Estimating the Reliability Function (2+1) Cascade Model for Inverse Chen Distribution

2.1 Inverse Chen distribution

The Inverse Chen distribution is a versatile tool for modeling non-monotonic hazard functions and positively skewed data. It is well-suited for time-to-failure data in industrial operations, biomedical research, and engineering systems. The distribution can handle light-tailed and heavy-tailed phenomena due to its shape and scale parameters. Its strong right-skewness and leptokurtic features make it a strong substitute for traditional models. The Inverse Chen distribution has been used in real-world applications, such as stress-strength reliability under Bayesian frameworks, survival probability in medical research, and simulation of electronic components in geological and environmental research. Its empirical adaptability and theoretical stability make it a valuable addition to lifetime distributions.^22–25

The Inverse Chen distribution is a two-parameter continuous probability distribution designed to model asymmetric lifetime data, especially in reliability analysis. Inverse Chen distribution with two parameters shape parameter and scale parameter.^22,23,25 The Probability Density Function (PDF) in Eq. (1) and Eq. (3) for $\mathrm{\chi }$ and $\mathrm{\gamma }$ , respectively. And the Cumulative Distribution Function (CDF) in Eq. (2) and Eq. (4) for $\mathrm{\chi }$ and $\mathrm{\gamma }$ , respectively. The Inverse Chen distribution, which is a different version of the original Chen distribution introduced by Chen (2000), is used to describe lifetime data with hazard rates that go up over time. It can handle a wide range of danger shapes and is statistically defined on the positive real line. The distribution is good for modelling data that isn’t symmetric, like failure times, stress-strength reliability, and other types of data. It has been utilized in reliability engineering, biomedical research, stress-strength modelling, and geological and environmental studies. It can handle skewed and heavy-tailed data in a lot of different ways.^22,23,25

2.2 The mathematical formula reliability

The Cascade Reliability model is a particular form of the Strength-Stress model that pertains to a certain (2+1) cascade arrangement, incorporating components $A$ and $B$ , in operation, while component C serves as a standby element. If any component is active, it indicates a failure; let component C be active, with ${\mathrm{\chi }}_3$ reflecting its strength and ${\mathrm{\gamma }}_3$ denoting the force exerted on it. When ${\mathrm{\chi }}_3=m{\mathrm{\chi }}_1\left(m{\mathrm{\chi }}_2\right)$ and ${\mathrm{\gamma }}_3=k{\mathrm{\gamma }}_1\left(k{\mathrm{\gamma }}_2\right),$ where $0<m<1$ and $k>1$ .¹²

Let ${\mathrm{\chi }}_i$ and ${\mathrm{\gamma }}_j$ be independent and identical random variables that are distributed Inverse Chen with unknown shape parameters ${\mathit{\alpha }}_i$ and ${\beta }_j$ , and a known value λ = 3, such that ( $i=1,2,3$ and $j=1,2,3$ ), let ${\mathrm{\chi }}_i$ indicate the strength with parameters ${\alpha }_i$ , and ${\mathrm{\gamma }}_j$ refer to stress with parameters ${\beta }_j$ , then the probability density functions are $f\left({\mathrm{\chi }}_i\right),$ $g\left({\mathrm{\gamma }}_j\right)$ and cumulative distribution functions are $F\left({\mathrm{\chi }}_i\right),$ $G\left({\mathrm{\gamma }}_j\right)$ respectively⁹;

The (2+1) cascade model’s reliability function is provided by.^13,14

Then, obtain

Upon resolving the equation, obtain:

In compensating Eqs. (8) and (9) within Eq. (12), $R_2$ is getting as follows:

Substitution Eqs. (6), (10) and (11) in Eq. (5) says, $R$

2.3 Model reliability estimation ( $\widehat{\mathsf{R}}$ )

2.3.1 Maximum Likelihood Estimator (MLE)

Let ${\mathrm{\chi }}_i$ for $i=1,2,3$ represent a strength random sample drawn from an IC ( $\alpha$ , λ) distribution with sample size $n$ , where $\alpha$ is an unknown shape parameter, and λ is a known parameter.

By taking the logarithm of the likelihood function $L$ ; $L=\prod f(x_i,i=1,2,\dots ,n;\alpha ,\lambda )$ . Next, derive L by $\alpha$ , and the results equal zero. The maximum likelihood estimator for $\alpha$ ( ${\widehat{\alpha }}_{\mathit{mle}}$ ) is obtained, which serves as the joint probability function, and has the following general form⁹:

Let ${\mathrm{\chi }}_{1i};i=1,2,\dots ,n_1,{\mathrm{\chi }}_{2i};i=1,2,\dots ,n_2$ and $,{\mathrm{\chi }}_{3i};i=1,2,\dots ,n_3$ strength random sample from IC( ${\alpha }_1,\lambda ),$ IC( ${\alpha }_2,\lambda ),$ IC( ${\alpha }_3,\lambda ),$ with sample size $n_1,n_2,\mathrm{\text{and}}{n}_3$ respectively:

For the stress random variable, let ${\mathit{\gamma }}_{1\mathrm{j}};j=1,2,\dots ,m_1,{\mathit{\gamma }}_{2j};j=1,2,\dots ,m_2$ and $,{\mathit{\gamma }}_{3j};j=1,2,\dots ,m_3$ stress random sample from IC( ${\beta }_1,\lambda ),$ IC( ${\beta }_2,\lambda ),$ IC( ${\beta }_3,\lambda ),$ with sample size $m_1,m_2,\mathrm{\text{and}}{m}_3$ respectively; the ML estimator for the unknown parameter ${\beta }_1,{\beta }_2\mathrm{\text{and}}{\beta }_3$ is as

By substituting ${\widehat{\alpha }}_{\mathit{\text{ξmle}}}$ and ${\widehat{\beta }}_{\mathit{\text{ξmle}}}$ in Eq. (12), then the estimation reliability of $R$ via the MLE can be summarized as follows:

2.3.2 Shrinkage Estimation Method

Let a classical shrinking estimator of the parameter α be $\widehat{\alpha }$ into a prior guess α₀ through the reduction weighting factor k( $\widehat{\mathit{\alpha }}$ ) where 0 ≤ k( $\widehat{\mathit{\alpha }}$ ) ≤1. Suppose that ${\alpha }_0$ is too close to the genuine value of α. Accordingly, the procedure of the shrinkage estimator introduced by Thompson⁹:

2.3.2.1 The Shrinking Weight Factor Estimators (sh₁)

The shrinkage weight factor is proposed as a function of sample sizes $n_{\xi }$ and $m_{\xi }$ ; respectively, which is assumed as the following formula.

i.e. $;k_{\xi }\left({\widehat{\alpha }}_{\xi }\right)$ = $\frac{n_{\xi }}{n_{\xi }+100}$ , $k_{\xi }\left({\widehat{\beta }}_{\xi }\right)$ = $\frac{m_{\xi }}{m_{\xi }+100}$ , such that $\xi =1,2,3$ , and this indicates in the resulting shrinkage estimators:

By substituting Eqs. (18) and (19) into Eq. (12), the reliability estimation can be derived approximately as follows:

2.3.2.2 The Trigonometric Shrinkage Weight Function (sh₂)

$k_{\xi }\left({\widehat{\alpha }}_{\xi }\right)$ = $\left|\frac{sin\left(n_{\xi }\right)}{n_{\xi }}\right|$ and $k_{\xi }\left({\widehat{\beta }}_{\xi }\right)$ = $\left|\frac{sin\left(m_{\xi }\right)}{m_{\xi }}\right|$ , that $\xi =1,2,3$ , implies the following shrinkage estimators:

Substituting Eqs. (21) and (22) into Eq. (12), the reliability estimation with the Trigonometric Shrinkage Weight Function estimator can be articulated approximately as follows:

2.3.3 Least Square Estimation Methods (LS)

The method of Least Square estimates the parameter by minimizing the Eq.^14,16

Such that $E\left(F\left({\mathrm{\chi }}_i\right)\right)$ equivalent to plotting position $P_i$ ; $P_i=\frac{i}{n+1}$ . Let’s ${\mathrm{\chi }}_i,i=1,2,\dots ,n,$ follow

After deriving Eq. (24) and equating the result to zero, get ${\widehat{\mathit{\alpha }}}_{\mathit{LS}}$ ;

The Least Square estimator ${\widehat{\mathit{\alpha }}}_{\mathit{\xi LS}}$ and ${\widehat{\mathit{\beta }}}_{\mathit{\xi LS}}$ , where $\xi =1,2,3$ and $P_j=\frac{j}{m+1}$

Substituting Eqs. (25) and (26) into Eq. (12), the reliability estimation obtained approximately by the least square approach is as follows:

2.3.4 Weighted Least Square Estimation Method (WLS)

The method of weighted least square estimators for the Inverse Chen distribution can be employed by minimizing the following equation:

From Eq. (24), it can be obtained:

By deriving (28) and make the result equal to zero, one can obtain the Weighted Least Square estimator ${\widehat{\mathit{\alpha }}}_{\mathit{WLS}}$ and ${\widehat{\mathit{\beta }}}_{\mathit{WLS}}$ , where $\xi =1,2,3$ ;

Substituting Eqs. (29) and (30) into Eq. (12), the reliability estimation obtained approximately by the weighted least squares method is as follows:

2.4 Estimators comparison

The behaviour of estimated $R$ can be studied via a simulation approach using different methods. A statistical criterion has been employed for comparison of the results, MSE (mean squared error) that follows the formula:

The simulation analysis is done after repeated (1000) times to get independent different sizes samples.²⁶

2.4.1 Random Sample Generated for Inverse Chen Distribution

Assume that a random variable U with a standard uniform distribution, IC data can be generated by the adoption of CDF inverse transformation, wherein if:

Then ${\mathrm{\chi }}_{\mathit{i\xi }}={[ln(1-\frac{\mathit{ln}{U}_{\mathit{i\xi }}}{{\alpha }_{\mathit{i\xi }}})]}^{-\frac{1}{\lambda }};i=1,2,3,\dots ,n_{\xi }\mathrm{\text{and}}{\mathit{\gamma }}_{\mathit{j\xi }}={[ln(1-\frac{\mathit{ln}{U}_{\mathit{j\xi }}}{{\beta }_{\mathit{j\xi }}})]}^{-\frac{1}{\lambda }};j=1,2,3,\dots ,m_{\xi }.$

2.4.2 Simulation analysis

The simulation program is developed using MATLAB 2023b software to facilitate the comparison of reliability estimators, which may be outlined by the following steps:

Step 1: Select random samples ${\mathrm{\chi }}_{11},{\mathrm{\chi }}_{12},\dots ,{\mathrm{\chi }}_{1n1},{\mathrm{\chi }}_{21},{\mathrm{\chi }}_{22},\dots ,{\mathrm{\chi }}_{2n2}$ , and ${\mathit{\gamma }}_{11},{\mathit{\gamma }}_{12},\dots ,{\mathit{\gamma }}_{1\mathit{m}1},{\mathit{\gamma }}_{21},{\mathit{\gamma }}_{22},\dots ,{\mathit{\gamma }}_{2\mathit{m}2}$ of sizes ( $n_1,n_2,m_1,m_2$ ) which represents a, b, c and d where a = (25,25,25,25), b = (50,50,50,50), c = (75,75,75,75) and d = (100,100,100,100) respectively, that are generated from Inverse Chen Distribution.

Step 2: Calculate the $R$ into Eq. (12) estimation value of all suggestion methods $$ in Eqs. (16), (20), (23), (27), and (31), correspondingly. The real parameter values are selected for 4 experiments ( ${\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2$ ) in Table 1.

Step 3: Calculate the MSE for all proposed estimators using $L$ =1000 duplicates.

Step 4: Compare the simulation results; the best estimation method for the reliability is the one that estimates R with the smallest MSE value.

2.5 Application of real dataset

An analysis based on a real dataset is carried out in this section. According to,²⁷ the application consists of 25 observations that show the size distribution of diamonds taken from a significant mining zone in South-West Africa. The diamond sizes, however, are as follows: 7.5, 7, 2.5, 4.5, 2, 2, 3, 2, 1, 1.5, 5, 7, 3, 1, and 2, 39, 358, 257.5, 137, 69.5, 40.5, 28, 20.5, 16.5, and 9. The information used in this application of diamonds with a sample size of 25 came from.²⁷ To demonstrate that the inverse Chen distribution appears to be a more competitive model for these data than the Log-logistic, Uniform, exponential, Rayleigh, Chi-Square Half-Normal, Gumbel and Cauchy distributions. Table 6 is a descriptive statistics table that is telling us about the dataset of 25 observations of diamonds taken.

4.1 Experiment 1

4.1.1 Within estimation method

For the maximum likelihood estimation (MLE): the mean squared error (MSE) is lower in sample size b and higher in sample size a. Generally, the MSE decreases from sample size a to b and thereafter increases from b to d.

4.1.2 Between estimation techniques

The estimate technique sh₂ exhibits the lowest mean squared error (MSE), indicating it is superior to the others. sh₁ ranks second across all sample sizes, while the ranks of LS, WLS, and MLE fluctuate between third and fifth, depending on the sample size.

4.2 Experiment 2

4.2.1 Within estimating technique

For the maximum likelihood estimation (MLE): the mean squared error (MSE) is lower for sample size b and higher for sample size c. Generally, the MSE decreases from sample size a to b and thereafter decreases from c to d.

4.2.2 Comparison of estimate methodologies

The estimate technique sh₂ exhibits the lowest mean squared error (MSE), indicating it is superior to the others. sh₁ ranks second across all sample sizes, while the ranks of LS, WLS, and MLE fluctuate between third and fifth, depending on the sample size.

4.3 Experiment 3

4.3.1 Internal estimation method

For the maximum likelihood estimation (MLE): the mean squared error (MSE) is lower in sample size b and higher in sample size d. Generally, the MSE decreases from sample size a to b and thereafter increases from b to d.

4.3.2 Comparison of estimate methodologies

The estimate technique sh₂ exhibits the lowest mean squared error (MSE), indicating it is superior to the others. Subsequently, sh₁ ranks second throughout all sample sizes, while the ranks of LS, WLS, and MLE fluctuate between third and fifth, depending on the sample size.

4.4 Experiment 4

4.4.1 Within estimation method

For the maximum likelihood estimation (MLE): the mean squared error (MSE) is lower in sample size b and higher in sample size d. Generally, the MSE decreases from sample size a to b and thereafter increases from b to d.

4.4.2 Between estimation methodologies

The estimate technique sh₂ exhibits the lowest mean squared error, indicating it is superior to the others. Following sh₂, sh₁ ranks second across all sample sizes, while maximum likelihood estimation (MLE) ranks third, least squares (LS) ranks fourth, and weighted least squares (WLS) ranks fifth. Consequently, we assert that the trigonometric shrinkage estimation approach outperformed the other presented estimation methods according to the minimal mean squared error criterion.