Suchergebnisse

Modelling an empirical distribution by means of a simple theoretical distribution is an interesting issue in applied statistics. A reasonable first step in this modelling process is to demand that measures for location, dispersion, skewness and kurtosis for the two distributions coincide. Up to now, the four measures used hereby were based on moments.In this paper measures are considered which are based on quantiles. Of course, the four values of these quantile measures do not uniquely determine the modelling distribution. They do, however, within specific systems of distributions, like Pearson's or Johnson's; they share this property with the four moment‐based measures.This opens the possibility of modelling an empirical distribution—within a specific system—by means of quantile measures. Since moment‐based measures are sensitive to outliers, this approach may lead to a better fit. Further, tests of fit—e.g. a test for normality—may be constructed based on quantile measures. In view of the robustness property, these tests may achieve higher power than the classical moment‐based tests.For both applications the limiting joint distribution of quantile measures will be needed; they are derived here as well.

The analysis and modeling of categorical time series requires quantifying the extent of dispersion and serial dependence. The dispersion of categorical data is commonly measured by Gini index or entropy, but also the recently proposed extropy measure can be used for this purpose. Regarding signed serial dependence in categorical time series, we consider three types of κ-measures. By analyzing bias properties, it is shown that always one of the κ-measures is related to one of the above-mentioned dispersion measures. For doing statistical inference based on the sample versions of these dispersion and dependence measures, knowledge on their distribution is required. Therefore, we study the asymptotic distributions and bias corrections of the considered dispersion and dependence measures, and we investigate the finite-sample performance of the resulting asymptotic approximations with simulations. The application of the measures is illustrated with real-data examples from politics, economics and biology.

The analysis and modeling of categorical time series requires quantifying the extent of dispersion and serial dependence. The dispersion of categorical data is commonly measured by Gini index or entropy, but also the recently proposed extropy measure can be used for this purpose. Regarding signed serial dependence in categorical time series, we consider three types of κ-measures. By analyzing bias properties, it is shown that always one of the κ-measures is related to one of the above-mentioned dispersion measures. For doing statistical inference based on the sample versions of these dispersion and dependence measures, knowledge on their distribution is required. Therefore, we study the asymptotic distributions and bias corrections of the considered dispersion and dependence measures, and we investigate the finite-sample performance of the resulting asymptotic approximations with simulations. The application of the measures is illustrated with real-data examples from politics, economics and biology.

The analysis and modeling of categorical time series requires quantifying the extent of dispersion and serial dependence. The dispersion of categorical data is commonly measured by Gini index or entropy, but also the recently proposed extropy measure can be used for this purpose. Regarding signed serial dependence in categorical time series, we consider three types of &kappa ; -measures. By analyzing bias properties, it is shown that always one of the &kappa ; -measures is related to one of the above-mentioned dispersion measures. For doing statistical inference based on the sample versions of these dispersion and dependence measures, knowledge on their distribution is required. Therefore, we study the asymptotic distributions and bias corrections of the considered dispersion and dependence measures, and we investigate the finite-sample performance of the resulting asymptotic approximations with simulations. The application of the measures is illustrated with real-data examples from politics, economics and biology.

Within the context of climate strength, this simulation study examines the validity of various dispersion indexes for detecting meaningful relationships between variability in group member perceptions and outcome variables. We used the simulation to model both individual-and group-level phenomena, vary appropriate population characteristics, and test the proclivity of standard and average deviation, interrater agreement indexes (rwg, r*wg, awg), and coefficient of variation (both normed and unnormed) for Type I and Type II errors. The results show that the coefficient of variation was less likely to detect interaction effects although it outperformed other measures when detecting level effects. Standard deviation was shown to be inferior to other indexes when no level effect is present although it may be an effective measure of dispersion when modeling strength or interaction effects. The implications for future research, in which dispersion is a critical component of the theoretical model, are discussed.

Let X, X1, ..., Xk be i.i.d. random variables, and for k∈ N let Dk(X) = E(X1 V ... V Xk+1) −EX be the kth centralized maximal moment. A sharp lower bound is given for D1(X) in terms of the Lévy concentration Ql(X) = supx∈ R P(X∈[x, x + l]). This inequality, which is analogous to P. Levy's concentration‐variance inequality, illustrates the fact that maximal moments are a gauge of how much spread out the underlying distribution is. It is also shown that the centralized maximal moments are increased under convolution.