Spatial and Seasonal Variations and Inter-Relationship in Fitted Model Parameters for Rainfall Totals across Australia at Various Timescales

Probabilistic models are useful tools in understanding rainfall characteristics, generating synthetic data and predicting future events. This study describes the results from an analysis on comparing the probabilistic nature of daily, monthly and seasonal rainfall totals using data from 1327 rainfall stations across Australia. The main objective of this research is to develop a relationship between parameters obtained from models fitted to daily, monthly and seasonal rainfall totals. The study also examined the possibility of estimating the parameters for daily data using fitted parameters to monthly rainfall. Three distributions within the Exponential Dispersion Model (EDM) family (Normal, Gamma and Poisson-Gamma) were found to be optimal for modelling the daily, monthly and seasonal rainfall total. Within the EDM family, Poisson-Gamma distributions were found optimal in most cases, whereas the normal distribution was rarely optimal except for the stations from the wet region. Results showed large differences between regional and seasonal ϕ-index values (dispersion parameter), indicating the necessity of fitting separate models for each season. However, strong correlations were found between the parameters of combined data and those derived from individual seasons (0.70–0.81). This indicates the possibility of estimating parameters of individual season from the parameters of combined data. Such relationship has also been noticed for the parameters obtained through monthly and daily models. Findings of this research could be useful in understanding the probabilistic features of daily, monthly and seasonal rainfall and generating daily rainfall from monthly data for rainfall stations elsewhere.

THE FULL TAILS GAMMA DISTRIBUTION APPLIED TO MODEL EXTREME VALUES

In this paper, we introduce the simplest exponential dispersion model containing the Pareto and exponential distributions. In this way, we obtain distributions with support (0, ∞) that in a long interval are equivalent to the Pareto distribution; however, for very high values, decrease like the exponential. This model is useful for solving relevant problems that arise in the practical use of extreme value theory. The results are applied to two real examples, the first of these on the analysis of aggregate loss distributions associated to the quantitative modelling of operational risk. The second example shows that the new model improves adjustments to the destructive power of hurricanes, which are among the major causes of insurance losses worldwide.

The Kendal- Ressel Exponential Dispersion Model: Some Statistical Aspects and Estimation

In this paper we revisit the NEF class of distributions generated by the Kendall-Ressel density. We study some of the statistical properties of this class (KR-NEF), as well as those of an associated class of Exponential Dispersion Model (KR-EDM). In particular, we discuss some of the immediate properties of these distributions, especially under the so called mean re-parameterization. The moments and other cumulants of these distributions are thoroughly presented as well as expressions for the measures of their skewness and kurtosis. Estimation procedures under the mean parameterization are also discussed where the maximum likelihood, second order minimax and a Bayes estimators are presented and illustrated.