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

Authors: Osama Abdulaziz Alamri; M. M. Abd El-Raouf; Eman Ahmed Ismail; Zahra Almaspoor; Basim S. O. Alsaedi; Saima Khan Khosa; M. Yusuf
Journal: Computational Intelligence and Neuroscience
 (2021)

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## Abstract

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 quantityPX>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.


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## Body

## 1. Introduction

The 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). The failure of a component is occurred instantaneously when the stress level applied to it exceeds the level of the strength. Thus, the component reliability is measured by R=PY<X. This 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. Therefore, the estimation of R=PY<X has a great importance in the practical applications. The 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 stress-strength. Many authors have adopted various distributions types for stress and strength. The works of Church and Harris, Surles and Padgett [2], Raqab and Kundu [3], Mokhlis [4], and Saraçoğlu 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. The estimation of R=PY<X is studied by AI-Hussaini [9] based on a finite mixture of lognormal components. For more reading, see [10–14].


## 2. Finite Mixture of Rayleigh and Half-Normal Distribution

The Rayleigh-half-normal distribution is denoted asRHNθ 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:(1)fx,θ=KfRx⋅12θ+1−KfHNx⋅12θ=K2θxe−θx2+1−K2θπe−θx2,where K=1/1+πθ.Thus, the Rayleigh-half normal distribution probability density function (pdf) is given by(2)fx,θ=2θx+1e−θx21+πθ,x,θ>0.The corresponding cumulative distribution function is given by(3)Fx,θ=1−e−θx2+πθerfθx1+πθ,x,θ>0,where erfu is the Gauss error function defined as(4)erfu=2π∫0ue−t2dt.


### 2.1. The Survival Function and the Hazard Function

The reliability function or the survival functionSx 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. The definition of the survival function is represented as follows:(5)Sx=1−Fx=e−θx2+πθerfcθx1+πθ,where erfcu is the complementary error function, and its definition is(6)erfcu=1−erfu=2π∫u∞e−t2dt.The 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(7)hx=fxSx=2θ1+x1+eθx2πθerfθx.


## 2.1. The Survival Function and the Hazard Function

The reliability function or the survival functionSx 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. The definition of the survival function is represented as follows:(5)Sx=1−Fx=e−θx2+πθerfcθx1+πθ,where erfcu is the complementary error function, and its definition is(6)erfcu=1−erfu=2π∫u∞e−t2dt.The 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(7)hx=fxSx=2θ1+x1+eθx2πθerfθx.


## 3. Stress-Strength Reliability Computations

In this section, the reliabilityR=PY<X was derived, where the random variables (X) and (Y) are the independent random variables, where the strength X follows Rayleigh-half 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 fx,θ; thus, the component reliability is(8)R=PY<X=∫−∞∞∫−∞xfx,ydydx.In case that the random variables are statistically independent, thenfx,y=fxgy so that(9)R=∫−∞∞∫−∞xfxgxdydx,where fx and gy are pdf’s of X and Y, respectively.


### 3.1. The Stress and the Strength Follows Rayleigh-Half-Normal Distribution

As the strengthX∼RHNθ and Y1∼RHNθ1, they are independent random variables with pdf fx and gy1, respectively:(10)fx=2θx+1e−θx21+πθ,0<θ⋅x,gy1=2θ1y1+1e−θ1y121+πθ1,0<θ1⋅y1.We derive the reliabilityR=PY<X as follows:(11)R1=PY<X=∫0∞∫0xfxgy1dydx=∫0∞∫0x2θ1y1+1e−θ1y121+πθ12θx+1e−θx21+πθdydx.And, we get after the simplification:(12)R1=1+πθ1+2θθ1Tan−1θ/θ1+πθ−θ1/θ+θ1−θ1/θ+θ11+πθ+πθ1+πθθ1.


### 3.2. The Strength Follows RHN Distribution and the Stress Follows Exponential Distribution

In this case, the probability density function (pdf) for the stressY2 that follows the exponential distribution is given by(13)gy2=θ2e−y1θ2,y2,θ2>0.Then, reliability functionR2 for the independent random variables X and Y2:(14)R2=∫0∞∫0xθ2e−y2θ22θx+1e−θx21+πθdydx,R2=π2θ+πθ2θ−2θ−θ2eθ22/4θerfcθ22θ,where the strength follows RHN distribution.


### 3.3. The Strength Follows RHN Distribution and the Stress Follows Rayleigh Distribution

In this case, the probability density function (pdf) for the stressY3 that follows the Rayleigh distribution is given by(15)gy3=y3θ22e−y32/2θ32,y,θ3>0.Then, reliability functionR3 for the independent random variables X and Y3 is(16)R3=∫0∞∫0xy3θ32e−y32/2θ322θx+1e−θx21+πθdydx=2θ1+πθ∫0∞x+1e−θx21−e−x2/2θ32dx,R3=θ1+πθ1θ+πθ−22θ+1/θ32−2π2θ+1/θ32,where the strength follows RHN distribution.


### 3.4. The Strength Follows RHN Distribution and the Stress Follows Half-Normal Distribution

In this case, the probability density function (pdf) for the stressY4 that follows half-normal distribution is given by(17)gy4=2θ4πe−y42/2θ42,y,θ4>0.Then, reliability functionR4 for the independent random variables X and Y4(18)R4=∫0∞∫0x2θ4πe−y422θ422θx+1e−θx21+πθdydx=2θ1+πθ∫0∞Erfx2θ4x+1e−θx2dx,R4=1θ1+πθ2θCot−1θ42θπ+1θ42+1/θθ42,where the strength follows RHN distribution.


## 3.1. The Stress and the Strength Follows Rayleigh-Half-Normal Distribution

As the strengthX∼RHNθ and Y1∼RHNθ1, they are independent random variables with pdf fx and gy1, respectively:(10)fx=2θx+1e−θx21+πθ,0<θ⋅x,gy1=2θ1y1+1e−θ1y121+πθ1,0<θ1⋅y1.We derive the reliabilityR=PY<X as follows:(11)R1=PY<X=∫0∞∫0xfxgy1dydx=∫0∞∫0x2θ1y1+1e−θ1y121+πθ12θx+1e−θx21+πθdydx.And, we get after the simplification:(12)R1=1+πθ1+2θθ1Tan−1θ/θ1+πθ−θ1/θ+θ1−θ1/θ+θ11+πθ+πθ1+πθθ1.


## 3.2. The Strength Follows RHN Distribution and the Stress Follows Exponential Distribution

In this case, the probability density function (pdf) for the stressY2 that follows the exponential distribution is given by(13)gy2=θ2e−y1θ2,y2,θ2>0.Then, reliability functionR2 for the independent random variables X and Y2:(14)R2=∫0∞∫0xθ2e−y2θ22θx+1e−θx21+πθdydx,R2=π2θ+πθ2θ−2θ−θ2eθ22/4θerfcθ22θ,where the strength follows RHN distribution.


## 3.3. The Strength Follows RHN Distribution and the Stress Follows Rayleigh Distribution

In this case, the probability density function (pdf) for the stressY3 that follows the Rayleigh distribution is given by(15)gy3=y3θ22e−y32/2θ32,y,θ3>0.Then, reliability functionR3 for the independent random variables X and Y3 is(16)R3=∫0∞∫0xy3θ32e−y32/2θ322θx+1e−θx21+πθdydx=2θ1+πθ∫0∞x+1e−θx21−e−x2/2θ32dx,R3=θ1+πθ1θ+πθ−22θ+1/θ32−2π2θ+1/θ32,where the strength follows RHN distribution.


## 3.4. The Strength Follows RHN Distribution and the Stress Follows Half-Normal Distribution

In this case, the probability density function (pdf) for the stressY4 that follows half-normal distribution is given by(17)gy4=2θ4πe−y42/2θ42,y,θ4>0.Then, reliability functionR4 for the independent random variables X and Y4(18)R4=∫0∞∫0x2θ4πe−y422θ422θx+1e−θx21+πθdydx=2θ1+πθ∫0∞Erfx2θ4x+1e−θx2dx,R4=1θ1+πθ2θCot−1θ42θπ+1θ42+1/θθ42,where the strength follows RHN distribution.


## 4. Estimation of Stress-Strength Reliability

In the literature, a discussion of the estimationR = 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. This 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.


### 4.1. Method of Moment (MOM) Estimation ofR

The estimation of reliability is very common in the statistical literature. Now, to computeR^, we need to estimate the parameters θ and θi,i=1,2,3,4, in four cases of stress.Since the strengthsX follow RHN θ,the stress have four cases:(i)
Y1 follows Rayleigh-half normal distribution with parameter θ1(ii)
Y2 follows exponential distribution with parameter θ2(iii)
Y3 follows Rayleigh distribution with parameter θ3(iv)
Y4 follows half-normal distribution with parameter θ4; then, their population means are given by(19)x¯=2θ+π2θ1+θπ,y¯1=2θ1+π2θ11+θ1π,y¯2=1θ2,y¯3=θ3π2,y¯4=θ42π.The ME’s ofθ,θ1,θ2,θ3,andθ4, denoted by θ^,θ^1,θ^2,θ^3, and θ^4, respectively, can be obtained by solving (x¯, y¯1, y¯2, y¯3, and y¯4) numerically:(20)θ^=∑i=1mxi−n2+nπ∑i=1mxi2π∑i=1mxi2+∑i=1mxi−n4+2nπ∑i=1mxi∑i=1mxi−n22π∑i=1mxi2,θ^1=∑j=1my1j−m2+mπ∑j=1my1j2π∑j=1my1j2+∑j=1my1j−n4+2mπ∑j=1my1j∑j=1my1j−m22π∑j=1my1j2,θ^2=m∑j=1my2j,θ^3=2π∑j=1my3jm,θ^4=π2∑j=1my4jm.The ME ofR, denoted by R^1⋅R^2⋅R^3 and R^4 is obtained by substitute θ^ with θ^1⋅θ^2⋅θ^3 and θ^4 in R1⋅R2⋅R3 and R4.


### 4.2. The Maximum Likelihood Estimators ofR

The maximum likelihood estimator (MLE) is the most popular method for reliability estimationR=pY<X because of its generality and flexibility. This method can be used if the joint distribution of the strength X and the stress Y is a known function with some unknown parameters.Supposex1⋅x2⋅⋯⋅xn is a random sample from RHN distribution with θ and y11⋅y12⋅⋯⋅y1m is a random sample from RHN distribution with θ1. Then, the likelihood function is given by(21)Lθ⋅θ1;x⋅y1=2n+mθnθ1m−1+πθn−1+πθ1m∏i=1nxi+1e−θxi2∏j=1my1j+1e−θ1y1j2.And, the log-likelihood function of the observed samples is(22)lnLθ⋅θ1=m+nln2+nlnθ+mlnθ1−nln1+πθ−mln1+πθ1−θ∑i=1nxi2−θ1∑j=1my1j2+∑i=1nlnxi+1+∑j=1mlny1j+1.By solving the following equations, the MLE ofθ and θ2 can be obtained:(23)∂lnLθ⋅θ1∂θ=nθ−nπ2θ1+πθ−∑i=1nxi2=0,∂lnLθ⋅θ1∂θ1=mθ1−mπ2θ11+πθ1−∑j=1my1j2=0.The MLEs ofθandθ1 can be obtained, respectively, as(24)θ^=16πA2B+2AA+nπ+A2n2π2+4AA−4nπB,θ^1=16πC2D+2CC+mπ+C2m2π2+4CC−4mπD,where A=∑i=0nxi2⋅C=∑j=0my1j2,(25)B=8A6−48πnA5+51π2n2A4−π3n3A3+33π3/2n3A7−16A2+71πnA−2π2n21/3,D=8C6−48πmC5+51π2m2C4−π3m3C3+33π3/2m3C7−16C2+71πmC−2π2m21/3.Then, the maximum likelihood estimator ofR when the strength X follows RHNθ distribution and stress Y follows RHNθ1 distribution is given as(26)R^1=1+πθ^1+2θ^θ^1Tan−1θ^/θ^1+πθ^−θ^1/θ^+θ^1−θ^1/θ^+θ^11+πθ^+πθ^1+πθ^θ^1.Similarly, we perform the same steps to find (MLE) in other cases; we can obtain(i)
When the stressY2 that follows the exponential distribution with parameter θ2, the MLE of R2 is given as(27)R^2=π2θ^+πθ^22θ^−2θ^−θ^2eθ^22/4θ^Erfcθ^22θ^,where the strengthX follows Rayleigh-half-normal distribution with parameter θ.(ii)
When the stressY3 that follows Rayleigh distribution with parameter θ3 and the strength X follows Rayleigh-half-normal distribution with parameter θ, the MLE of R3 is given as(28)R^3=θ^1+πθ^1θ^+πθ^−22θ^+1/θ^32−2π2θ^+1/θ^32.(iii)
When the stressY4 that follows half-normal distribution with parameter θ4, the M. L. E of R2 is given as(29)R^4=1θ^1+πθ^2θ^Cot−1θ^22θ^π+1θ^42+1/θ^θ^42,where the strength X follows Rayleigh-half-normal distribution with parameter θ.


## 4.1. Method of Moment (MOM) Estimation ofR

The estimation of reliability is very common in the statistical literature. Now, to computeR^, we need to estimate the parameters θ and θi,i=1,2,3,4, in four cases of stress.Since the strengthsX follow RHN θ,the stress have four cases:(i)
Y1 follows Rayleigh-half normal distribution with parameter θ1(ii)
Y2 follows exponential distribution with parameter θ2(iii)
Y3 follows Rayleigh distribution with parameter θ3(iv)
Y4 follows half-normal distribution with parameter θ4; then, their population means are given by(19)x¯=2θ+π2θ1+θπ,y¯1=2θ1+π2θ11+θ1π,y¯2=1θ2,y¯3=θ3π2,y¯4=θ42π.The ME’s ofθ,θ1,θ2,θ3,andθ4, denoted by θ^,θ^1,θ^2,θ^3, and θ^4, respectively, can be obtained by solving (x¯, y¯1, y¯2, y¯3, and y¯4) numerically:(20)θ^=∑i=1mxi−n2+nπ∑i=1mxi2π∑i=1mxi2+∑i=1mxi−n4+2nπ∑i=1mxi∑i=1mxi−n22π∑i=1mxi2,θ^1=∑j=1my1j−m2+mπ∑j=1my1j2π∑j=1my1j2+∑j=1my1j−n4+2mπ∑j=1my1j∑j=1my1j−m22π∑j=1my1j2,θ^2=m∑j=1my2j,θ^3=2π∑j=1my3jm,θ^4=π2∑j=1my4jm.The ME ofR, denoted by R^1⋅R^2⋅R^3 and R^4 is obtained by substitute θ^ with θ^1⋅θ^2⋅θ^3 and θ^4 in R1⋅R2⋅R3 and R4.


## 4.2. The Maximum Likelihood Estimators ofR

The maximum likelihood estimator (MLE) is the most popular method for reliability estimationR=pY<X because of its generality and flexibility. This method can be used if the joint distribution of the strength X and the stress Y is a known function with some unknown parameters.Supposex1⋅x2⋅⋯⋅xn is a random sample from RHN distribution with θ and y11⋅y12⋅⋯⋅y1m is a random sample from RHN distribution with θ1. Then, the likelihood function is given by(21)Lθ⋅θ1;x⋅y1=2n+mθnθ1m−1+πθn−1+πθ1m∏i=1nxi+1e−θxi2∏j=1my1j+1e−θ1y1j2.And, the log-likelihood function of the observed samples is(22)lnLθ⋅θ1=m+nln2+nlnθ+mlnθ1−nln1+πθ−mln1+πθ1−θ∑i=1nxi2−θ1∑j=1my1j2+∑i=1nlnxi+1+∑j=1mlny1j+1.By solving the following equations, the MLE ofθ and θ2 can be obtained:(23)∂lnLθ⋅θ1∂θ=nθ−nπ2θ1+πθ−∑i=1nxi2=0,∂lnLθ⋅θ1∂θ1=mθ1−mπ2θ11+πθ1−∑j=1my1j2=0.The MLEs ofθandθ1 can be obtained, respectively, as(24)θ^=16πA2B+2AA+nπ+A2n2π2+4AA−4nπB,θ^1=16πC2D+2CC+mπ+C2m2π2+4CC−4mπD,where A=∑i=0nxi2⋅C=∑j=0my1j2,(25)B=8A6−48πnA5+51π2n2A4−π3n3A3+33π3/2n3A7−16A2+71πnA−2π2n21/3,D=8C6−48πmC5+51π2m2C4−π3m3C3+33π3/2m3C7−16C2+71πmC−2π2m21/3.Then, the maximum likelihood estimator ofR when the strength X follows RHNθ distribution and stress Y follows RHNθ1 distribution is given as(26)R^1=1+πθ^1+2θ^θ^1Tan−1θ^/θ^1+πθ^−θ^1/θ^+θ^1−θ^1/θ^+θ^11+πθ^+πθ^1+πθ^θ^1.Similarly, we perform the same steps to find (MLE) in other cases; we can obtain(i)
When the stressY2 that follows the exponential distribution with parameter θ2, the MLE of R2 is given as(27)R^2=π2θ^+πθ^22θ^−2θ^−θ^2eθ^22/4θ^Erfcθ^22θ^,where the strengthX follows Rayleigh-half-normal distribution with parameter θ.(ii)
When the stressY3 that follows Rayleigh distribution with parameter θ3 and the strength X follows Rayleigh-half-normal distribution with parameter θ, the MLE of R3 is given as(28)R^3=θ^1+πθ^1θ^+πθ^−22θ^+1/θ^32−2π2θ^+1/θ^32.(iii)
When the stressY4 that follows half-normal distribution with parameter θ4, the M. L. E of R2 is given as(29)R^4=1θ^1+πθ^2θ^Cot−1θ^22θ^π+1θ^42+1/θ^θ^42,where the strength X follows Rayleigh-half-normal distribution with parameter θ.


## 5. Numerical Evaluation

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


### 5.1. Case 1: Strength and Stress Follows RHN Distribution

From Table1 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.Table 1
Variation inR1 when strength and stress has RHN distribution.
θθ10.10.20.30.40.50.60.70.80.910.10.2080.3590.4720.5580.6240.6770.7180.7520.7800.8030.20.1490.2670.3640.4420.5070.5610.6060.6450.6780.7060.30.1210.2220.3070.3790.4400.4930.5380.5770.6110.6410.40.1050.1940.2710.3370.3950.4450.4900.5290.5630.5940.50.0930.1740.2450.3070.3620.4100.4530.4910.5260.5560.60.0850.1590.2250.2840.3360.3820.4240.4610.4950.5260.70.0780.1480.2100.2650.3150.3600.4000.4370.4700.5000.80.0730.1380.1970.2500.2970.3410.3800.4160.4480.4780.90.0680.1300.1860.2370.2830.3250.3630.3980.4300.45910.0650.1240.1770.2260.2700.3110.3480.3820.4130.443Figure 1
Variation inR1 for constant stress.Figure 2
Variation inR1 for constant strength.


### 5.2. Case 2: Strength Follows RHN Distribution and Stress Follows Exponential Distribution

From Table2 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.Table 2
Variation inR2 when strength has RHN distribution and stress has Exponential distribution.
θθ20.10.20.30.40.50.60.70.80.910.10.5000.3490.2740.2280.1970.1750.1570.1440.1330.1240.20.6510.5000.4120.3540.3120.2800.2560.2360.2190.2050.30.7260.5880.5000.4380.3920.3570.3280.3040.2850.2680.40.7720.6460.5620.5000.4530.4150.3840.3590.3370.3180.50.8030.6880.6080.5470.5000.4620.4300.4030.3800.3610.60.8250.7200.6430.5850.5380.5000.4680.4410.4170.3960.70.8430.7440.6720.6160.5700.5320.5000.4720.4480.4270.80.8560.7640.6960.6410.5970.5590.5280.5000.4760.4550.90.8670.7810.7150.6630.6200.5830.5520.5240.5000.47910.8760.7950.7320.6820.6390.6040.5730.5450.5210.500Figure 3
Variation inR2 for constant stress.Figure 4
Variation inR2 for constant strength.


### 5.3. Case 3: Strength Follows RHN Distribution and Stress Follows Rayleigh Distribution

From Table3 and Figures 5 and 6, it is noticed that, with increasing the value of the strength and stress parameter, the reliability value decreases.Table 3
Variation inR3 when strength has RHN distribution and stress has Rayleigh distribution.
θθ30.10.20.30.40.50.60.70.80.910.10.9830.9630.9410.9170.8910.8640.8350.8060.7770.7470.20.9700.9360.8980.8580.8160.7730.7300.6870.6450.6040.30.9590.9130.8630.8100.7560.7020.6500.6000.5520.5080.40.9490.8920.8310.7680.7060.6450.5870.5330.4830.4380.50.9400.8740.8040.7320.6620.5960.5350.4790.4290.3850.60.9320.8570.7780.7000.6250.5550.4920.4360.3860.3430.70.9240.8410.7550.6710.5910.5190.4550.3990.3510.3090.80.9170.8260.7340.6440.5620.4880.4240.3680.3210.2810.90.9100.8120.7140.6200.5350.4600.3960.3420.2960.25710.9030.7990.6950.5980.5110.4350.3720.3180.2740.238Figure 5
Variation inR3 for constant stress.Figure 6
Variation inR3 for constant strength.


### 5.4. Case 4: Strength Follows RHN Distribution and Stress Follows Half-Normal Distribution

From Table4 and Figures 7 and 8, it is noticed that, with increasing the value of the strength and stress parameter, the reliability value decreases.Table 4
Variation inR4 when strength has RHN distribution and stress has half-normal distribution.
θθ40.10.20.30.40.50.60.70.80.910.10.9890.9770.9640.9500.9340.9180.9010.8840.8660.8480.20.9810.9600.9370.9130.8880.8620.8350.8080.7810.7550.30.9740.9460.9150.8830.8500.8160.7830.7500.7180.6870.40.9680.9330.8960.8570.8170.7780.7400.7030.6680.6340.50.9630.9210.8780.8340.7890.7450.7040.6640.6270.5920.60.9570.9110.8620.8120.7640.7170.6720.6300.5920.5570.70.9520.9010.8470.7930.7410.6910.6440.6020.5620.5270.80.9480.8910.8330.7750.7200.6680.6200.5760.5370.5010.90.9430.8830.8200.7590.7010.6470.5980.5540.5140.47910.9390.8740.8080.7440.6830.6280.5780.5330.4940.459Figure 7
Variation inR4 for constant stress.Figure 8
Variation inR4 for constant strength.


### 5.5. 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. The 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 R1 is obtained. The average biases of R1 is reported in Table 5 and mean squared errors (MSEs) of R1 are in Table 6.Table 5
Average MSE of the simulated estimates ofR1.
θ⋅θ1(n, m)(5, 5)(10, 10)(20, 20)(30, 30)(40, 40)(50, 50)(100, 100)(1, 1)0.01960.0180−0.0181−0.0073−0.0041−0.00340.0005−0.0720−0.0318−0.01930.0138−0.00840.0076−0.0029(1, 0.5)−0.0594−0.0476−0.0132−0.0120−0.0101−0.00430.0048−0.0790−0.0747−0.0643−0.0662−0.0617−0.0658−0.0555(1, 1.5)−0.04190.02620.01720.0146−0.00630.00240.00110.06780.06430.05570.04780.04230.03720.0370(1, 2)−0.01250.01150.01110.0113−0.00730.0044−0.00150.0800−0.07770.06870.06630.06560.06490.0638(0.5, 1)0.02520.01460.0122−0.0097−0.0030−0.00180.00110.07430.07370.06860.06560.06540.06390.0627(1.5, 1)0.0594−0.00900.0076−0.00640.00620.00190.0019−0.0807−0.0413−0.0402−0.0364−0.0352−0.01640.0125(2, 1)0.01530.00660.00660.00610.00480.00480.0045−0.0997−0.0726−0.0719−0.0647−0.0624−0.0618−0.0584Table 6
Average MSE of the simulated estimates ofR1.
θ⋅θ1(n, m)(5, 5)(10, 10)(20, 20)(30, 30)(40, 40)(50, 50)(100, 100)(1, 1)0.00510.00200.00200.00130.00100.00050.00030.01590.01040.00900.00350.00240.00200.0013(1, 0.5)0.01230.00950.00370.00330.00250.00180.00090.01480.01120.00510.00490.00460.00440.0044(1, 1.5)0.00770.00720.00440.00300.00220.0020−0.00010.01350.00910.00540.00510.00340.00210.0020(1, 2)0.00550.00500.00480.00330.00220.0019−0.00030.01150.00860.00590.00590.00490.00490.0009(0.5, 1)0.00900.00380.00380.00360.00310.00240.00100.00920.00870.00700.00500.00500.00480.0046(1.5, 1)0.01470.00970.00240.0024−0.00180.0014−0.00080.0221−0.0124−0.00900.00620.00270.00220.0014(2, 1)0.01210.00640.00590.00300.00240.0018−0.00030.0478−0.02720.0256−0.0145−0.00760.00500.0010The first row includes the average bias ofR1 using the MLE and second row includes the average bias of R1 using the MOM, in each cell.The first row includes the average MSE ofR1 using the MLE and second row includes the average MSE of R1 using the MOM, in each cell.


## 5.1. Case 1: Strength and Stress Follows RHN Distribution

From Table1 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.Table 1
Variation inR1 when strength and stress has RHN distribution.
θθ10.10.20.30.40.50.60.70.80.910.10.2080.3590.4720.5580.6240.6770.7180.7520.7800.8030.20.1490.2670.3640.4420.5070.5610.6060.6450.6780.7060.30.1210.2220.3070.3790.4400.4930.5380.5770.6110.6410.40.1050.1940.2710.3370.3950.4450.4900.5290.5630.5940.50.0930.1740.2450.3070.3620.4100.4530.4910.5260.5560.60.0850.1590.2250.2840.3360.3820.4240.4610.4950.5260.70.0780.1480.2100.2650.3150.3600.4000.4370.4700.5000.80.0730.1380.1970.2500.2970.3410.3800.4160.4480.4780.90.0680.1300.1860.2370.2830.3250.3630.3980.4300.45910.0650.1240.1770.2260.2700.3110.3480.3820.4130.443Figure 1
Variation inR1 for constant stress.Figure 2
Variation inR1 for constant strength.


## 5.2. Case 2: Strength Follows RHN Distribution and Stress Follows Exponential Distribution

From Table2 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.Table 2
Variation inR2 when strength has RHN distribution and stress has Exponential distribution.
θθ20.10.20.30.40.50.60.70.80.910.10.5000.3490.2740.2280.1970.1750.1570.1440.1330.1240.20.6510.5000.4120.3540.3120.2800.2560.2360.2190.2050.30.7260.5880.5000.4380.3920.3570.3280.3040.2850.2680.40.7720.6460.5620.5000.4530.4150.3840.3590.3370.3180.50.8030.6880.6080.5470.5000.4620.4300.4030.3800.3610.60.8250.7200.6430.5850.5380.5000.4680.4410.4170.3960.70.8430.7440.6720.6160.5700.5320.5000.4720.4480.4270.80.8560.7640.6960.6410.5970.5590.5280.5000.4760.4550.90.8670.7810.7150.6630.6200.5830.5520.5240.5000.47910.8760.7950.7320.6820.6390.6040.5730.5450.5210.500Figure 3
Variation inR2 for constant stress.Figure 4
Variation inR2 for constant strength.


## 5.3. Case 3: Strength Follows RHN Distribution and Stress Follows Rayleigh Distribution

From Table3 and Figures 5 and 6, it is noticed that, with increasing the value of the strength and stress parameter, the reliability value decreases.Table 3
Variation inR3 when strength has RHN distribution and stress has Rayleigh distribution.
θθ30.10.20.30.40.50.60.70.80.910.10.9830.9630.9410.9170.8910.8640.8350.8060.7770.7470.20.9700.9360.8980.8580.8160.7730.7300.6870.6450.6040.30.9590.9130.8630.8100.7560.7020.6500.6000.5520.5080.40.9490.8920.8310.7680.7060.6450.5870.5330.4830.4380.50.9400.8740.8040.7320.6620.5960.5350.4790.4290.3850.60.9320.8570.7780.7000.6250.5550.4920.4360.3860.3430.70.9240.8410.7550.6710.5910.5190.4550.3990.3510.3090.80.9170.8260.7340.6440.5620.4880.4240.3680.3210.2810.90.9100.8120.7140.6200.5350.4600.3960.3420.2960.25710.9030.7990.6950.5980.5110.4350.3720.3180.2740.238Figure 5
Variation inR3 for constant stress.Figure 6
Variation inR3 for constant strength.


## 5.4. Case 4: Strength Follows RHN Distribution and Stress Follows Half-Normal Distribution

From Table4 and Figures 7 and 8, it is noticed that, with increasing the value of the strength and stress parameter, the reliability value decreases.Table 4
Variation inR4 when strength has RHN distribution and stress has half-normal distribution.
θθ40.10.20.30.40.50.60.70.80.910.10.9890.9770.9640.9500.9340.9180.9010.8840.8660.8480.20.9810.9600.9370.9130.8880.8620.8350.8080.7810.7550.30.9740.9460.9150.8830.8500.8160.7830.7500.7180.6870.40.9680.9330.8960.8570.8170.7780.7400.7030.6680.6340.50.9630.9210.8780.8340.7890.7450.7040.6640.6270.5920.60.9570.9110.8620.8120.7640.7170.6720.6300.5920.5570.70.9520.9010.8470.7930.7410.6910.6440.6020.5620.5270.80.9480.8910.8330.7750.7200.6680.6200.5760.5370.5010.90.9430.8830.8200.7590.7010.6470.5980.5540.5140.47910.9390.8740.8080.7440.6830.6280.5780.5330.4940.459Figure 7
Variation inR4 for constant stress.Figure 8
Variation inR4 for constant strength.


## 5.5. 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. The 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 R1 is obtained. The average biases of R1 is reported in Table 5 and mean squared errors (MSEs) of R1 are in Table 6.Table 5
Average MSE of the simulated estimates ofR1.
θ⋅θ1(n, m)(5, 5)(10, 10)(20, 20)(30, 30)(40, 40)(50, 50)(100, 100)(1, 1)0.01960.0180−0.0181−0.0073−0.0041−0.00340.0005−0.0720−0.0318−0.01930.0138−0.00840.0076−0.0029(1, 0.5)−0.0594−0.0476−0.0132−0.0120−0.0101−0.00430.0048−0.0790−0.0747−0.0643−0.0662−0.0617−0.0658−0.0555(1, 1.5)−0.04190.02620.01720.0146−0.00630.00240.00110.06780.06430.05570.04780.04230.03720.0370(1, 2)−0.01250.01150.01110.0113−0.00730.0044−0.00150.0800−0.07770.06870.06630.06560.06490.0638(0.5, 1)0.02520.01460.0122−0.0097−0.0030−0.00180.00110.07430.07370.06860.06560.06540.06390.0627(1.5, 1)0.0594−0.00900.0076−0.00640.00620.00190.0019−0.0807−0.0413−0.0402−0.0364−0.0352−0.01640.0125(2, 1)0.01530.00660.00660.00610.00480.00480.0045−0.0997−0.0726−0.0719−0.0647−0.0624−0.0618−0.0584Table 6
Average MSE of the simulated estimates ofR1.
θ⋅θ1(n, m)(5, 5)(10, 10)(20, 20)(30, 30)(40, 40)(50, 50)(100, 100)(1, 1)0.00510.00200.00200.00130.00100.00050.00030.01590.01040.00900.00350.00240.00200.0013(1, 0.5)0.01230.00950.00370.00330.00250.00180.00090.01480.01120.00510.00490.00460.00440.0044(1, 1.5)0.00770.00720.00440.00300.00220.0020−0.00010.01350.00910.00540.00510.00340.00210.0020(1, 2)0.00550.00500.00480.00330.00220.0019−0.00030.01150.00860.00590.00590.00490.00490.0009(0.5, 1)0.00900.00380.00380.00360.00310.00240.00100.00920.00870.00700.00500.00500.00480.0046(1.5, 1)0.01470.00970.00240.0024−0.00180.0014−0.00080.0221−0.0124−0.00900.00620.00270.00220.0014(2, 1)0.01210.00640.00590.00300.00240.0018−0.00030.0478−0.02720.0256−0.0145−0.00760.00500.0010The first row includes the average bias ofR1 using the MLE and second row includes the average bias of R1 using the MOM, in each cell.The first row includes the average MSE ofR1 using the MLE and second row includes the average MSE of R1 using the MOM, in each cell.


## 6. Conclusions

The 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 half-normal 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. The 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=PX>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.



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*Source: 7653581-2021-07-05.xml*