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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.
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=P(Y < 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=P(Y < 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=P(Y < X) is studied by AI-Hussaini [9] based on a finite mixture of lognormal components. For more reading, see [10–14].
The 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 / 2 θ is used to represent this model:
f x , θ = K f R x · 1 2 θ + 1 − K f HN x · 1 2 θ = K 2 θ x e − θ x 2 + 1 − K 2 θ π e − θ x 2 ,where K = 1 / 1 + π θ .
Thus, the Rayleigh-half normal distribution probability density function (pdf) is given by
f x , θ = 2 θ x + 1 e − θ x 2 1 + π θ , x , θ > 0.The corresponding cumulative distribution function is given by
F x , θ = 1 − e − θ x 2 + π θ erf θ x 1 + π θ , x , θ > 0 ,where erf(u) is the Gauss error function defined as
erf u = 2 π ∫ 0 u e − t 2 d t .The 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. The definition of the survival function is represented as follows:
S x = 1 − F x = e − θ x 2 + π θ erfc θ x 1 + π θ ,where erfc(u) is the complementary error function, and its definition is
erfc u = 1 − erf u = 2 π ∫ u ∞ e − t 2 d t .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
h x = f x S x = 2 θ 1 + x 1 + e θ x 2 π θ erf θ x .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 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 f(x, θ); thus, the component reliability is
R = P Y < X = ∫ − ∞ ∞ ∫ − ∞ x f x , y d y d x .In case that the random variables are statistically independent, then f(x, y)=f(x)g(y) so that
R = ∫ − ∞ ∞ ∫ − ∞ x f x g x d y d x ,As the strength X ~ RHN(θ) and Y1 ~ RHN(θ1), they are independent random variables with pdf f (x) and g(y1), respectively:
f x = 2 θ x + 1 e − θ x 2 1 + π θ , 0 < θ · x , g y 1 = 2 θ 1 y 1 + 1 e − θ 1 y 1 2 1 + π θ 1 , 0 < θ 1 · y 1 .
We derive the reliability R=P(Y < X) as follows:
R 1 = P Y < X = ∫ 0 ∞ ∫ 0 x f x g y 1 d y d x = ∫ 0 ∞ ∫ 0 x 2 θ 1 y 1 + 1 e − θ 1 y 1 2 1 + π θ 1 2 θ x + 1 e − θ x 2 1 + π θ d y d x .
And, we get after the simplification:
R 1 = 1 + π θ 1 + 2 θ θ 1 Tan − 1 θ / θ 1 + π θ − θ 1 / θ + θ 1 − θ 1 / θ + θ 1 1 + π θ + π θ 1 + π θ θ 1 .
In this case, the probability density function (pdf) for the stress Y2 that follows the exponential distribution is given by
g y 2 = θ 2 e − y 1 θ 2 , y 2 , θ 2 > 0.Then, reliability function R2 for the independent random variables X and Y2:
R 2 = ∫ 0 ∞ ∫ 0 x θ 2 e − y 2 θ 2 2 θ x + 1 e − θ x 2 1 + π θ d y d x , R 2 = π 2 θ + π θ 2 θ − 2 θ − θ 2 e θ 2 2 / 4 θ erfc θ 2 2 θ ,
where the strength follows RHN distribution.
In this case, the probability density function (pdf) for the stress Y3 that follows the Rayleigh distribution is given by
g y 3 = y 3 θ 2 2 e − y 3 2 / 2 θ 3 2 , y , θ 3 > 0.Then, reliability function R3 for the independent random variables X and Y3 is
R 3 = ∫ 0 ∞ ∫ 0 x y 3 θ 3 2 e − y 3 2 / 2 θ 3 2 2 θ x + 1 e − θ x 2 1 + π θ d y d x = 2 θ 1 + π θ ∫ 0 ∞ x + 1 e − θ x 2 1 − e − x 2 / 2 θ 3 2 d x , R 3 = θ 1 + π θ 1 θ + π θ − 2 2 θ + 1 / θ 3 2 − 2 π 2 θ + 1 / θ 3 2 ,
where the strength follows RHN distribution.
In this case, the probability density function (pdf) for the stress Y4 that follows half-normal distribution is given by
g y 4 = 2 θ 4 π e − y 4 2 / 2 θ 4 2 , y , θ 4 > 0.Then, reliability function R4 for the independent random variables X and Y4
R 4 = ∫ 0 ∞ ∫ 0 x 2 θ 4 π e − y 4 2 2 θ 4 2 2 θ x + 1 e − θ x 2 1 + π θ d y d x = 2 θ 1 + π θ ∫ 0 ∞ Erf x 2 θ 4 x + 1 e − θ x 2 d x , R 4 = 1 θ 1 + π θ 2 θ Cot − 1 θ 4 2 θ π + 1 θ 4 2 + 1 / θ θ 4 2 ,
where the strength follows RHN distribution.
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. 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.
The estimation of reliability is very common in the statistical literature. Now, to compute R ^ , we need to estimate the parameters θ and θi, i=1,2,3,4, in four cases of stress.
Since the strengths X follow RHN (θ),the stress have four cases:
Y1 follows Rayleigh-half normal distribution with parameter θ1 Y2 follows exponential distribution with parameter θ2 Y3 follows Rayleigh distribution with parameter θ3 Y4 follows half-normal distribution with parameter θ4; then, their population means are given byx ¯ = 2 θ + π 2 θ 1 + θ π , y ¯ 1 = 2 θ 1 + π 2 θ 1 1 + θ 1 π , y ¯ 2 = 1 θ 2 , y ¯ 3 = θ 3 π 2 , y ¯ 4 = θ 4 2 π .
The ME's of θ, θ1, θ2, θ3, and θ4, 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:
θ ^ = ∑ i = 1 m x i − n 2 + n π ∑ i = 1 m x i 2 π ∑ i = 1 m x i 2 + ∑ i = 1 m x i − n 4 + 2 n π ∑ i = 1 m x i ∑ i = 1 m x i − n 2 2 π ∑ i = 1 m x i 2 , θ ^ 1 = ∑ j = 1 m y 1 j − m 2 + m π ∑ j = 1 m y 1 j 2 π ∑ j = 1 m y 1 j 2 + ∑ j = 1 m y 1 j − n 4 + 2 m π ∑ j = 1 m y 1 j ∑ j = 1 m y 1 j − m 2 2 π ∑ j = 1 m y 1 j 2 , θ ^ 2 = m ∑ j = 1 m y 2 j , θ ^ 3 = 2 π ∑ j = 1 m y 3 j m , θ ^ 4 = π 2 ∑ j = 1 m y 4 j m .
The ME of R, 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.
The maximum likelihood estimator (MLE) is the most popular method for reliability estimation R=p(Y < 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.
Suppose x1 · 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
L θ · θ 1 ; x · y 1 = 2 n + m θ n θ 1 m − 1 + π θ n − 1 + π θ 1 m ∏ i = 1 n x i + 1 e − θ x i 2 ∏ j = 1 m y 1 j + 1 e − θ 1 y 1 j 2 .
And, the log-likelihood function of the observed samples is
ln L θ · θ 1 = m + n ln 2 + n ln θ + m ln θ 1 − n ln 1 + π θ − m ln 1 + π θ 1 − θ ∑ i = 1 n x i 2 − θ 1 ∑ j = 1 m y 1 j 2 + ∑ i = 1 n ln x i + 1 + ∑ j = 1 m ln y 1 j + 1 .
By solving the following equations, the MLE of θ and θ2 can be obtained:
∂ ln L θ · θ 1 ∂ θ = n θ − n π 2 θ 1 + π θ − ∑ i = 1 n x i 2 = 0 , ∂ ln L θ · θ 1 ∂ θ 1 = m θ 1 − m π 2 θ 1 1 + π θ 1 − ∑ j = 1 m y 1 j 2 = 0.
The MLEs of θ and θ1 can be obtained, respectively, as
θ ^ = 1 6 π A 2 B + 2 A A + n π + A 2 n 2 π 2 + 4 A A − 4 n π B , θ ^ 1 = 1 6 π C 2 D + 2 C C + m π + C 2 m 2 π 2 + 4 C C − 4 m π D ,
B = 8 A 6 − 48 π n A 5 + 51 π 2 n 2 A 4 − π 3 n 3 A 3 + 3 3 π 3 / 2 n 3 A 7 − 16 A 2 + 71 π n A − 2 π 2 n 2 1 / 3 ,
D = 8 C 6 − 48 π m C 5 + 51 π 2 m 2 C 4 − π 3 m 3 C 3 + 3 3 π 3 / 2 m 3 C 7 − 16 C 2 + 71 π m C − 2 π 2 m 2 1 / 3 .
Then, the maximum likelihood estimator of R when the strength X follows RHN(θ) distribution and stress Y follows RHN(θ1) distribution is given as
R ^ 1 = 1 + π θ ^ 1 + 2 θ ^ θ ^ 1 Tan − 1 θ ^ / θ ^ 1 + π θ ^ − θ ^ 1 / θ ^ + θ ^ 1 − θ ^ 1 / θ ^ + θ ^ 1 1 + π θ ^ + π θ ^ 1 + π θ ^ θ ^ 1 .
Similarly, we perform the same steps to find (MLE) in other cases; we can obtain