640.5580	252.2402
R	0.5974		0.8003

. The MLE and BE of $R=P(Z_2<Z_1)$ are $\widehat{R}=$ 0.5974124 and $\widehat{R}=$ 0.8003251, respectively, based on the entire dataset.

Table 8. MLE of $P(Z_2<Z_1)$ for different models.

	${\mathit{b}}_1$	${\mathit{c}}_1$	${\mathit{\omega }}_1$	${\mathit{b}}_2$	${\mathit{c}}_2$	${\mathit{\omega }}_2$	$\mathit{\rho }$	$\mathit{P}({\mathit{Z}}_2<\mathit{Z}{}_1)$
TTLILo (New)	13.6892	−0.5568	0.3726	3.6497	−0.2449	0.7876	207.0288	0.5974
Exponential [44]	356.7297	-		340.74	-			0.5177
Exponentiated Gumbel Distribution [45]	3.360852	-		223.3009	-			0.51816
Weibull [46]	389.3	1.71	-	372.17	1.36	-	2.21	0.5384
Exponentiated Inverted Weibull [47]	441.8773	-		315.0805	-		1.1569	0.5838
Alpha Power Exponential [48]	23.44078	-		5.6544	-		0.004932	0.4019

shows the MLE of $P(Z_2<Z_1)$ for the different models, and the TTLILo model has a larger value of $P(Z_2<Z_1)$ for the MLE by comparing it with the exponential SS, exponentiated Gumbel SS, alpha power exponential SS, exponentiated inverted Weibull SS, and Weibull SS.

9. Conclusions

In this study, we investigate a brand new four-parameter model called the TTLILo distribution. By adding shape b and transmuted c parameters, where $b,\omega >0$ are the shape parameters, $\rho >0$ is the scale parameter, and $\left|c\right|\le 1$ is the transmuted parameters, the newly supplied distribution, known as the TTLILo distribution, is recommended. The additional parameters involved in the TTLILo model provide greater flexibility for modeling several types of data. Some statistical and mathematical properties of the TTLILo distribution are implemented, including moments, inverse moments, stochastic ordering, the quantile function, the SS model, and various information measures. Based on two truncated moments, several important characterization findings are provided. The distribution parameters are estimated using both Bayesian and non-Bayesian estimation techniques. Bayesian credible intervals, ACIs, and BCIs are the four different types of CIs that may be constructed. The BCIs have the smallest length. The MCMC technique is employed for a few challenging calculations. Simulation experiments based on various sample sizes have been conducted to evaluate how various estimates behave.

The outcomes of the simulated research show that the BEs are superior to the corresponding MLEs. The smallest CIs are those of the bootstrap method. Regarding the shortest length, Bayesian credible intervals are superior to ACIs. Additionally, we use a real-world data analysis to demonstrate how the suggested model differs from some of its competitors. The results illustrate that the new distribution matches the data more accurately than the prior rival distributions. According to our analysis, it is the best option among all of its rivals because it has the lowest accuracy measure values and the greatest p-values. To ensure that the findings are unique and have maximum points, we also graph the contour plots of the parameters for the TTLILo distribution for the two sets of data. For SS application, we note that the MLE of $P(Z_2<Z_1)$ has a larger value for the TTLILo model compared with the different models, such as the exponential SS, exponentiated Gumbel SS, alpha power exponential SS, exponentiated inverted Weibull SS, and Weibull SS.

We will be considering the reliability of a generalized stress-strength model in a future paper that consists of a serial system with one stress and multiple strengths. For more examples, see [20,49,50,51,52,53].