Expanded Fr é chet Model: Mathematical Properties, Copula, Di ﬀ erent Estimation Methods, Applications and Validation Testing

: The extreme value theory is expanded by proposing and studying a new version of the Fr é chet model. Some new bivariate type extensions using Farlie–Gumbel–Morgenstern copula, modiﬁed Farlie–Gumbel–Morgenstern copula, Clayton copula, and Renyi’s entropy copula are derived. After a quick study for its properties, di ﬀ erent non-Bayesian estimation methods under uncensored schemes are considered, such as the maximum likelihood estimation method, Anderson–Darling estimation method, ordinary least square estimation method, Cram é r–von-Mises estimation method, weighted least square estimation method, left-tail Anderson–Darling estimation method, and right-tail Anderson–Darling estimation method. Numerical simulations were performed for comparing the estimation methods using di ﬀ erent sample sizes for three di ﬀ erent combinations of parameters. The Barzilai–Borwein algorithm was employed via a simulation study. Three applications were presented for measuring the ﬂexibility and the importance of the new model for comparing the competitive distributions under the uncensored scheme. Using the approach of the Bagdonavicius–Nikulin goodness-of-ﬁt test for validation under the right censored data, we propose a modiﬁed chi-square goodness-of-ﬁt test for the new model. The modiﬁed goodness-of-ﬁt statistic test was applied for the right censored real data set, called leukemia free-survival times for autologous transplants. Based on the maximum likelihood estimators on initial data, the modiﬁed goodness-of-ﬁt test recovered the loss in information while the grouping data and followed chi-square distributions. All elements of the modiﬁed goodness-of-ﬁt criteria tests are explicitly derived and given. Simulations were performed by comparing the estimation methods using di ﬀ erent sample sizes for three di ﬀ erent combinations of parameters. The Barzilai–Borwein algorithm is employed via simulation studies, assessing the performance of the estimators with di ﬀ erent sample sizes as sample size tends to ∞ . Three applications for measuring the ﬂexibility and the importance of the new model are presented and used for comparing the competitive distributions under the uncensored scheme. Using the approach of the Bagdonavicius–Nikulin goodness-of-ﬁt test for validation under the right censored data, we propose a modiﬁed chi-square goodness-of-ﬁt test for the new model. The modiﬁed goodness-of-ﬁt statistic test is applied for right censored real data set called leukemia free-survival times (in months) for autologous transplants. Based on the maximum likelihood estimators on initial data, the modiﬁed goodness-of-ﬁt test recovers the loss of information. All elements of the modiﬁed GOF criteria tests are explicitly derived and given.


Introduction
The extreme value theory (EVT) was firstly introduced by [1] then followed by [2] and completed by [3,4]. The EVT is devoted to stochastic series of independent and identical distributed (iid) random variables (RVs). The EVT was originally devoted toward studying the behavior of extreme values (EVs), even though these extremes have a very low chance to appear, these EVs can turn out to have a very high effect to the observed experiment. Insurance and finance are the best fields of research to observe the importance of extreme events in EVT. The EVT started in the last century as an equivalent theory to the central limit theory (CLT), which was devoted toward studying the asymptotic distribution of the average of a sequence of RVs. Assume that Z 1 , Z 2 , . . ., Z n is a sequence of iid-RVs with common cumulative distribution function (CDF) F(·). One of the most interesting statistics is the sample maximum S 1:n = max{Z 1 , Z 2 , . . . , Z n }. One is interested in the behavior of M 1:n as the sample size n increases to infinity.
Then, for any non-degenerate CDF, it will belong to one of the three main types of classic extreme value families, which include the Gumbel, Fréchet, and Weibull distributions. The Fréchet (F) model is one of the most important distributions in modeling extreme values. The F model was originally proposed by [1]. It has many applications, in ranging, accelerated life testing, earthquakes, floods, wind speed, horse racing, rainfall, queues in supermarkets, and sea waves. One can find more details about the F model in the literature, for example [5] investigated the exponentiated F distribution. Moreover, [6] discussed the odd Chen F random variables (RVs), [7] defined and applied a new version of the F distribution for relief times and survival times data. Eliwa et al. [8] proposed the exponentiated odd Chen F distribution, [9] proposed some applications of the Marshall-Olkin F (MO-F) distribution, [10] investigated the odd flexible Weibull F distribution, among others. Many other useful F model extensions are defined and studied by [11][12][13][14][15][16].
In this article, we expanded the EVT with proposing and studying a new version the F model, called the odd-Burr generalized Fréchet (OB-F) model. The new model is derived based on compiling the standard F model with the odd-Burr generalized (OB-G) family (see [17]). Many new bivariate type extensions using the Farlie-Gumbel-Morgenstern (FGM) copula, modified FGM copula, Clayton copula, and Renyi's entropy copula, are derived. The bivariate type extensions may e useful in modeling the bivariate real data sets.
After a quick studying for its properties, different classical estimation methods under uncensored schemes are considered, such as the maximum likelihood estimation (MLE) method, Anderson-Darling estimation (ADE) method, ordinary least square estimation (OLSE) method, Cramér-von Mises estimation (CVME) method, weighted least square estimation (WLSE) method, left-tail Anderson-Darling estimation (LT (ADE) ) method, and right-tail Anderson-Darling estimation (RT (ADE) ) method. Numerical simulations are performed for comparing the estimation methods using different sample sizes for the three different combinations of parameters. The Barzilai-Borwein (BB) algorithm is employed via a simulation study for assessing the performance of the estimators with different sample sizes, as sample size tends to ∞. Three applications for measuring the flexibility and the importance of the OB-F model are presented and are also used for comparing the competitive distributions under the uncensored scheme. Using the approach of the Bagdonavicius-Nikulin goodness-of-fit (BN-GOF) test for validation under the right censored data, we propose a modified chi-square GOF tests for the OB-F model. The modified GOF statistic test is applied for the right censored real data set, called leukemia free-survival times (in months) for autologous transplants. Based on the maximum likelihood estimators (MLEs) on initial data, the modified BN-GOF test recovers the loss in information, while the grouping data follow chi-square distributions. All elements of the modified BN-GOF criteria tests are explicitly derived and given.
A RV Z is said to have the F distribution if its probability density function (PDF) and CDF are given by Mathematics 2020, 8,1949 3 of 29 and G θ (z) = exp −z −θ z≥0 , (5) where θ > 0 is a shape parameter. For θ = 2 we get the Inverse Rayleigh (IR) model. Due to [17], the CDF of the OB-G family is given by where G Φ (z) = 1 − G Φ (z). The PDF corresponding to (6) is given by For b = 1, the OB-G family reduces to the Odd G (O-G) family (see [18]). For a = 1, the OB-G family reduces to the proportional reversed hazard rate (PRHR) family (see [19]). In this work, we define and study a new Fréchet model based on OB-G family, called generalized odd log-logistic F (OB-F) model. The OB-F survival function (SF) is given by where S P (z) = 1 − F P (z) (P=a,b,θ) . For b = 1, the OB-F reduces to the O-F. For a = 1, the OB-F reduces to the PRHR-F. The PDF corresponding to (8) is given by The hazard rate function (HRF) for the new model can be given from f P (z)/S P (z). Let c = inf z| G(z;Φ)>0 , the asymptotics of the CDF, PDF, and HRF as z → c are given by and The asymptotes of CDF, PDF, and HRF as z → ∞ are given by and Mathematics 2020, 8, 1949 4 of 29 For simulation of this new model, we obtain the quantile function (QF) of Z (by inverting F P (z) based on (8)), say z u = h(u) = F −1 (u), as Equation (16) is used for simulating the new model (see Section 5). The now OB-F model can be used in modeling extreme data, such as extreme floods, maximum sizes of ecological populations, the size of freak waves, the amount of large insurance losses, equity risks, day-to-day market risk, side effects of drugs (e.g., ximelagatran), survival times, large wildfires, repair data, and estimate fastest time of running (e.g., 100 m sprint) (see [11][12][13][14][15][16]).

Copula
For modeling the bivariate real data sets, we can consider the bivariate OB-F type generated via the FGM copula, modified FGM copula, Clayton copula, and Renyi's entropy copula. Many other types of copula could be considered in separate works. In this section, we derive a new bivariate type OB-F (BOB-F) model using the theorems of FGM Copula, modified FGM Copula, Clayton Copula and Renyi's entropy. The Multivariate OB-F (MvOB-F) type is also presented. However, future works may be allocated to study these new models. First, we consider the joint CDF of the FGM family, where C ρ (m, w) = mw(1 + ρm * w * ) m * =1−m , where the marginal function m = F 1 , w = F 2 , ρ ∈ (−1, 1) is a dependence parameter and for every m, w ∈ (0, 1), C(m, 0) = C(0, m) = 0 which is "grounded minimum" and C(m, 1) = m and C(1, w) = w which is "grounded maximum",

Via FGM Copula
Due to [20][21][22][23], a Copula is continuous in m and w where is the stronger Lipschitz condition. For 0 ≤ m 1 ≤ m 2 ≤ 1 and 0 ≤ w 1 ≤ w 2 ≤ 1, we have Then, setting We can easily obtain the joint CDF of the FGM family from The joint PDF can then be derived from or from
Type-II Let ∨(m) * and ∧(w) * be two functional forms satisfying all the conditions stated earlier where Then, the corresponding BOB-F-FGM (Type-II) can be derived from Type-III and In this case, one can also derive a closed form expression for the associated CDF of the BOB-F-FGM (Type-III) from C ρ (m, w) = mw 1 + ρ ..

Type-IV
The CDF of the BOB-F-FGM (Type-IV) model can be derived from where F −1 P 1 (m) and F −1 P 2 (w) can be derived using (16).

Via Clayton Copula
The Clayton Copula can be considered as Setting w 1 ∈ (0, 1) = F P 1 (t) and w 2 ∈ (0, 1) = F P 2 (x). Then, the BOB-F type can be derived from C(w 1 , w 2 ) = C F P 1 (t), F P 2 (x) . Similarly, the MvOB-F (m-dimensional extension) from the above can be derived from

Moments and Incomplete Moments
The ρ th ordinary moment of Z is given by then we obtain where ∆ (ρ,θ) κ where E(Z) = µ 1 is the mean of Z. The ρ th incomplete moment, say φ ρ (τ), of Z can be expressed, from (42), as where γ(δ, ρ) is the incomplete gamma function.
The first incomplete moment given by (11) with ρ = 1 as

Residual Life and Reversed Residual Life Functions
The h th moment of the residual life can be derived using Z>τ, h=1,2,...

. (52)
Then, the h th moment of the residual life of Z is given also by Therefore, using (40), we have where ∆ Analogously, the h th moment of the reversed residual life is Z≤τ, τ>0 and h=1,2,...
uniquely determines F(z). We obtain W h (τ) = 1 . Then, the h th moment of the reversed residual life of Z becomes where ∆

The MLE Method
Let z 1 , z 2 , . . . , z h be a random sample from size h from the OB-F distribution with parameters a, b and θ. Let P T be the 3 × 1 parameter vector. For determining the MLE of P, we have the log-likelihood function Mathematics 2020, 8,1949 10 of 29 The components of the score vector is available if needed. Setting L a = L b = L θ = 0 and solving them simultaneously yields the MLE, where L s = ∂ /∂s. To solve these equations, it is usually more convenient to use nonlinear optimization methods, such as the quasi-Newton algorithm to numerically maximize . For interval estimation of the parameters, we obtain the 3 × 3 observed information matrix whose elements can be computed numerically.

The CVME Method
The CVME of the parameters P are obtained via minimizing the following expression with respect to (WRT) to the parameters P respectively, where where c [1] (h,i) = 2i−1 2h and Then, the CVME of the parameters P are obtained by solving the two following non-linear equations and where

OLS Method
Let F P z [i,h] denotes the CDF of OB-F model and let z 1 < z 2 < · · · < z h be the h ordered RS. The OLSEs are obtained upon minimizing then, we have where c [2] (h,i) = i h+1 .The LSEs are obtained via solving the following non-linear equations where

WLSE Method
The WLSE is obtained by minimizing the function WLSE(P) WRT P where c The WLSEs are obtained by solving and where

The ADE Method
The ADE are obtained by minimizing the function where The parameter estimates follow by solving the nonlinear equations The RT (ADE) are obtained by minimizing The parameter estimates follow by solving the nonlinear equations

The LT (ADE) Method
The LT (ADE) are obtained by minimizing The parameter estimates follow by solving the nonlinear equations

Simulation Studies for Comparing Estimation Methods under Uncensored Scheme
A numerical simulation is performed in to compare the classical estimation methods. The simulation study is based on N = 1000 generated data sets from the OB-F version where n = 50, 100, 300, and 500, and under the following three scenarios: Scenario 1 : (a = 0.80, b = 0.80, θ = 0.80), Scenario 2 : (a = 1.25, b = 0.80, θ = 0.50) and Scenario 3 : Figures 1a, 1b and 1c gives the density functions for three scenarios. The estimates are compared in terms of 1.
The mean of the absolute difference between the theoretical and the estimates D (abs) ; and 4. The maximum absolute difference between the true parameters and estimates D (max) . compared in terms of 1. Bias (BIAS ( ) ) ; 2. Root mean-standard error (RMSE ( ) ) ; 3. The mean of the absolute difference between the theoretical and the estimates ( (abs) ); and 4. The maximum absolute difference between the true parameters and estimates ( (max) ).
The BIAS (P) (BIAS(a), BIAS(b) and BIAS(θ)) tends to 0 when n increases and tends to ∞, which means that all estimators are non-biased.

4.
The worst estimation method cannot be determined obviously since all other estimation methods perform well, especially when n increases and tends to ∞, where in Tables 3-5 we have RT = RT (ADE) and LT = LT (ADE) (81) Table 5. Simulation results for parameters a = 0.5, b = 0.75, and θ = 1.5. Overall, MLE is providing better estimation compared to other methods for all sample sizes and for all initials "a = b = θ = 0.8", "a = 1.25, b = 0.8 and θ = 0.5", and "a = 0.5, b = 0.75 and θ = 1.5".

Modeling Uncensored Real Data for Comparing the Competitive Models
For illustrating the wide applicability of the new OB-F model, we consider the Cramér-von Mises (C1) statistic, the Anderson-Darling (C2) statistic, the Kolmogorov-Smirnov (K-S) statistic, and its corresponding p-value (P − V). Table 6 below gives the competitive models.

Stress Data
The first data set is an uncensored data set consisting of 100 observations on breaking stress of carbon fibers (in Gba) given by [31] and these data are used by [32][33][34][35]. For exploring the empirical HRF of data set I, the total time test (TTT) (see [36]) plot is plotted in Figure 2a. Due to Figure 2a, the empirical HRF of data sets I is "monotonically increasing". For exploring the initial shape of real data set I, the nonparametric Kernel density estimation (KDE) is provided in Figure 2b. In Figure 2b, it is noted that the nonparametric density of data I is "asymmetric right skewed with heavy tail".
For exploring the extreme value, the box plot is plotted in Figure 2c. Based on Figure 2c, we note that some extreme values were found and the quantile-quantile (Q-Q) plot (Figure 2d) confirms this fact. The statistics C1, C2, K-S, and P-V for all fitted models are presented in Table 7. The MLEs and corresponding standard errors (SEs) are given in Table 8. From Table 7, the OB-F model gives the lowest values the C1, C2, K-S and the biggest P-V statistics (bold values) as compared to further F models, therefore, the OB-F can be chosen as the best model. Figures 3a, 3b, 3c and 3d give the estimated (E-PDF), estimated CDF (E-CDF), probability-probability (P-P) plot and estimated HRF (E-HRF), respectively, for data set I. Based on Table 8

Glass Fibers Data
The second data set is generated data to simulate the strengths of glass fibers, given by [37]. For exploring the empirical HRF of data set II, the TTT plot is plotted in Figure 4a. Due to Figure 4a, the empirical HRF of data sets II is "monotonically increasing". For exploring the initial shape of real

Glass Fibers Data
The second data set is generated data to simulate the strengths of glass fibers, given by [37]. For exploring the empirical HRF of data set II, the TTT plot is plotted in Figure 4a. Due to Figure 4a, the empirical HRF of data sets II is "monotonically increasing". For exploring the initial shape of real data II non-parametrically, the KDE is provided in Figure 4b. In Figure 4b, it is noted that the nonparametric density of data II is "asymmetric right skewed with heavy tail". For exploring the extreme value, the box plot is plotted in Figure 4c. Based on Figure 4c, we note that some extreme values were found and the Q-Q plot (Figure 4d) confirms this fact. The C1, C2, K-S, and P-V are listed in Table 9. The MLEs and SEs are given in Table 10. From Table 9, the OB-F model gives the lowest values the C1, C2, K-S, and the biggest value of the P-V, therefore, the OB-F can be chosen as the best model. Figures 5a, 5b, 5c and 5d gives the E-PDF, E-CDF, P-P plot, and E-HRF for data set II. Based on Table 10 and for a = 7.5535, b = 0.4841 and θ = 1.1588 we have, mean = 1.611337, variance = 0.2134206, skewness = 4.838406 > 0 and kurtosis = 232.3354 > 3. Table 9. C1, C2, K-S, and P-V for data set II.  Figure 4c. Based on Figure 4c, we note that some extreme values were found and the Q-Q plot (Figure 4d) confirms this fact. The C1, C2, K-S, and P-V are listed in Table 9. The MLEs and SEs are given in Table 10. From Table 9, the OB-F model gives the lowest values the C1, C2, K-S, and the biggest value of the P-V, therefore, the OB-F can be chosen as the best model. Figure 5a, Figure 5b, Figure 5c and Figure 5d gives the E-PDF, E-CDF, P-P plot, and E-HRF for data set II. Based on Table 10 and for a = 7.5535, b = 0.4841 and θ = 1.1588 we have, mean = 1.611337, variance = 0.2134206, skewness = 4.838406 > 0 and kurtosis = 232.3354 > 3.   Table 9. C1, C1, K-S, and P-V for data set II.

Relief Time Data
The 3rd data set (Wingo data) represents a complete sample from a clinical trial describe a relief time (in hours) for 50 arthritic patients [38]. For exploring the empirical HRF of data set III, the TTT plot is plotted in Figure 6 a. Due to Figure 6a the empirical HRF of data sets III is "monotonically increasing". For exploring the initial shape of real data III, the nonparametric KDE is provided in Figure 6b. Figure 6b it is noted that the nonparametric density of data III is "nearly symmetric". The box plot is presented in Figure 6c. Based on Figure 6c, we note no extreme values were found and the Q-Q plot (Figure 6d) confirms this fact. The C1, C2, K-S, and P-V are listed in Table 11. The MLEs and SEs are given in Table 12. From Table 11, the OB-F model gives the lowest values the C1, C2, K-S and the biggest value of the P-V, therefore the OB-F can be chosen as the best model. Figure 7a, Figure 7b, Figure 7c and Figure 7d gives the E-PDF, E-CDF, P-P plot and E-HRF for data set III. Based on Table 12 and for a = 29.50356, b = 0.64127 and θ = 0.14316 we have, mean = 16.20257, variance = 67.31756, skewness = 9.75092> 0 and kurtosis = 1520.895> 3.

Relief Time Data
The 3rd data set (Wingo data) represents a complete sample from a clinical trial describe a relief time (in hours) for 50 arthritic patients [38]. For exploring the empirical HRF of data set III, the TTT plot is plotted in Figure 6 a. Due to Figure 6a the empirical HRF of data sets III is "monotonically increasing". For exploring the initial shape of real data III, the nonparametric KDE is provided in Figure 6b. Figure 6b it is noted that the nonparametric density of data III is "nearly symmetric". The box plot is presented in Figure 6c. Based on Figure 6c, we note no extreme values were found and the Q-Q plot (Figure 6d) confirms this fact. The C1, C2, K-S, and P-V are listed in Table 11. The MLEs and SEs are given in Table 12. From Table 11, the OB-F model gives the lowest values the C1, C2, K-S and the biggest value of the P-V, therefore the OB-F can be chosen as the best model. Figures 7a, 7b, 7c and 7d gives the E-PDF, E-CDF, P-P plot and E-HRF for data set III. Based on Table 12 and for a = 29.50356, b = 0.64127 and θ = 0.14316 we have, mean = 16.20257, variance = 67.31756, skewness = 9.75092> 0 and kurtosis = 1520.895> 3.
Q-Q plot (Figure 6d) confirms this fact. The C1, C2, K-S, and P-V are listed in Table 11. The MLEs and SEs are given in Table 12. From Table 11, the OB-F model gives the lowest values the C1, C2, K-S and the biggest value of the P-V, therefore the OB-F can be chosen as the best model. Figure 7a, Figure 7b, Figure 7c and Figure 7d gives the E-PDF, E-CDF, P-P plot and E-HRF for data set III. Based on Table 12 and for a = 29.50356, b = 0.64127 and θ = 0.14316 we have, mean = 16.20257, variance = 67.31756, skewness = 9.75092> 0 and kurtosis = 1520.895> 3.

Maximum Likelihood Estimation for Censored Data
In reliability studies and survival analysis, data are often censored. If 1 , 2 , . . . . . , ℎ is a

Maximum Likelihood Estimation for Censored Data
In reliability studies and survival analysis, data are often censored. If Z 1 , Z 2 , . . . . ., Z h is a censored sample from the OB-F distribution, each observation can be written as where Z i are failure times and C i censoring times. The likelihood function is The right censoring is assumed to be non-informative, so the log-likelihood function can be written as: we pose The MLEs of the unknown parameters a, b and θ are derived from the nonlinear following score equations: The explicit form of cannot be obtained, so we use numerical methods.

Test Statistic for Right Censored Data
Let Z 1 , Z 2 , . . . . ., Z h be h i.i.d. random variables grouped into r classes I j . To assess the adequacy of a parametric model H 0 : Pr(Z i ≤ z | H 0 ) = F 0 (z; P), z ≥ 0 (P=(P 1 ,P 1 ,...,P s ) T ∈P⊂R s ) , when data are right censored and the parameter vector P is unknown, [39] proposed a statistic test T 2 based on the vector j=1,2,...,r and r s) .
This one represents the differences between observed and expected numbers of failures (V j and e j ) to fall into these grouping intervals where τ is a finite time. The authors considered ρ j as random data functions such as the r intervals chosen to have equal expected numbers of failures e j . The statistic test T 2 is defined by where e j and V j are the expected and the observed numbers of failure in grouping intervals, other elements were defined in [38][39][40][41][42][43][44], where The limits a j of r random gouging intervals I j = a j−1 , a j were chosen, such as the expected failure times to fall into these intervals, which were the same for each; j = 1, . . . , r − 1,â r = max x (l) , τ . The estimatedρ j is defined bŷ where HP(x l ) is the cumulative HRF (CHRF) of the new distribution. This statistic test Y 2 n follows a chi-squared distribution. To calculate the quadratic form Q of the statistic Y 2 n , and, as its distribution doesn't depend on the parameters, we can use the estimated matricesŴ andĈ and the estimated information matrixÎ. The elements ofĈ defined bŷ

Criteria Test for OB-F
For testing the null hypothesis H 0 that data belong to the OB-F model, we construct a modified chi-squared type goodness-of-fit test based on the statistic T 2 . Suppose that τ is a finite time, and observed data are grouped into r > s sub-intervals I j = ρ j−1 , ρ j of [0, τ]. The limit intervals ρ j are considered as random variables, such that the expected number of failures in each interval I j are the same, so the expected number of failures e j are obtained as Therefore, the quadratic form of the test statistic can be obtained easily: where matricesŴ,Ĉ and the estimated information matrixÎ are derived as given in [40][41][42][43][44][45]. The matrix C is defined in (99) and its elements areĈ 1 j ,Ĉ 2 j andĈ 2 j are numerically evaluated in Section 8.

Simulations
An important simulation study is carried out to show the performance of the techniques used and the feasibility of the goodness-of-fit test developed in this work. At this end, we generated N = 10, 000 right censored samples with different sizes (h = 15, 25, 50, 130, 350, 500) from the OB-F model with parameter values (a = 2, b = 0.5, θ = 1.5). Firstly, we compute the MLEs of the unknown parameters, and then the criteria T 2 of the corresponding samples are provided. Using R statistical software and the BB algorithm (see [46]), we calculate the MLEs of the unknown parameters, the corresponding bias and mean square errors (MSE). The BB algorithm is recently applied by [47][48][49][50]. The results are given in Table 13. For testing the null hypothesis H 0 that right censored data become from OB-F model, we compute the criteria statistic T 2 h (P) as defined above for 10, 000 simulated samples from the hypothezised distribution with different sizes (h = 50, 130, 350, 500).Then, we calculate empirical levels of significance, when T 2 > χ 2 ε (r), correponding to theoretical levels of significance (ε = 0.10, 0.05, 0.01), We choose r = 5. The results are reported in Table 14. The null hypothesis H 0 for which simulated samples are fitted by OB-F distribution, is widely validated for the different levels of significance. Therefore, the test proposed in this work, can be used to fit data from this new distribution.

Application to Leukemia Free-Survival Times
We consider sample data of 51 patients with advanced acute myelogenous leukemia reported to the International Bone Marrow Transplant Registry. These patients had received an autologous (auto) bone marrow transplant in which, after high doses of chemotherapy, their own marrow was reinfused to replace their destroyed immune system. Leukemia free-survival times ( Data are grouped into r = 5 intervals I j . We give the necessary calculus in the following Table 15. Then we obtain the value of the statistic test Y 2 h : For significance level ε = 0.05, the critical value χ 2 5 = 11.0705 is superior than the value of Y 2 h = 8.0231, so we can say that the proposed OB-F model fit these data.

Concluding Remarks
In this article, we expanded the extreme value theory by proposing and studying a new version of the Fréchet model called the odd-Burr generalized Fréchet model. The new model is based on compiling the standard Fréchet model with the odd-Burr generalized family. Many new bivariate type extensions using Farlie-Gumbel-Morgenstern Copula, modified Farlie-Gumbel-Morgenstern, Clayton Copula, and Renyi's entropy Copula are derived. After a quick study for its properties, different classical estimation methods under an uncensored scheme are considered, such as the maximum likelihood estimation method, Anderson-Darling estimation method, ordinary least square estimation method, Cramér-von Mises estimation method, weighted least square estimation method, left-tail Anderson-Darling estimation method, and right-tail Anderson-Darling estimation method.
Simulations were performed by comparing the estimation methods using different sample sizes for three different combinations of parameters. The Barzilai-Borwein algorithm is employed via simulation studies, assessing the performance of the estimators with different sample sizes as sample size tends to ∞. Three applications for measuring the flexibility and the importance of the new model are presented and used for comparing the competitive distributions under the uncensored scheme. Using the approach of the Bagdonavicius-Nikulin goodness-of-fit test for validation under the right censored data, we propose a modified chi-square goodness-of-fit test for the new model. The modified goodness-of-fit statistic test is applied for right censored real data set called leukemia free-survival times (in months) for autologous transplants. Based on the maximum likelihood estimators on initial data, the modified goodness-of-fit test recovers the loss of information. All elements of the modified GOF criteria tests are explicitly derived and given.