A Generalization of Reciprocal Exponential Model: Clayton Copula, Statistical Properties and Modeling Skewed and Symmetric Real Data Sets

We introduce a new extension of the reciprocal Exponential distribution for modeling the extreme values. We used the Morgenstern family and the clayton copula for deriving many bivariate and multivariate extensions of the new model. Some of its properties are derived. We assessed the performance of the maximum likelihood estimators (MLEs) via a graphical simulation study. The assessment was based on the sample size. The new reciprocal model is employed for modeling the skewed and the symmetric real data sets. The new reciprocal model is better than some other important competitive models in statistical modeling.


Introduction
A random variable (RV) is said to have the reciprocal Exponential (RE) distribution if its probability density function (P-D-F) and cumulative distribution function (C-D-F) are given by and where 3 > 0 is a scale parameter. The P-D-F in (1) is a special case from the well-known reciprocal Weibull (RW) model. The RW model is also known as the Fréchet distribution (see Fréchet (1927) ) and also contains the reciprocal Rayleigh (RR) model as a special case. Cordeiro  (4) For 2 = 1 we get the OLL-G family Gleaton and Lynch (2006). For 1 = 1 we get the Proportional reversed hazard rate G family Gupta and Gupta (2007). Here, we define a new RE model based on Cordeiro The P-D-F corresponding to (5) is given by  A for all Tables and Appendix B for all Figures) to illustrate some of its characteristics. For simulation of this new model, we obtain the quantile function (QF) of (by inverting (5)), say = ( ) = −1 ( ), as Equation (7)

The Multivariate extension
A straightforward -dimensional extension IRom the above will be where where ( 1 + 2 ) ≥ 0, , 1 > 0, 2 > 0, 3 > 0. Future works may be allocated for studying the new bivariate and multivariate extensions of the GOLL-RE model.

Properties 3.1 Representations
Based on Cordeiro et al. (2016), the P-D-F in (6) can be expressed as and (1+ 1 ) ( ; 3 ) is the P-D-F of the Fr model with scale parameter 3 (1 + 1 ) and shape parameter . So, the new P-DF in (6) can be expressed as a double linear mixture of the RE P-D-F. Then, several structural properties of the new model can be obtained from Equation (8) and those properties of the RE model. By integrating Equation (8), the C-D-F of becomes

Moments and incomplete moments
The ℎ ordinary moment of is given by The ℎ incomplete moment, say ( ), of can be expressed, from (9), as where ( 1 , 2 ) is the incomplete gamma function.

Residual life and reversed residual life functions
The ℎ moment of the residual life ( ) = [( − ) | > , =1,2,… ] the ℎ moment of the residual life of is given by Therefore, ).
The components of the score vector , whose elements can be computed numerically.
4. compute the biases and mean squared errors given for = 1 , 2 , 3 . We repeated these steps for = 50,100, … ,500 with 1 = 1, 2 = 1, 3 = 1, so computing biases(Bias ( )), mean squared errors (MSEs) (MSE ( )) for 1 , 2 , 3 and = 50,100, … ,500 where The parameters of the above densities are all positive real numbers except for the T-RW distribution for which | 1 | ≤ 1. The 1 st data set is an uncensored data set consisting of 100 observations on breaking stress of carbon fibers (in Gba) given by Nichols and Padgett (2006). Figure 5 gives the total time test (TTT) plot (See Aarset (1987)) for data set I. It indicates that the empirical H-R-Fs of data sets I is increasing. The 2 nd data set is generated data to simulate the strengths of glass fibers which was given by Smith and Naylor (1987). Figure 6 gives the TTT plot for data set II. It indicates that the empirical H-R-Fs of data sets II is increasing.
The 3 rd data set (wingo data) represents a complete sample from a clinical trial describe a relief time (in hours) for 50 arthritic patients. Many other usefull real data sets can be found and analyzed (see Basikhasteh . (2020a, b)). Figure 7 gives the TTT plot for data set III. It indicates that the empirical H-R-Fs of data sets III is increasing.
The statistics (C-V-M, A-D, K-S and P-value) of all fitted models are presented in Table 1, Table 3 and Table 5 for data set I, II and III respectively. The MLEs and corresponding standard errors (SEs) are given in Table 2, Table 4 and Table 6 for data set I, II and III respectively. Figure 8 gives the estimated density (E-P-D-F), Figure 9 gives the estimated C-D-F (E-C-D-F), Figure 10 gives the estimated H-R-F (E-H-R-F) and Figure 8 gives the P-P (P-P) plots. The GOLL-RE distribution in Tables 1, 3 and 5 give the lowest values the C-V-M, A-D, K-S and the biggest value of the P-value statistics as compared to other extensions of the RW models, and therefore the GOLL-RE can be chosen as the best model.

Concluding remarks
We introduce a new extension of the reciprocal Exponential distribution for modeling the extreme values. We used the Morgenstern family and the clayton copula for deriving many bivariate and multivariate extensions of the new model. Some of its mathematical properties are derived. We assessed the performance of the maximum likelihood estimators via a graphical simulation study. Based on the graphical simulations, the biases for each parameter are generally negative and decrease to zero as → ∞, the MSEs for each parameter decrease to zero as → ∞. The assessment was based on the sample size. The new reciprocal model is better than some other important competitive models in modeling the breaking stress data, the glass fibers data and the relief time data.
A Generalization of Reciprocal Exponential Model: Clayton Copula, Statistical Properties and Modeling skewed and symmetric Real Data Sets 381 Appendix: A