Bivariate general exponential models with stress-strength reliability application

In this paper, we introduce two families of general bivariate distributions. We refer to these families as general bivariate exponential family and general bivariate inverse exponential family. Many bivariate distributions in the literature are members of the proposed families. Some properties of the proposed families are discussed, as well as a characterization associated with the stress-strength reliability parameter, R, is presented. Concerning R, the maximum likelihood estimators and a simple estimator with an explicit form depending on some marginal distributions are obtained in case of complete sampling. When the stress is censored at the strength, an explicit estimator of R is also obtained. The results obtained can be applied to a variety of bivariate distributions in the literature. A numerical illustration is applied on some well-known distributions. Finally a real data example is presented to fit one of the proposed models.


Introduction
Mokhlis et al. [1] presented two forms of survival functions, given by where g 1 (u; c) does not contain θ, θ ∈ Θ, and g 2 (u; c) does not contain β ∈ β, c ∈ C, { Θ, β, and C} are the parametric spaces, where g 1 (u; c) is a continuous, monotone increasing, and differential function such that g 1 (u; c) → 0 as u → 0 and g 1 (u; c) → ∞ as u → ∞, while g 2 (u; c) is continuous, monotone decreasing and differential function such that g 2 (u; c) → 0 as u → ∞ and g 2 (u; c) → ∞ as u → 0. With appropriate choices of g i (u; c), i = 1, 2, in (1) and (2), many distributions in the literature can be obtained, such as exponential distribution, Weibull distribution, Rayleigh distribution, Pareto, Lomax, and others from the first form (1), and inverse exponential distribution, inverse Weibull distribution, inverse Rayleigh, Burr type III distribution and others from the second form (2), see Mokhlis et al. [1]. For facilitation we will denote the forms (1) and (2) by EF(θ, c) and IEF(β; c) and denote its survival functions and probability density function by F EF (u; θ, c), f EF (u; θ, c) and F IEF (u; β, c), f IEF (u; β, c), respectively.
In the area of stress-strength models, there have been a large amount of work regarding estimation of the reliability parameter, R = P(Y < X), when X and Y independent random variables belonging to the same univariate family, see for example Mokhlis [2], Kundu and Gupta [3], Singh et al. [4] and others. Recently, Mokhlis et al. [1] discussed R, when the variables are independent with survival functions having forms (1) and (2), respectively. Indeed, many real situations entail that X and Y are related in some way. However, some authors have studied the stressstrength reliability parameter, R, for some specified bivariate distributions, see for example Kotz et al. [5], Mokhlis [6], Nadarajah and Kotz [7], Nguimkeu et al. [8], Pak et al. [9] and, Abdel-Hamid [10].
There are many methods in the literature for obtaining bivariate distribution. Some of the popular methods are the copula type, the bivariate pseudo type and the Marshall-Olkin type. Recently many attempts of obtaining generalized bivariate distributions using these types are presented in the literature. Among those are Kolesarova et al. [11], Arnold and Arvanitis [12], El-Bassiouny et al. [13] and Sarhan [14].
In the present paper, we introduce two bivariate models of distributions which are types of the bivariate Marshall-Olkin distribution. We call these models bivariate exponential and bivariate inverse exponential models. Some properties of the proposed models are discussed. Many bivariate distributions in the literature can be considered as special cases or members of our models, for example, Marshall and Olkin (M-O) bivariate exponential distribution and M-O bivariate Weibull introduced by Marshall and Olkin [15] and the bivariate Rayleigh distribution introduced by Pak et al. [9].
An explicit expression of the stress-strength parameter R is obtained showing that it is not a function of the parameter c (c could be a vector parameter). The maximum likelihood estimator of R is obtained as well as simple estimators of R are obtained in a closed form depending on the marginal distribution of X and the distribution of min{X, Y} or depending on the marginal distribution of Y and the distribution of max{X, Y}. Since many bivariate distributions in the literature belong to the proposed families, the results obtained could be applicable to a variety of bivariate distributions.
The remaining part of the paper is organized as follows: In the "Proposed families of bivariate distributions" section, we introduce two new families (models) of bivariate distributions. Some characterization of the proposed models such as marginals and the distribution of min{X, Y} and max {X, Y} are also discussed. The stress-strength reliability parameter, R, concerning the new models is considered in the "Stress-strength reliability" section. In the "Point estimation of R" section, we obtain maximum likelihood estimators of R as well as simple estimators of R depending on some marginal distributions in case of complete sampling. When the stress is censored at the strength, an explicit estimator of R is also obtained. Some bivariate members of the proposed family are presented in the "Special cases" section. In the "Numerical illustrations" section, a numerical illustration using some well-known distributions is performed to highlight the theoretical results. Also an application is introduced using real data example. Finally conclusions of the results obtained are introduced in the "Conclusions" section.

Proposed families of bivariate distributions
In this section, we introduce two new families of bivariate distributions with marginals having distributions with forms (1) or (2). We apply a similar technique of that proposed by Marshall and Olkin [15], for obtaining these families.
The construction of the families (models)

Lifetime model
Suppose that a system consists of two subsystems, say A and B. Subsystem A contains two components, say A 1 , and C, connected in series (parallel) with lifetimes U 1 and U 0 , respectively. Subsystem B contains the two components, say B 1 and C, connected in series (parallel), where the lifetime of component B 1 is U 2 . Suppose that U i , i = 0, 1, 2. , are independent random variables following EF(θ i , c), i = 0, 1, 2 for the series case and IEF(β i ; c), i = 0, 1, 2. , for the parallel case, i.e., If X and Y are the lifetimes of the two subsystems A and B, respectively, then we have X = min {U 0 , U 1 } and Y = min {U 0 , U 2 }., for the series case, while X = max {U 0 , U 1 } and Y = max {U 0 , U 2 }, for the parallel case.

Stress model
Consider a two-component system and consider three independent stresses say U 0 , U 1 , and U 2 . Each component is subject to an individual stress say U 1 and U 2 , respectively, while U 0 is an overall stress transmitted to both the components equally. Then, 1. The observed stress on the two components is X = max {U 0 , U 1 } and Y = max {U 0 , U 2 }., respectively. 2. If the stresses are always fatal, then the lifetime of the two components are X = min {U 0 , U 1 } and Y = min {U 0 , U 2 }.
We can observe that in the two models there is the possibility of having X = Y; thus, the two models have both an absolute continuous part and a singular part, similar to M-O bivariate exponential model. Theorems 1-3 present the survival functions and the probability density functions of the proposed bivariate families.
We will denote the bivariate distribution with cumulative function with the form (5) by BIEF(β 0 , β 1 , β 2 ; c). Clearly, X and Y are independent iff β 0 = 0. The joint cumulative function can also be written as , then their joint pdf is given by Proof Clearly, for the two models, f 1 (x, y) and f 2 (x, y) can be easily obtained by using Similarly for the BIEF, we have Hence, the proof is complete. Notice that both distribution BEF(θ 0 , θ 1 , θ 2 ; c) and BIEF (β 0 , β 1 , β 2 ; c) are singular on the line X = Y, since P(X = Y) ≠ 0. Thus the two models have a singular part and an absolute continuous part, similar to Marshall and Olkin's model. The following theorem provides explicitly the absolute continuous part and the singular part of BEF and BIEF.
Proof (i) For the BEF, using the fact that F BEF ðx; yÞ ¼ α F BEFðaÞ ðx; yÞ þ ð1−αÞ F BEFðsÞ ðx; yÞ Hence α may be obtained as The marginal distributions of X and Y and the conditional distributions are given by Theorems 5 and 6, while the distributions of min{X, Y}, for the BEF, and max{X, Y}, for the BIEF, are given by Theorem 7.
Notice that the marginal distributions of X and Y can also be obtained using the next lemma.
Here "∼" means follows or has the distribution.
(ii) Similarly for the BIEF.
Consequently, from Theorems 1 and 2 and Lemma 1, we have the following lemma, Lemma 2.
Theorem 6 The conditional distribution of X given Y = y, is given by khames and Mokhlis Journal of the Egyptian Mathematical Society (2020) 28:9 Page 6 of 15 for the BEF, while for the BIEF is given by Proof The proof is trivial so it is omitted.
is a bivariate vector of continuous random variables, then Similarly by using (5) for the BIEF, we can show that max{X, Y} ∼ IEF(β; c).

Stress-strength reliability
In this section, we present the stress-strength reliability of the two bivariate models. Many bivariate distributions in the literature have forms of the proposed models, for example, M-O bivariate exponential distribution, Marshal and Olkin [15], and the bivariate Rayleigh distribution introduced by Pak et al. [9] for the BEF and bivariate inverse Weibull and bivariate Burr type III for the BIEF. So the following theorem can be applied to many distributions possessing BEF or BIEF. Theorem 8 Let (X, Y) be a bivariate vector. Then, the stress-strength reliability function, R, is given by iff (X, Y) BEF(θ 0 , θ 1 , θ 2 ; c), where θ = θ 0 + θ 1 + θ 2 .

Point estimation of R
Let (X 1 , Y 1 ), …, (X n , Y n ) be a random sample of size n from either BEF(θ 0 , θ 1 , θ 2 ; c) or BIEF (β 0 , β 1 , β 2 ; c), assuming c is known. Let n 1 be the number of observations having y i > x i and n 2 be the number of observations having y i < x i and n 0 be the number of observations having y i = x i in the sample of size n, where n = n 0 + n 1 + n 2 . Then, the non-parametric estimator of R is given by Ř ¼ n 2 n , where n 2 is binomial (n, R). Thus, EðŘÞ ¼ R and variance V ðŘÞ ¼ R n ð1−RÞ:
Again replacing the parameters in (12) by their estimators given by (21) and (22), we obtain the simple estimator of R for the BIEF.
Estimation of R when the stress is censored at the strength Sometimes, obtaining the estimate of R based on complete sample is neither possible nor desirable on account of lack of time or minimization of the experiment cost. Thus, there are some situations where the stress is censored at the strength (see Hanagel [16]).
Let (X 1 , Y 1 ), …, (X n , Y n ) be a random sample of size n from BEF(θ 0 , θ 1 , θ 2 ; c); then, the strength and stress associated with the ith pair of sample is and the likelihood function can be written as Thus, the MLE's θ 0 þ θ 1 -; θ 2 of θ 0 + θ 1 and θ 2 , respectively, are then, the MLE, R, of R when the stress is censored at the strength is given by Special cases Table 1 present some well-known bivariate distributions as members of the BEF(θ 0 , θ 1 , θ 2 ; c) or BIEF(β 0 , β 1 , β 2 ; c) and some other distributions for some choices of θ i , β i , i = 0, 1, 2, g 1 (x; c) and g 2 (x; c). Clearly putting c = 1 and 2 in the bivariate inverse Weibull, we get bivariate inverse exponential and bivariate inverse Rayleigh distributions, respectively.
It is to be noted that these values are chosen arbitrary just for illustrating the results obtained. Tables 2 and 3 show the true value of R and its corresponding estimate by using From Tables 2 and 3, we see that all estimates converge to R, when n increases and MSE decreases. In Table 2, we see MSE (M) < MSE (C) < MSE (S) < MSE (N) , while in Table  3, MSE (M) < MSE (S) < MSE (N) . However, the R (C) , when the stress is censored at the strength for the BEF, and R (s) , for the BEF and BIEF, both estimators are simple, easier in computation and provide sufficient results for biasedness and mean square error.

Real data example
In real life, there are many situations where we have X < Y, Y < X or X = Y, such as nuclear reactor safety, competing risks, (see Kotz et al. [17]). In the medical field X and Y can represent the blood pressure or count of the white blood cells for patients before and after a certain operation.
The following data set is from the American Football (National Football League) matches for three consecutive weekends in 1986. The data was first published in the "Washington Post" and available in Csörgő and Welsh [18] Table 4.
The bivariate variables X and Y are as follows: X represents the game time to the first points scored by kicking the ball between goal posts and Y represents the game time to the first points scored by moving the ball into the end zone. This data was first analyzed by Csörgő and Welsh [18], by converting the seconds to decimal minutes. Also Kundu and Gupta [19] and Jamalizadeh and Kundu [20] analyzed this data.
We consider BEF and BIEF for fitting this data set. First, we fit each EF and IEF to X and Y separately. The data fit two cases, namely exponential which is special case of the EF and inverse exponential which is special case of the IEF, respectively. In case of exponential distribution, the MLEs of the scale parameters of X and Y are 0.1102 and 0.0745, respectively, while for the inverse exponential the MLEs of the scale parameters are 4.4000 and 5.0214, respectively.  The Kolmogorov-Smirnov distances between the fitted distribution and the empirical distribution function for X and Y are 0.14997 and 0.1182 respectively for the exponential case, while those for the inverse exponential case are 0.1530 and 0.1955. The above values are less than the critical value D 0.05 ≅ 0.2099, for n = 42, so that each of exponential distribution and inverse exponential distribution is an appropriate fit for the given data. This means that there may exist three independent random variables, say U i , i = 1, 2, 3, with EF or IEF thus X = min {U 0 , U 1 } or max{U 0 , U 1 } and Y = min {U 0 , U 2 } or max{U 0 , U 2 }. Now, we try to test whether M-O bivariate exponential distribution or bivariate inverse exponential distribution provides better fit to the above data set. We use the Akaike information criterion (AIC) to check the model validity. Based on the above data, the MLEs of parameters for the M-O bivariate exponential distribution θ 0 = 0.0715, θ 1 = 0.0456, and θ 2 = 0.0030, and the MLEs of parameters for the bivariate inverse exponential distribution are β 0 = 4.2769, β 1 = 0.1746, and β 2 = 2.0715. Thus, for the case of M-O bivariate exponential the log-likelihood value is − 227.9347 and the corresponding AIC is 461.8694, while for bivariate inverse exponential distribution the log-likelihood value is − 249.6874 and the AIC is 505.3748. Therefore, M-O bivariate exponential provides better fit than bivariate inverse exponential distribution. We estimate the reliability parameter R using the corresponding MLEsθ i θ i , i = 0, 1, 2 for the M-O bivariate exponential distribution is R = 0.0248, while using the proposed simple estimators, we have R S = 0.0235 and R C = 0.0238 and the non-parametric estimator R N = 0.0238.

Conclusions
In this paper, we have suggested two forms of bivariate distributions, BEF and BIEF, with marginal distributions having a general exponential form or inverse exponential form. Some distributions in the literature belong to these families, such as the M-O bivariate exponential distribution, Marshall and Olkin [15], and bivariate Rayleigh distribution, Pak et al. [9]. Other bivariate distributions could belong to these families such as bivariate Weibull and bivariate Burr type III and others according to the form of g 1 (x; c) or g 2 (x; c). We discussed some properties of the proposed families and studied the stress-strength reliability parameter, R = P(Y < X). The MLEs of the distribution parameters are derived and simple estimators of R based on some marginal distributions are introduced in case of complete sampling. When the stress is censored at the strength, an explicit estimator of R is also obtained for the BEF distribution. Some bivariate members of the proposed families are presented. A simulation study is performed showing that the proposed simple estimators of R are easier in computation and provide sufficient results with respect to biasedness and mean square error. An example of a real data of bivariate variables (X, Y) belonging to the proposed family is also introduced.