A New Probability Model for Hydrologic Events: Properties and Applications

Abstract

Upon the motivation of unstable climatic conditions of the world like excess of rains, drought and huge floods, we introduce a versatile hydrologic probability model with two scale parameters. The proposed model contains Lindley and exponentiated exponential (Lindley in J R Stat Soc Ser B 20:102–107, 1958; Gupta and Kundu in Biom J 43(1):117–130, 2001) distributions as special cases. Various properties of the distribution are obtained, such as shapes of the density and hazard functions, moments, mean deviation, information-generating function, conditional moments, Shannon entropy, L-moments, order statistics, information matrix and characterization via hazard function. Parameters are estimated via maximum likelihood estimation method. A simulation scheme is provided for generating the random data from the proposed distribution. Four data sets are used for comparing the proposed model with a set of well-known hydrologic models, such as generalized Pareto, log normal (3), log Pearson type III, Kappa(3), Gumbel, generalized logistic and generalized Lindley distributions, using some goodness-of-fit tests. These comparisons render the proposed model suitable and representative for hydrologic data sets with least loss of information attitude and a realistic return period, which render it as an appropriate alternate of the existing hydrologic models. Supplementary materials for this paper are available online.

Appendix

In this appendix, we state the partial derivatives of the log-likelihood function (5) with respect to the unknown parameters $\theta$, $\alpha$ and $\beta$ as

$$\begin{aligned} \frac{\partial \ln L(\theta ,\alpha ,\beta )}{\partial \theta }= & {} \frac{2n}{\theta }-\frac{n}{(\theta +\beta )}+\sum _{i=1}^{n}\frac{ x_{i}\alpha (\alpha -1)(1-\hbox {e}^{-\theta x_{i}})^{\alpha -1}\hbox {e}^{-\theta x_{i}}}{ \alpha ((1-\hbox {e}^{-\theta x_{i}})^{\alpha -1})+\beta x_{i}} -\sum _{i=1}^{n}x_{i}=0, \\ \frac{\partial \ln L(\theta ,\alpha ,\beta )}{\partial \alpha }= & {} \sum _{i=1}^{n} \frac{(1-\hbox {e}^{-\theta x_{i}})^{\alpha -2}+\alpha (1-\hbox {e}^{-\theta x_{i}})^{\alpha -1}ln(1-\hbox {e}^{-\theta x_{i}})}{\alpha ((1-\hbox {e}^{-\theta x_{i}})^{\alpha -1})+\beta x_{i}}=0, \\ \frac{\partial \ln L(\theta ,\alpha ,\beta )}{\partial \beta }= & {} \sum _{i=1}^{n} \frac{x_{i}}{\alpha ((1-\hbox {e}^{-\theta x_{i}})^{\alpha -1})+\beta x_{i}}-\frac{n }{\theta +\beta }=0. \end{aligned}$$

The second derivatives that exist in the elements of the $3\times 3$ information matrix are given as

$$\begin{aligned} \frac{\partial ^{2}\ln L(\theta ,\alpha ,\beta )}{\partial \theta ^{2}}= & {} \sum _{i=1}^{n}\frac{x_{i}^{2}\alpha (\alpha -1)(1-\hbox {e}^{-\theta x_{i}})^{\alpha -2}\hbox {e}^{-\theta x_{i}}((\alpha -2)(1-\hbox {e}^{-\theta x_{i}})^{-1}\hbox {e}^{-\theta x_{i}}-1)}{\alpha ((1-\hbox {e}^{-\theta x_{i}})^{\alpha -1})+\beta x_{i}} \\&-\,\dfrac{2n}{\theta ^{2}}-\sum _{i=1}^{n}\frac{x_{i}^{2}\alpha ^{2}(\alpha -1)^{2}((1-\hbox {e}^{-\theta x_{i}})^{2\alpha -4}\hbox {e}^{-2\theta x_{i}}}{(\alpha ((1-\hbox {e}^{-\theta x_{i}})^{\alpha -1})+\beta x_{i})^{2}}-\dfrac{n}{(\theta +\beta )^{2}}, \\ \frac{\partial ^{2}\ln L(\theta ,\alpha ,\beta )}{\partial \alpha ^{2}}= & {} \sum _{i=1}^{n}\frac{2(1-\hbox {e}^{-\theta x_{i}})^{\alpha -1}\ln (1-\hbox {e}^{-\theta x_{i}})+\alpha (1-\hbox {e}^{-\theta x_{i}})^{\alpha -1}\ln ((1-\hbox {e}^{-\theta x_{i}})^{2})}{\alpha ((1-\hbox {e}^{-\theta x_{i}})^{\alpha -1})+\beta x_{i}} \\&-\,\sum _{i=1}^{n}\frac{((1-\hbox {e}^{-\theta x_{i}})^{\alpha -1}\hbox {e}^{-2\theta x_{i}}+\alpha (1-\hbox {e}^{-\theta x_{i}})^{\alpha -1}\ln (1-\hbox {e}^{-\theta x_{i}}))^{2}}{ (\alpha ((1-\hbox {e}^{-\theta x_{i}})^{\alpha -1})+\beta x_{i})^{2}}, \\ \frac{\partial ^{2}\ln L(\theta ,\alpha ,\beta )}{\partial \beta ^{2}}= & {} \sum _{i=1}^{n}\frac{x_{i}^{2}}{(\alpha ((1-\hbox {e}^{-\theta x_{i}})^{\alpha -1})+\beta x_{i})^{2}}-\frac{n}{(\theta +\beta )^{2}}. \end{aligned}$$

Table 1 Data sets.