A Unified General Class of Interdistributional Income Inequality Measures Based on Weighted Incomplete Moments

Many authors have proposed measures for between groups income inequalities. Mostly, these measures are based on functional of the income distribution. Others are based on Gini index, measures of entropies or additive functions. Butler and McDonald (1987) developed a class of between groups income inequality measures based on incomplete moments and showed its applicability. In this article, A unified class of interdistributional inequality measures are introduced. Most of previous measures are special cases from the new class, such as Butler-McDonald measures. These new measures are estimated and studied. Also, the new general class is based on probability weighted moments which can be given any values as the upper value. A real data application is presented to compare among all these measures and show the benefits of the new measures.

Goodness-of-fit Measures Based on the Mellin Transform for Beta Generalized Lifetime Data

In recent years various probability models have been proposed for describing lifetime data. Increasing model flexibility is often sought as a means to better describe asymmetric and heavy tail distributions. Such extensions were pioneered by the beta-G family. However, efficient goodness-of-fit (GoF) measures for the beta-G distributions are sought. In this paper, we combine probability weighted moments (PWMs) and the Mellin transform (MT) in order to furnish new qualitative and quantitative GoF tools for model selection within the beta-G class. We derive PWMs for the Fr\’{e}chet and Kumaraswamy distributions; and we provide expressions for the MT, and for the log-cumulants (LC) of the beta-Weibull, beta-Fr\’{e}chet, beta-Kumaraswamy, and beta-log-logistic distributions. Subsequently, we construct LC diagrams and, based on the Hotelling’s $T^2$ statistic, we derive confidence ellipses for the LCs. Finally, the proposed GoF measures are applied on five real data sets in order to demonstrate their applicability.

On Modelling Extreme Damages from Natural Disasters in Kenya

We seek to develop a distribution to model the extreme damages resulting from Natural Disasters in Kenya.The distribution is based on the Compound Extreme Value Distribution, which takes into account both the distributions of the frequency of occurrence and magnitude of the events. Threshold modelling is employed, where the extreme damages are identified as the points that lie above a sufficiently high threshold. The distribution of the number of the exceedance is found to be Negative Binomial, while that of the severity is approximated by a Generalised Pareto Distribution. Maximum likelihood estimation is used to estimate the parameters, and the log-likelihood is maximised using numerical methods. Probability weighted moments estimation is used to determine the starting values for the iterations. Prediction study is then carried out to investigate the performance of the proposed distribution in predicting future events.

Assessment of techniques to include historical information in flood frequency distributions for hydrological dam safety assessment

<p>Flood peak quantiles for return periods up to 10 000 years are required for dam design and safety assessment, though flood series usually have a record length of around 20-40 years that leads to a high uncertainty. The utility of historical data of flooding is generally recognised for estimating the magnitude of extreme events with return periods in excess of 100 years. Therefore, historical information can be incorporated in flood frequency analyses to reduce uncertainties in high return period flood quantile estimates that are used in hydrological dam safety analyses.</p><p>This study assesses a set of existing techniques to incorporate historical information of flooding in extreme frequency analyses, focusing on their reliability and uncertainty reduction for high return periods that are used for dam safety analysis. Monte Carlo simulations are used to assess both the reliability and uncertainty in high return period quantile estimates. Varying lengths in the historical (Nh = 100 and 200 years) and systematic (Ns = 20, 40 and 60 years) periods are considered. In addition, a varying number of known flood magnitudes that exceed a given perception threshold in the historical period are also considered (k = 1-2). The values of Nh, Ns and k used in the study are the most usual in practice.</p><p>The reliability and uncertainty reduction in flood quantile estimates for each technique depend on the statistical properties of flood series. Therefore, a set of feasible combinations of L-coefficient of variation (L-CV) and skewness (L-CS) values should be considered. The analysis aims to understand how each technique behaves in terms of flood quantile reliability and uncertainty reduction depending on the L-moment statistics of flood series. In this study, L-CV and L-CS regional values in the 29 homogeneous regions identified in Spain for developing the national map of flood quantiles by the Centre for Hydrographic Studies of CEDEX are considered.</p><p>The results show that the maximum likelihood estimator (MLE) and weighted moments (WM) techniques show the best results in the regions with small L-CS values. However, the biased partial probability weighted moments (BPPWM) technique shows the best results in the regions with high L-CS values. While the expected moments algorithm (EMA) tends to underestimate flood quantiles for high return periods, the unbiased partial probability weighted moments (UPPWM) technique tends to overestimate them. In addition, including historical information of flooding in flood frequency analyses improves flood quantile estimates in most cases regardless the technique that is used. Uncertainty reduction in high return period flood quantile estimates are higher for short systematic time series, regions with high L-CS values and long historical periods.</p><p><strong>Acknowledgments:</strong> This research has been supported by the project SAFERDAMS (PID2019-107027RB-I00) funded by the Spanish Ministry of Science and Innovation.</p>

A Novel Global Sensitivity Measure Based on Probability Weighted Moments

Global sensitivity analysis (GSA) is a useful tool to evaluate the influence of input variables in the whole distribution range. Variance-based methods and moment-independent methods are widely studied and popular GSA techniques despite their several shortcomings. Since probability weighted moments (PWMs) include more information than classical moments and can be accurately estimated from small samples, a novel global sensitivity measure based on PWMs is proposed. Then, two methods are introduced to estimate the proposed measure, i.e., double-loop-repeated-set numerical estimation and double-loop-single-set numerical estimation. Several numerical and engineering examples are used to show its advantages.

The Marshall–Olkin Transmuted-G Family of Distributions

We introduce a new family of univariate continuous distributions called the Marshall–Olkin transmuted-G family which extends the transmuted-G family pioneered by Shaw and Buckley (2007). Special models for the new family are provided. Some of its mathematical properties including quantile measure, explicit expressions for the ordinary and incomplete moments, generating function, Rényi and Shannon entropies, order statistics and probability weighted moments are derived. The estimation of the model parameters is performed by maximum likelihood. The flexibility of the proposed family is illustrated by means of two applications to real data sets.