Dependence uncertainty bounds for the energy score and the multivariate Gini mean difference

The energy distance and energy scores became important tools in multivariate statistics and multivariate probabilistic forecasting in recent years. They are both based on the expected distance of two independent samples. In this paper we study dependence uncertainty bounds for these quantities under the assumption that we know the marginals but do not know the dependence structure. We find some interesting sharp analytic bounds, where one of them is obtained for an unusual spherically symmetric copula. These results should help to better understand the sensitivity of these measures to misspecifications in the copula.

Use of Nonconventional Dispersion Measures to Improve the Efficiency of Ratio-Type Estimators of Variance in the Presence of Outliers

The use of auxiliary information in survey sampling to enhance the efficiency of the estimators of population parameters is a common phenomenon. Generally, the ratio and regression estimators are developed by using the known information on conventional parameters of the auxiliary variables, such as variance, coefficient of variation, coefficient of skewness, coefficient of kurtosis, or correlation between the study and auxiliary variable. The efficiency of these estimators is dubious in the presence of outliers in the data and a nonsymmetrical population. This study presents improved variance estimators under simple random sampling without replacement with the assumption that the information on some nonconventional dispersion measures of the auxiliary variable is readily available. These auxiliary variables can be the inter-decile range, sample inter-quartile range, probability-weighted moment estimator, Gini mean difference estimator, Downton’s estimator, median absolute deviation from the median, and so forth. The algebraic expressions for the bias and mean square error of the proposed estimators are obtained and the efficiency conditions are derived to compare with the existing estimators. The percentage relative efficiencies are used to numerically compare the results of the proposed estimators with the existing estimators by using real datasets, indicating the supremacy of the suggested estimators.

Notes on Cumulative Entropy as a Risk Measure

Di Crescenzo and Longobardi [Di Crescenzo and Longobardi, On cumulative entropies, J. Statist. Plann. Inference 139 2009, 12, 4072–4087] proposed the cumulative entropy (CE) as an alternative to the differential entropy. They presented an estimator of CE using empirical approach. In this paper, we consider a risk measure based on CE and compare it with the standard deviation and the Gini mean difference for some distributions. We also make empirical comparisons of these measures using samples from stock market in members of the Organization for Economic Co-operation and Development (OECD) countries.

Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units

We describe a method of calculating sharp lower and upper bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units of the original i.i.d. observations. In particular, we provide sharp lower and upper bounds on the expectations of kth records and their differences. We also present the families of distributions which attain the bounds in the limit.