The Generalized Odd Generalized Exponential Fréchet Model: Univariate, Bivariate and Multivariate Extensions with Properties and Applications to the Univariate Version

A new univariate extension of the Fréchet distribution is proposed and studied. Some of its fundamental statistical properties such as stochastic properties, ordinary and incomplete moments, moments generating functions, residual life and reversed residual life functions, order statistics, quantile spread ordering, Rényi, Shannon and q-entropies are derived. A simple type Copula based construction using Morgenstern family and via Clayton Copula is employed to derive many bivariate and multivariate extensions of the new model. We assessed the performance of the maximum likelihood estimators using a simulation study. The importance of the new model is shown by means of two applications to real data sets.

(2) The cumulative distribution function (CDF) and probability density function (PDF) of the Generalized Odd Generalized Exponential G (GOGE-G) family are given, respectively, by where ( ) is the baseline CDF depending on a parameter vector , ( ) = ( ) is the corresponding PDF and , > 0 are two additional shape parameters. Using (2) and (3) the CDF of the GOGEFr can be derived as where ℰ ; , , = [− ( ) ]and the corresponding PDF of (5) can be expressed as Henceforth, ∼ GOGEFr ( )| = , , , denotes a RV with density function in (6). The hazard rate function (HRF) of can be derived using the well-known relationship where ∼ (0,1). Now, we provide a useful representation for (6). , which holds for | | < 1 and ϒ > 0 real non-integer and using the power series, the PDF of the GOGEFr density in (6) can be expressed as where ℎ * ( ) is the Fr density with scale parameter √ * and shape parameter and parameter √ * and shape parameter . Figure 1 shows that the new density function can take unimodal, symmetric and right skewed shapes. Figure 2 shows that the HRF may be "increasing-constant", "decreasing", "increasing", "upside-down" or "constant" failure rate function. Many useful mathematical tools can be found in Cordeiro and

Mathematical properties 2.1 Moments and cumulants
The rth ordinary moment of is given by Then, we obtain where is the complete gamma function. Setting = 1 in (9), we have the mean of . The last integration can be computed numerically for most parent distributions. The skewness and kurtosis measures can be calculated from the ordinary moments using the well-known relationships. The skewness and kurtosis measures can also be calculated from the ordinary moments using the well-known relationships. The mean, variance, skewness and kurtosis of the GOGEFr distribution are computed numerically for some selected parameter values using the R software. The numerical values displayed in Table 1 indicate that the skewness of the GOGEFr distribution is always positive and can range in the interval (0.23,149.38). The spread of its kurtosis is much larger ranging from 1.085 to 22316.32.

Incomplete moment
The r ℎ incomplete moment, say ( ) , of can be expressed, from (8), as where ( , ) is the incomplete gamma function.
and 1 1 [⋅,⋅,⋅] is a confluent hypergeometric function. The first incomplete moment can be calculated by setting = 1 in ( ) as

The moment generating function (MGF)
The MGF ( ) = ( ) of can be derived from equation (4) Then, the MGF of (1) can be defined as By combining expressions (8) and (10), we obtain the MGF of GOGEFr as Equations (9) and (11) can be easily evaluated by scripts of the Maple, Matlab and Mathematica platforms.

Residual life and reversed residual life functions
The

Entropies
The Rényi entropy of a RV represents a measure of uncertainty and defined by Using the PDF in (6), we obtain Then, the Rényi entropy of the GOGEFr model is given by The q-entropy, say ( ), can be defined as Where is a special case of the Rényi entropy, ( )| ( >0 and ≠1) , when ↑ 1.

Order statistics
be a random sample (RS) from the GOGEFr distribution and let 1: , 2: , … , : be the corresponding order statistics. The PDF of the th order statistic, say : , can be written as where B (⋅,⋅) is the beta function. Substituting (1) and (2) in (12) and using a power series expansion, we have Then, the PDF of : can be expressed as The density function of the GOGEFr order statistics is a mixture of Fr densities. Based on the previous equation, the moments of : can be expressed as

BivGOGEFr-FGM (Type I) model
The

Estimation
Let 1 , … , be a RS from the GOGEFr distribution with parameters , , and . Let Ψ be the 4 × 1 parameter vector. For determining the MLE of , the log-likelihood function is . The score vector components are easy to be derived, setting the nonlinear system of equations = = = 0 and = 0 and solving them simultaneously yields the MLE.

Graphical assessment
Graphically, we can perform the simulation experiments to assess of the finite sample behavior of the MLEs. The assessment was based on the following algorithm: • Use (7) we generate = 10000 samples of size from the GOGEFR distribution; • Compute the MLEs for the 1000 samples; • Compute the SEs of the MLEs for the 1000 samples; • Compute the biases and mean squared errors given for Ψ = , , 1 , 2 . We repeated these steps for = 50,100, … ,150 with = = = = 1 , so computing biases ( ( )) , mean squared errors ( ) ( ℎ ( )) for , , , and = 50,100, … ,150.      Nichols and Padgett (2006). The 2 nd data set consists of 63 observations of the strengths of 1.5 cm glass fibers (see Smith and Naylor (1987)). In order to compare the distributions, we consider the AIC (Akaike Information Criterion), CAIC (Consistent Akaike Information Criterion), BIC (Bayesian Information Criterion) and HQIC (Hannan-Quinn Information Criterion) for comparing models. Total time test (TTT) plot (see Figure 7) is an important graphical approach to verify whether our data can be applied to a specific model. The TTT plots of the two real data sets are presented in Figure 7 (first row). These plots indicate that the empirical HRFs of the two data sets are "increasing HRF". The box plots of the two real data sets are presented in Figure 7 (second row). The normal Q-Q plots of the two real data sets are presented in Figure 7 (third row).         Figures 8-10, respectively, display the plots of estimated CDFs, estimated PDFs and Kaplan-Meier survival plots for the two data sets. These plots reveal that the proposed distribution gives adequate fit for both data sets.

Conclusions
A new extension of the Fréchet model is proposed and studied. Some of its fundamental statistical properties such as, some stochastic properties, ordinary and incomplete moments, moments generating functions, residual life and reversed residual life functions, order statistics, quantile spread ordering, Rényi, Shannon and q-entropies are derived. A simple type Copula based construction via Renyi's entropy Copula Farlie Gumbel Morgenstern Copula, modified Farlie Gumbel Morgenstern Copula, Clayton Copula is employed to derive many bivariate and multivariate extensions of the new model. We assessed the performance of the maximum likelihood estimators using a simulation study. The importance of the new model is shown via two applications of real data sets.