The Generalized Odd Log-Logistic Fr´echet Distribution for Modeling Extreme Values

We introduce a new extension of the Fr ´ echet distribution for modeling the extreme values. The new model generalizes eleven distributions at least, ﬁve of them are quite new. Some important mathematical properties of the new model are derived. We assess the performance of the maximum likelihood estimators (MLEs) via a simulation study. The new model is better than some other important competitive models in modeling the breaking stress data, the glass ﬁbers data and the relief time data. data and the relief time data. In our upcoming work, we can apply many new useful goodness-of-ﬁt tests for validation such as


Introduction
A statistical field known as extreme value theory or extreme value analysis (EVA) studies extreme departures from the median of probability distributions. It aims to determine the likelihood of events that are more extreme than any previously recorded events from a given ordered sample of a given random variable. Extreme value analysis is frequently employed in a variety of fields, including geological engineering, finance, earth sciences, and structural and structural engineering. The EVA, for instance, might be used in the hydrology profession to calculate the likelihood of an exceptionally significant flooding occurrence, like the 100-year flood. Similar to this, a coastal engineer would aim to determine the 50-year wave and construct the structure appropriately while designing a breakwater (for more details, see Kotz and Nadarajah (2000), Afify et al. (2016a,b) and Salah et al. (2020)). The Fisher-Tippet-Gnedenko theorem only states that if the distribution of a normalised maximum converges, then the limit has to be one of a specific class of distributions. The role of the extremal types theorem for maxima is similar to that of the central limit theorem for averages, with the exception that the central limit theorem applies to the average of a sample from any distribution with finite variance. It doesn't say that the normalised maximum distribution converges.
The Fisher-Tippett-Gnedenko theorem in statistics is a broad conclusion of the extreme value theory concerning the asymptotic distribution of extreme order statistics. It is also known as the Fisher-Tippett theorem or the extreme value theorem.Only one of three alternative distributions, the Gumbel distribution, the Fréchet distribution, or the Weibull distribution, can be reached by the maximum of a sample of iid random variables after sufficient renormalization. The Fréchet (Fr) model is one of the most important distributions in modeling extreme values. The Fr model was originally proposed by Fréchet (1927). It has many applications in ranging, accelerated life testing, earthquakes, the floods, the wind speeds, the horse racing, the rainfall, queues in supermarkets and sea waves. .Some new Fréchet versions can be cited by Jahanshahi   A RV X is said to have the Fr distribution if its probability density function (PDF) and cumulative distribution function (CDF) are given by and where a > 0 is a scale parameter and b > 0 is a shape parameters. For b = 2.we get the Inverse Rayleigh (IR) model. For a = 1 we get the Inverse Exponential (IEx) model. Recently, Cordeiro et al. (2016) proposed a new class of distributions called the generalized odd log-logistic-G (GOLL-G) family with two extra shape parameters. For an arbitrary baseline CDF G ξ (x), the CDF of the the GOLL-G family is given by The PDF corresponding to (3) is given by For θ = 1 we get the OLL-G family (Gleaton and Lynch (2006)). For α = 1 we get the Proportional reversed hazard rate G family (Gupta and Gupta (2007) The new CDF in (5) (5) is given by The new model in (6) , the asymptotics of the CDF, PDF and HRF as x → τ are given by The asymptotics of CDF, PDF and HRF as x → ∞ are given by    Some plots of the GOLLFr PDF and HRF are given in Figure 1 to illustrate some of its characteristics.
(a) (b) Figure 1: Plots of the GOLLFr PDF and HRF.
For simulation of this new model, we obtain the quantile function (QF) of X (by inverting (5)), say Equation (7) is used for simulating the new model (see Section 5).

Useful representations
Based on generalized binomial expansions and after some algebra, the PDF in (6) can be expressed as and π (1+k) (x; a, b) is the PDF of the Fr model with scale parameter a [(1 + k)] 1 b and shape parameter b. So, the new density (6) can be expressed as a double linear mixture of the Fr density.Then, several structural properties of the new model can be obtained from Equation (8) and those properties of the Fr model. By integrating Equation (8), the CDF of X becomes The Generalized Odd Log-Logistic Fréchet Distribution for Modeling Extreme Values where Π (1+k) (x; a, b) is the CDF of the Fr distribution with scale parameter a (ck) 1 b and shape parameter b. The r th ordinary moment of X is given by then we obtain where Setting r = 1, 2, 3 and 4 in (10), we have where E(X) = µ 1 is the mean of X. The skewness (Skew(X)) and kurtosis (Kur(X)) measures can be calculated via (10) using well-known relationships. E(X), variance (Var(X)), Skew(X) and Kur(X) of the GOLLFr distribution are computed numerically for some selected values of parameter α, θ, a and b using the R software. We conclude that, the Skew (X) can range in the interval (1,19706) and always positive, whereas the Kur(X) varies in the interval (1, 388338). The parameters θ and a don't control neither the Skew(X)) nor the Kur(X) as illustrated below in Table  2, on the other hand parameters α and b control Skew(X)) and Kur(X) . E(X) decreases as α and b increases. E(X) increases as a and θ increases.  The r th incomplete moment, say ϕ r (t), of X can be expressed, from (9), as where γ (ω, q) is the incomplete gamma function.
The Generalized Odd Log-Logistic Fréchet Distribution for Modeling Extreme Values and 1 F 1 [·, ·, ·] is a confluent hypergeometric function. The first incomplete moment given by (11) with r = 1 as

Moment generating function (MGF)
The MGF M X (t) = E e t X of X can be derived from equation (8) as We aslo can determine the generating function of g a,b (x) by setting y = x −1 , the MGF can be written as By expanding the first exponential and calculating the integral, we have Consider the Wright generalized hypergeometric function (Wright (1935))defined by Then, M (t; a, β) can be written as Combining expressions (10) and (12), we obtain the MGF of X, say M (t),as

Residual life and reversed residual life functions
The n th moment of the residual life the n th moment of the residual life of X is given by The Generalized Odd Log-Logistic Fréchet Distribution for Modeling Extreme Values Therefore, The n th moment of the reversed residual life, say uniquely determines F (x). We obtain Then, the n th moment of the reversed residual life of X becomes (−1) r n r t n−r .

Maximum likelihood estimation (MLE)
Let x 1 , . . . , x n be a random sample from the GOLLFr distribution with parameters α, θ, a and b. Let Θ =(α, θ, a, b) be the 4 × 1 parameter vector. For determining the MLE of Θ, we have the log-likelihood function The components of the score vector, , are available if needed. Setting L α = L θ = L a = and L b = 0 and solving them simultaneously yields the MLE Θ = ( α, θ, a, b) . 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 5 × 5 observed information matrix J(Θ) = {∂ 2 /∂r ∂s} (∀r, s = α, θ, a, b), whose elements can be computed numerically.

Simulation studies
We simulate the GOLLFr model by taking n=50, 100, 200, 500 and 1000 using (7). For each sample size, we evaluate the sample means and standard deviations (SDs) using the optim function of the R software. Then, we repeat this  Table 3 indicate that the empirical means approach to the true parameter values when the sample size n increases. The SDs decrease when the sample size n increases as expected. These results are in agreement with first-order asymptotic theory.
where z i = F (y j ) and the y j 's values are the ordered observations. We compare the fits of the GOLLFr distribution with other models such as Fréchet ; TFr : ; McFr : ; OLLEFr: The parameters of the above densities are all positive real numbers except for the TFr distribution for which |α| ≤ 1.

Breaking stress data
The 1st 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) and these data are used by Mahmoud and Mandouh (2013) Figure 2 gives the total time test (TTT) plot for data set I. It indicates that the empirical HRFs of data sets I is increasing.  Table 4. The MLEs and corresponding standard errors are given in Table 5. The GOLLFr distribution in Table 4 gives the lowest values theW , A , K-S and the biggest value of the p-value statistics as compared to other extensions of the Fr models, and therefore the new one can be chosen as the best model. Figure 3 gives the estimated density, estimated CDF, P-P plot and estimated HRF for data set I.    Figure 3: Estimated density, estimated CDF, P-P plot and estimated HRF for data set I

Glass fibers data
The 2nd data set is generated data to simulate the strengths of glass fibers which was given by Smith and Naylor (1987 Figure 4 gives the TTT plot for data set II. It indicates that the empirical HRFs of data sets II is increasing.  The statistics (W , A , K-S and p-value) of the fitted models are provided in Table 6. The MLEs and corresponding standard errors are given in Table 7. From Table 6, the GOLLFr distribution gives the lowest values the W , A , K-S and the biggest value of the p-value statistics as compared to further Fr models, and therefore the new one can be chosen as the best model. Figure 5 gives the estimated density, estimated CDF, P-P plot and estimated HRF for data set II.
The Generalized Odd Log-Logistic Fréchet Distribution for Modeling Extreme Values   Figure 5: Estimated density, estimated CDF, P-P plot and estimated HRF 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.  Figure 6 gives the TTT plot for data set III. It indicates that the empirical HRFs of data sets III is increasing.     Figure 7: Estimated density, estimated CDF, P-P plot and estimated HRF for data set III The statistics (W , A , K-S and p-value) of all fitted models are presented in Table 8. The MLEs and corresponding standard errors are given in Table 9. From Table 8, the GOLLFr distribution gives the lowest values theW , A , K-S and the biggest value of the p-value statistics as compared to further Fr models, and therefore the new one can be chosen as the best model. Figure 5 gives the estimated density, estimated CDF, P-P plot and estimated HRF for data set III.

Concluding remarks
We introduce a new distribution called GOLLFr distribution for modeling the extreme values.