A new bivariate Gamma distribution generated from functional scale parameter with application to drought data

Abstract

Univariate and bivariate Gamma distributions are among the most widely used distributions in hydrological statistical modeling and applications. This article presents the construction of a new bivariate Gamma distribution which is generated from the functional scale parameter. The utilization of the proposed bivariate Gamma distribution for drought modeling is described by deriving the exact distribution of the inter-arrival time and the proportion of drought along with their moments, assuming that both the lengths of drought duration (X) and non-drought duration (Y) follow this bivariate Gamma distribution. The model parameters of this distribution are estimated by maximum likelihood method and an objective Bayesian analysis using Jeffreys prior and Markov Chain Monte Carlo method. These methods are applied to a real drought dataset from the State of Colorado, USA.

Appendix

The Fisher information matrix $Q\left( {\hat{g}} \right)$ is given by:

$$ \begin{aligned} Q\left( {\hat{g}} \right) & = - E\left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} L\left( g \right)}}{{\partial \alpha^{2} }}} & {\frac{{\partial^{2} L\left( g \right)}}{\partial \alpha \,\partial \beta }} & 0 & 0 \\ {} & {\frac{{\partial^{2} L\left( g \right)}}{{\partial \beta^{2} }}} & 0 & 0 \\ {} & {} & {\frac{{\partial^{2} L\left( g \right)}}{{\partial \gamma^{2} }}} & {\frac{{\partial^{2} L\left( g \right)}}{\partial \gamma \,\partial \delta }} \\ {} & {} & {} & {\frac{{\partial^{2} L\left( g \right)}}{{\partial \delta^{2} }}} \\ \end{array} } \right] \\ & = n\left[ {\begin{array}{*{20}c} {\psi^{\prime}\left( \alpha \right)} & { - \frac{1}{\beta }} & 0 & 0 \\ { - \frac{1}{\beta }} & {\frac{\alpha }{{\beta^{2} }}} & 0 & 0 \\ 0 & 0 & {\psi^{\prime}\left( \gamma \right)} & { - \frac{1}{\delta }} \\ 0 & 0 & { - \frac{1}{\delta }} & {\frac{\gamma }{{\delta^{2} }}} \\ \end{array} } \right] \\ \end{aligned} $$

Here $\,\psi^{\prime}\left( . \right)$ is the first derivative of the Psi function (also called trigamma function).