Improved parameter estimation of the Chaudhry and Ahmad distribution with climate applications

The Chaudhry-Ahmad distribution is a two-parameter continuous probability distribution obtained as a solution to a generalized Pearson system of differential equation. Although its probability density curve resembles the inverse-Gaussian, gamma, log-normal, Weibull and other distributions, it has been neglected in the analysis of right-skewed data. The purpose of this paper is three folded. Firstly, to reparametrize the Chaudhry and Ahmad distribution and present some of its basic properties. Secondly to derive the analytical bias-corrected maximum likelihood estimators applying the Cox-Snell methodology and thirdly to study, by MC simulations, the small-sample properties of the maximum likelihood estimators and their bias-corrected versions, obtained from the Cox-Snell formula and by parametric bootstrap technique. The numerical results show, for both parameters, that the maximum likelihood estimators are highly biased, especially in small samples. On the other hand, both, the analytical and bootstrap methodologies, significantly reduce the biases and the mean-squared errors. It is apparent from the results that the analytical bias-correction is more efficient than bootstrap resamples. Finally, wind speed data from six weather stations distributed in the state of Tocantins in Brazil is used to illustrate the applicability of the proposed methods.