A Probabilistic Model for the Distribution of GDP per Capita in NUTS 3 Zones of Europe

A macroeconomic indicator of productivity and economic development, used to obtain information on the economic and social conditions of a country, is the GDP per capita, which is also used as an indicator of social welfare. By construction it can be used directly to compare areas of interest. It is an indicator of great variability to which it is difficult to assign a probabilistic model to describe its distribution. In fact, it usually appears as a strongly asymmetric and frequently multimodal variable, which directly indicates a strong non-normality. In this work we propose to deal with the problem of finding a probabilistic model for this variable through the estimation of a model of finite mixtures of normal distributions. As an application example, we present the model obtained through the finite mixture for GDP per capita data from the NUTS 3 zones in the nomenclature of the European Union, EU countries and neighbouring countries. Thus, the model is estimated, its validity is checked and the results obtained are analysed, both for the GDP per capita variable and as a function of the countries to which the studied areas belong.

Finite mixtures of normal distributions in the study of the error in altimetry

The modelling of the altimetric error is proposed by means of the mixture of normal distributions. This alternative allows to avoid the problems of lack of normality of the altimetric error and that have been indicated numerous times. The conceptual bases of the mixture of distributions are presented and its application is demonstrated with an applied example. In the example, the altimetric errors existing between a DEM with 5 × 5 m resolution and another DEM with 2 × 2 m resolution are modelled, which is considered as a reference. The application demonstrates the feasibility and power of analysis of the proposal made.

Bayesian sparse convex clustering via global-local shrinkage priors

Sparse convex clustering is to group observations and conduct variable selection simultaneously in the framework of convex clustering. Although a weighted $$L_1$$ L 1 norm is usually employed for the regularization term in sparse convex clustering, its use increases the dependence on the data and reduces the estimation accuracy if the sample size is not sufficient. To tackle these problems, this paper proposes a Bayesian sparse convex clustering method based on the ideas of Bayesian lasso and global-local shrinkage priors. We introduce Gibbs sampling algorithms for our method using scale mixtures of normal distributions. The effectiveness of the proposed methods is shown in simulation studies and a real data analysis.

Normal approximation for mixtures of normal distributions and the evolution of phenotypic traits

Explicit bounds are given for the Kolmogorov and Wasserstein distances between a mixture of normal distributions, by which we mean that the conditional distribution given some $\sigma$ -algebra is normal, and a normal distribution with properly chosen parameter values. The bounds depend only on the first two moments of the first two conditional moments given the $\sigma$ -algebra. The proof is based on Stein’s method. As an application, we consider the Yule–Ornstein–Uhlenbeck model, used in the field of phylogenetic comparative methods. We obtain bounds for both distances between the distribution of the average value of a phenotypic trait over n related species, and a normal distribution. The bounds imply and extend earlier limit theorems by Bartoszek and Sagitov.