Al-Essa Laila A, Saboor Abdus, Tahir Muhammad H, Khan Sadaf, Jamal Farrukh, Elhassanein Ahmed
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia.
Institute of Numerical Sciences, Kohat University of Science & Technology, Kohat 26000, Pakistan.
Heliyon. 2024 Apr 17;10(8):e27659. doi: 10.1016/j.heliyon.2024.e27659. eCollection 2024 Apr 30.
The premise of extreme value theory focuses on the stochastic behaviour and occurrence of extreme observations in an event that is random. Traditionally for univariate case, the behaviour of the maxima is described either by the types-I, types-II or types-III extreme value distributions, primarily known as the Gumbel, Fréchet or reversed Weibull models. These are all particular cases of the generalized extreme value ( ) model. However, in real-world scenario, these incidents take place as a consequence of concurrent dependent random events, where the relationship between the two variables is unidirectional or asymmetrical. [1] introduced a rigorous univariate extension of distribution involving an additional parameter, the - generalized extreme value ( ) distribution, as well as the - Gumbel distribution. The prime interest of this paper lies in conceptualizing a novel approach to model bi-variate ( ) data, arising naturally from independent random variables. This is achieved via the transformation of variables technique by establishing the resulting supports. Concisely, a technique is developed to model interdependent bivariate observations consisting of extreme values in terms of probability density functions. Besides, we employed the suggested technique to a bivariate flood data set and demonstrate the competitiveness of the proposed bivariate . Additionally, conventional method to propose the newly defined bivariate ( ) distribution with bivariate - Gumbel distribution (a special case for ) has also been established with related inferences and application to climate data.
极值理论的前提聚焦于随机事件中极端观测值的随机行为和出现情况。传统上对于单变量情形，最大值的行为由I型、II型或III型极值分布来描述，主要被称为耿贝尔、弗雷歇或逆威布尔模型。这些都是广义极值（GEV）模型的特殊情况。然而，在现实世界场景中，这些事件是由并发的相关随机事件导致的，其中两个变量之间的关系是单向或不对称的。[1]引入了一种严格的单变量GEV分布扩展，涉及一个额外参数，即ξ - 广义极值（ξ - GEV）分布，以及ξ - 耿贝尔分布。本文的主要兴趣在于构思一种新颖的方法来对由独立的ξ随机变量自然产生的双变量（ξ，ξ）数据进行建模。这是通过变量变换技术并建立所得支撑来实现的。简而言之，开发了一种技术，根据概率密度函数对由极值组成的相互依赖的双变量观测值进行建模。此外，我们将所建议的技术应用于一个双变量洪水数据集，并展示了所提出的双变量ξ的竞争力。另外，还建立了用双变量ξ - 耿贝尔分布（ξ的一种特殊情况）来提出新定义的双变量（ξ，ξ）分布的传统方法，并给出了相关推断以及在气候数据中的应用。
