Size Distribution of States, Counties, and Cities in the USA: New Inequality Form Information

In this article, we propose a new approach for studying the patterns of size distribution in settlement systems, based on the analysis of the shape of the Pareto curve (PC). To study the shape of the PC, we used the Gini coefficient, the asymmetry coefficient, and, by analogy with the physics of phase transitions, critical exponent — the index of the PC degree in the neighborhood of zero. An empirical analysis of the PC of various levels of aggregation in the US settlement system has been performed. The form of size distribution of states was studied by decades from 1790 to 2010. The spatial analysis of the PC shape for counties and cities was performed for 2010. The results of an empirical study showed that the PC of the states had left-hand asymmetry over 220 years. The PC of districts and cities had both right-hand and left-hand asymmetries. The obtained results explain in which cases the Pareto distribution having a PC with right-hand asymmetry, and the lognormal distribution with a symmetric PC may not correspond objectively to real settlement systems. As an alternative to power-series distribution and lognormal distribution, we considered an analytically simple two-parameter model with a wide range of PC asymmetry that combines the properties of power-series distribution and lognormal distribution. Verification of the model showed that it adequately described the size of settlements in homogeneous settlement systems.

Complementary Kumaraswamy Weibull Power Series Distribution: Some Properties and Application

In this paper, we propose Complementary Kumaraswamy Weibull Power Series (CKWPS) Distributions. The method is obtained by compounding the Kumaraswamy-G distribution and Power Series distribution on a latent complementary distance problem base. The mathematical properties of the proposed class of distribution are studied. The method of Maximum Likelihood Estimation is used for obtaining the estimates of the model parameters. A member of the family is investigated in detail. Finally an application of the proposed class is illustrated using a real data set.

Some Reliability and Other Properties of Beta-Transformed Random Variables

The reliability properties of beta-transformed random variables are discussed. A necessary and sufficient condition for a beta-transformed geometric random variable to follow a power series distribution is derived. It is shown that a beta-transformed member of the Katz family does not belong to the Katz family unless it is a geometric distribution, thereby getting a characterization.