Bivariate q- generalized extreme value distribution: A comparative approach with applications to climate related data

Abstract

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 (GEV) 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 GEV distribution involving an additional parameter, the q− generalized extreme value (q−GEV) distribution, as well as the q− Gumbel distribution. The prime interest of this paper lies in conceptualizing a novel approach to model bi-variate (EV) data, arising naturally from independent q−GEV 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 q−GEV probability density functions. Besides, we employed the suggested technique to a bivariate flood data set and demonstrate the competitiveness of the proposed bivariate q−GEV. Additionally, conventional method to propose the newly defined bivariate (q−GEV) distribution with bivariate q− Gumbel distribution (a special case for ξ→0) has also been established with related inferences and application to climate data.

Keywords

• Generalized extreme value distributions
• Goodness–of–fit statistic
• Gumbel distribution
• Bi-variate distribution
• Moments
• Monte Carlo simulation