Abstract
In this paper, a system of nonlinear equations for the maximum likelihood estimators as wel as the exact forms of the Fisher information matrix for Crovelli's bivariate gamma distribution and bivariate gamma beta distribution of the second kind are determined. An application of the results to the rainfall data from the city of Passo Fundo are provided.
Highlights
Motivaded by their crescent use, especially in the analysis of non normal data, several bivariate gamma models can be found in literature, see, for example, Balakrishnan and Lai (2009)
The Fisher information matrices (FIM) is used to calculate the covariance matrices associated with maximumlikelihood estimators
The calculations of the FIM and its inverse matrix were obtained by using the software R (R CORE TEAM, 2019) and its package VGAM (YEE et al, 2015)
Summary
Motivaded by their crescent use, especially in the analysis of non normal data, several bivariate gamma models can be found in literature, see, for example, Balakrishnan and Lai (2009). It can be seen in Silva et al, (2013a, 2013b) that the exact distributions of the variables U = X + Y , P = XY and Q = X/(X + Y ) when X and Y follow (1) and (2) were deducted and successfully implemented on modeling hydrological processes. The FIM is used to calculate the covariance matrices associated with maximumlikelihood estimators It can be used in the formulation of test statistics, such as the Wald test (LIU et al, 2014); for intervals estimation and hypothesis tests on the distribution parameters (BARROS et al, 2017; LOUZADA et al , 2018), and it is applicable to unknown systematic errors (FISCHER, 2018). The properties of these special functions can be found in Oldham et al, (2009) and Beals and Wong (2010)
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