Abstract
The extreme value theory is expanded by proposing and studying a new version of the Fréchet 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, different 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, Cramér–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 different sample sizes for three different 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.
Highlights
The extreme value theory (EVT) was firstly introduced by [1] followed by [2] and completed by [3,4]
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 Mathematics 2020, 8, 1949; doi:10.3390/math8111949
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 odd-Burr generalized Fréchet (OB-F) model
Summary
The extreme value theory (EVT) was firstly introduced by [1] followed by [2] and completed by [3,4]. [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. We expanded the EVT with proposing and studying a new version the F model, called the odd-Burr generalized Fréchet (OB-F) model. We define and study a new Fréchet model based on OB-G family, called generalized odd log-logistic F (OB-F) model. The asymptotes of CDF, PDF, and HRF as z → ∞ are given by h ib 1 − FP (z) ∼ ab 1 − exp −z−θ z→∞ , fP (z) ∼ bab θ and hP ( z ) ∼ For simulation of this new model, we obtain the quantile function (QF) of Z (by inverting FP (z) based on (8)), say zu = h(u) = F−1 (u), as. The 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])
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