A Copula Based Investigation of Reliability for the Multivariate Exponential Family of Distributions

This paper explores the possibility for using the copulas in the context of evaluating the reliability for the exponential family of distributions.

The exponential family of distributions have many applications in our lives. For example, our life span follows the exponential pattern. The life-time of electronic gadgets follow the exponential pattern. The exponential family of distributions are used in both statistical and engineering quality control analysis. These distributions are used in reliability evaluations. There has been steady interest in the reliability evaluations since the time of Marshall and Olkin Failure Model (1966). Bemis (1971) has chronicled the literature with regard to the exponential distribution prior to 1971. Ristic and Kundu (2015) summarize the literature about the research work done after 1971. Recently, the focus is on the generalized exponential family of distributions. However, there is very little known work related to copulas in the context of exponential distributions. This article somewhat seeks to fill the void in the context of exponential distributions.
The idea of using copulas to approximate the joint probability distributions originated after Sklar's Theorem (1959). Gumbel (1960), Clayton (1978) and Joe (1993) developed Copula models bearing their names. There were a sudden influx of research papers beginning from the 1980's in Economics, Finance, and Engineering. Frees and Valdez (1998) discuss in detail the copula construction and the applications in topics such as quantile regression, actuarial science and stochastic ordering. The interested readers are referred to Nelson (1999) for the literature review.
In this paper, we investigate the reliability in the context of exponential families by using the Copula models. We are interested in identifying the Copula model which yields a better approximation for the joint distribution. In this regard, we begin our investigation first with the Bivariate Exponential distribution. Next, we extend it to the Tri-variate situation. Overall, the Clayton Copula model seemed to perform better while approximating the Bivariate and Tri-variate Exponential distributions. Moreover, there is no difference whether we use the arithmetic mean or geometric mean or harmonic mean of the dependence parameter based on the pairwise copulas for the dependence parameter in the three variable situation.

Methodology
Here, we will use the Archimedean Copulas such as Clayton, Gumbel, and Frank and the non-Archimedean Copula such as the Gaussian Copula for modeling the joint distribution.

Bivariate Exponential Distribution
First, we will look to see the how the bivariate-exponential arises in nature. Let us consider the situation where there are three independent exponential variables such that


Next, we will derive the joint distribution of The theoretical reliability can be shown to be Next, we will use some Copula models for computing the reliability.

Note: As previously indicated
X is the stress endured by a glass panel and Y is the strength of the glass material.

Clayton Copula
The Clayton Copula is defined as and the Copula density is For the Clayton Copula, we can estimate the dependence parameter  by using the equation So, the theoretical reliability Where n means the number of very small partitions of the interval   0, y . (13)

Gumbel Copula
The Gumbel Copula is defined as and the Copula density is For the Gumbel Copula, we can estimate the dependence parameter  by using the equation So, again, the theoretical reliability Where n means the number of very small partitions of the interval   0, y .

Farlie-Gumbel Morgenstern (FGM) Copula
The Farlie-Gumbel Morgenstern Copula is defined as For the Farlie-Gumbel Morgenstern Copula, we can estimate the dependence parameter  by using the equation, The reliability Where n means the number of very small partitions of the interval   0, y

Frank Copula
The Frank Copula is defined as Note: In order to use the copula based approximations, we used the following results. , , Where n means the number of very small partitions of the interval   0, y

Frank Copula:
The Frank Copula is defined as  x y e x y x y x Gumbel Copula, we can estimate the dependence parameter  by using the equation ( 1 ) Where n means the number of very small partitions of the interval   0, y

Farlie-Gumbel Morgenstern (FGM) Copula
The Farli-Gumbel Morgenstern Copula is defined as For the Farlie-Gumbel Morgenstern Copula, we can estimate the dependence parameter  by using the equation, The reliability   R P X Y   is approximated by the numerical integral as in the other copulas. Again note that in the context of bivariate normal distribution we are using the parameters,   As you can see from the numerical results, the Archimedean Copulas such as Clayton, Gumbel, Frank, and Farlie-Gumbel Morgenstern (FGM) are not suitable for computing the reliability based on a bivariate normal distribution.
Next, we will investigate the tri-variate exponential distribution.

Tri-variate Exponential Distribution
In this section, we deal with the tri-variate exponential distribution.
Let us define the following independent variables as follows First, we will discuss this in the context of Clayton Copula.

 .
i Simple Reliability (probability) based approach where 2 1      and  is as described above in equation (51).
For other approaches listed below, in the case of Clayton Copula ii  is the harmonic mean of 12 23 13 ,, iii  is the geometric mean of 12 23 13 ,, iv  is the arithmetic mean of 12 23 13 ,, Similarly, we propose methods to estimate the dependence parameter  . We will discuss this in the context of Gumbel Again along the same lines, we propose methods to estimate the dependence parameter  . We will discuss this in the context of Frank Copula.

 .
i Simple Reliability (probability) based approach where is the "Debye" function and For other approaches given below, we will use the correlation coefficient ii  is the harmonic mean of 12 23 13 ,, iii  is the geometric mean of 12 23 13 ,, iv  is the arithmetic mean of 12 23 13 ,,    .

Three variate Pairwise Hierarchical Copula
As seen from the accompanying hierarchical copula diagram, at the top level, the generator function is 1  . At the next level, the generator is 12  . There is a hierarchy in the level arrangement. Note that 1  and 12  are the dependence parameters at the first level and second level respectively. The variables that exhibit the higher order of correlation are placed at the higher level.
So, one can write the hierarchical copula as follows.

Numerical Simulation
Here, we compare the performance of Clayton, Gumbel, and Frank Copulas in approximating the Tri-variate Exponential Distribution. The results are based on 10000 simulation runs.
In order to compare the performance, we will use absolute percentage error where xyz F represents the cumulative distribution of the Tri-variate Exponential and C is the Copula. Table 3

Discussion and Conclusion
At first this paper studied the use of Bivariate Copulas in the context of approximating the Bivariate Exponential distribution and the Bivariate Normal distribution. The Archimedean Copulas did reasonably well in approximating the Bivariate Exponential distribution while not doing well with respect to the Bivariate Normal distribution. So, we decided to use the Archimedean Copulas to approximate the Tri-variate Exponential distribution. In Archimedean Copula constructions in the context of higher dimensions, the question arises as to how one should estimate the dependence parameter. This paper aims to seek an answer for this question based on some commonly used Archimedean Copula models such as Clayton, Gumbel, and Frank models in the case of a three dimensional problem. Overall, the Clayton Copula model is seen to perform better in approximating the Tri-variate Exponential distribution. As you can see from the numerical results for the absolute percentage error, Clayton Copula gives the smallest percentage error.
Moreover, there is not much of a difference between the absolute percentage error calculated based on the dependence parameter estimates using the arithmetic mean or geometric mean or harmonic mean. The dependence parameter estimate based on the probability   P X Y Z   yields a high error rate and therefore should not be recommended. Also, the use of hierarchical copulas should be avoided as these also yield high error rates.