Estimate Stress-Strength Reliability Model Using Rayleigh and Half-Normal Distribution

In the field of life testing, it is very important to study the reliability of any component under testing. One of the most important subjects is the “stress-strength reliability” term which always refers to the quantity P (X > Y) in any statistical literature. It resamples a system with random strength (X) that is subjected to a random strength (Y) such that a system fails in case the stress exceeds the strength. In this study, we consider stress-strength reliability where the strength (X) follows Rayleigh-half-normal distribution and stress (Y1, Y2, Y3, and Y4) follows Rayleigh-half-normal distribution, exponential distribution, Rayleigh distribution, and half-normal distribution, respectively. This effort comprises determining the general formulations of the reliabilities of a system. Also, the maximum likelihood estimation approach and method of moment (MOM) will be utilized to estimate the parameters. Finally, reliability has been attained utilizing various values of stress and strength parameters.


Introduction
e life of a component is described using the stress-strength models, in reliability theory, that is including a random strength (X) which is subjected to a random stress (Y). e failure of a component is occurred instantaneously when the stress level applied to it exceeds the level of the strength.
us, the component reliability is measured by R � P(Y < X). is measurement has a variety of applications, most notably in the engineering industry, such as the degradation of rocket motors and structures, the fatigue failure of aircraft structures, the ageing of concrete pressure vessels, and static fatigue of ceramic components. erefore, the estimation of R � P(Y < X) has a great importance in the practical applications. e literature demonstrates that reliability estimation (R) has already been performed when the distributions of (X) and (Y) are Weibull, exponential, or log normal.
Church and Harris [1] firstly introduced the term stressstrength. Many authors have adopted various distributions types for stress and strength. e works of Church and Harris, Surles and Padgett [2], Raqab and Kundu [3], Mokhlis [4], and Saraçoglu et al. [5] contain the discussion of the estimation problems of the stress-strength reliability model for different distributions. Recently, a review of all methods and results on the stress-strength reliability have presented by Kotz et al. [6]. Bayes estimators and reliability function and the parameters of the Consul, Geeta, and size-biased Geeta distributions are obtained by Khan Adil and Jan [7]. Akman et al. [8] studied the estimation of reliability using a finite mixture of inverse Gaussian distributions. e estimation of R � P(Y < X) is studied by AI-Hussaini [9] based on a finite mixture of lognormal components. For more reading, see [10][11][12][13][14].

Finite Mixture of Rayleigh and Half-Normal Distribution
e Rayleigh-half-normal distribution is denoted as RHN(θ) by Abd El-Monsef and Abd El-Raouf [15]. A mixture of Rayleigh and half-normal distribution with a parameter (1/ �� 2θ √ ) is used to represent this model: . us, the Rayleigh-half normal distribution probability density function (pdf ) is given by e corresponding cumulative distribution function is given by where erf(u) is the Gauss error function defined as

e Survival Function and the Hazard Function.
e reliability function or the survival function S(x) tests the chance of occurring of a breakdown of units beyond certain given point in time. For monitoring, a unit lifetime across the support of its lifetime distribution; generally, the probability that an item will work properly for a specified time period with no failure is the survival function. e definition of the survival function is represented as follows: where erfc(u) is the complementary error function, and its definition is e definition of the hazard rate function is the ratio between the density function and its survival function, which measures the tendency to die or to fail depending on the reached age, and therefore, it has a critical role in the classification of the distributions of lifetime, so the hazard rate function of the RHN distribution is given by

Stress-Strength Reliability Computations
In this section, the reliability R � P(Y < X) was derived, where the random variables (X) and (Y) are the independent random variables, where the strength X follows Rayleighhalf normal distribution and the stress (Y) takes different cases (Rayleigh-half normal distribution, exponential distribution, Rayleigh distribution, and half-normal distribution). Let (X) and (Y) be two independent random variables, where (X) represents "strength" and (Y) represents "stress" and (X), and (Y) follows a joint pdf f(x, θ); thus, the component reliability is In case that the random variables are statistically independent, then f(x, y) � f(x)g(y) so that where f(x) and g(y) are pdf's of X and Y, respectively.

e Stress and the Strength Follows Rayleigh-Half-Normal
Distribution. As the strength X ∼ RHN(θ) and Y 1 ∼ RHN(θ 1 ), they are independent random variables with pdf f (x) and g(y 1 ), respectively: We derive the reliability R � P(Y < X) as follows: 2 Computational Intelligence and Neuroscience And, we get after the simplification:

e Strength Follows RHN Distribution and the Stress
Follows Exponential Distribution. In this case, the probability density function (pdf ) for the stress Y 2 that follows the exponential distribution is given by en, reliability function R 2 for the independent random variables X and Y 2 : where the strength follows RHN distribution.

e Strength Follows RHN Distribution and the Stress Follows Rayleigh Distribution.
In this case, the probability density function (pdf ) for the stress Y 3 that follows the Rayleigh distribution is given by en, reliability function R 3 for the independent random variables X and Y 3 is where the strength follows RHN distribution.

e Strength Follows RHN Distribution and the Stress Follows Half-Normal Distribution.
In this case, the probability density function (pdf ) for the stress Y 4 that follows half-normal distribution is given by en, reliability function R 4 for the independent random variables X and Y 4

Estimation of Stress-Strength Reliability
In the literature, a discussion of the estimation R � P(Y < X) when random variables (X) and (Y) are following the specified distributions have been presented including engineering statistics, quality control, medicine, reliability, biostatistics, and psychology. is quantity for a limited number of cases could be calculated in a closed form (Nadarajah [16] and Barreto-Souza et al. [17]). Several authors including Milan and Vesna [18] have considered the estimation of (R) for independent variables and normally distributed (X) and (Y). Later, a list of papers related to the estimation problem of (R) were reported by Greco and Venture [19] when (X) and (Y) are independent and follow a class of lifetime distributions containing Gamma distributions, exponential, generalized exponential, bivariate exponential, Weibull distribution, Burr type t model, and others.

Method of Moment (MOM) Estimation of R.
e estimation of reliability is very common in the statistical literature. Now, to compute R, we need to estimate the parameters θ and θ i , i � 1, 2, 3, 4, in four cases of stress.

e Maximum Likelihood Estimators of R.
e maximum likelihood estimator (MLE) is the most popular method for reliability estimation R � p(Y < X) because of its generality and flexibility. is method can be used if the joint distribution of the strength (X) and the stress (Y) is a known function with some unknown parameters.
Suppose x 1 · x 2 · · · · · x n is a random sample from RHN distribution with θ and y 11 · y 12 · · · · · y 1m is a random sample from RHN distribution with θ 1 . en, the likelihood function is given by 4 Computational Intelligence and Neuroscience And, the log-likelihood function of the observed samples is By solving the following equations, the MLE of θ and θ 2 can be obtained: e MLEs of θ and θ 1 can be obtained, respectively, as where en, the maximum likelihood estimator of R when the strength X follows RHN(θ) distribution and stress Y follows RHN(θ 1 ) distribution is given as Similarly, we perform the same steps to find (MLE) in other cases; we can obtain (i) When the stress Y 2 that follows the exponential distribution with parameter θ 2 , the MLE of R 2 is given as where the strength X follows Rayleigh-half-normal distribution with parameter θ.
(ii) When the stress Y 3 that follows Rayleigh distribution with parameter θ 3 and the strength X follows Rayleigh-half-normal distribution with parameter θ, the MLE of R 3 is given as (iii) When the stress Y 4 that follows half-normal distribution with parameter θ 4 , the M. L. E of R 2 is given as Computational Intelligence and Neuroscience where the strength X follows Rayleigh-half-normal distribution with parameter θ.

Numerical Evaluation
In different cases, the system reliability R has evaluated for some specific values of the parameters involved in the expression of R.

Case 1: Strength and Stress Follows RHN Distribution.
From Table 1 and Figures 1 and 2, it is noticed that, with the increase in the strength parameter values, the reliability value decreases. If the stress parameter increases, then the value of reliability increases.

Case 2: Strength Follows RHN Distribution and Stress
Follows Exponential Distribution. From Table 2 and Figures 3 and 4, it is observed that if the strength parameter increases then the value of reliability increases. If the stress parameter increases, then the value of reliability increases.

Case 3: Strength Follows RHN Distribution and Stress
Follows Rayleigh Distribution. From Table 3 and Figures 5  and 6, it is noticed that, with increasing the value of the strength and stress parameter, the reliability value decreases.

Case 4: Strength Follows RHN Distribution and Stress
Follows Half-Normal Distribution. From Table 4 and Figures 7 and 8, it is noticed that, with increasing the value of the strength and stress parameter, the reliability value decreases.

Simulation Study.
In this section, some results are represented depending on Monte-Carlo simulation, for comparing the estimates of (R) performance using MLE and MOM estimators fundamentally for many sample sizes. e following sample sizes are considered; (n, m) � (5, 5), (10,10), (20, 20), (30, 30), (40, 40), (50, 50), and (100, 100). From each sample, the estimates are computed for the parameters using MLE and method of moment estimation. Once the              parameters are estimated, the estimates of R 1 is obtained. e average biases of R 1 is reported in Table 5 and mean squared errors (MSEs) of R 1 are in Table 6. e first row includes the average bias of R 1 using the MLE and second row includes the average bias of R 1 using the MOM, in each cell. e first row includes the average MSE of R 1 using the MLE and second row includes the average MSE of R 1 using the MOM, in each cell.

Conclusions
e proposed model in this paper, the stress-strength reliability has been studied for Rayleigh-half normal when the strength (X) follows Rayleigh-half normal distribution, and the stress (Y) takes Rayleigh-half normal distribution, exponential distribution, Rayleigh distribution, and halfnormal distribution. Based on the computations and graphs, (i) it has been noticed that when the stress parameter is increased, the reliability value lowers, and when the strength parameter is increased, the reliability value increases.
e numerical assessment demonstrates that increasing the stress parameter decreases the dependability value in case (ii), whereas increasing the strength parameter increases the reliability value. In cases (iii) and (vi), increasing the stress parameter decreases the reliability value, whereas increasing the strength parameter increases it. A comparison is carried out between two methods of reliability estimation R � P(X > Y) when (Y) and (X) both follow Rayleigh-half normal distributions for various parameters scale. We provide MLE and MOM procedure for estimating the unknown parameters that are used for reliability estimation (R). Based on the simulation findings, we can conclude that MLE outperforms MOM in terms of average bias and average MSE for a variety of parameter choices.

Data Availability
All data used to support the findings of the study are available within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.