Abstract
We propose a statistical duality among the preferences and endowments of the agents. Under this duality, the logarithmic prices of random trades among agents in a decentralized economy converge in expectation to the logarithm of the Walrasian equilibrium price in a centralized economy.

Abstract
This research determines how to integrate factors related to evacuation in emergency preparedness using techniques for Multicriteria Decision-Making (MCDM). A distinctive MCDM technique that incorporates human behavior into evacuation models enhances decision-making and safety during emergencies, especially in vulnerable populations. For this purpose, a hybrid combination of MCDM methods-CRiteria Importance Through Intercriteria Correlation (CRITIC), Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), and Weighted Aggregated Sum Product Assessment (WASPAS)-is used to rank the vulnerability of Chilean regions by considering various factors. First, the related factors are ranked by CRITIC, and the result is that the psychosocial problem factor has the highest priority and weight. Then, according to the hybrid methods and CRITIC, all regions of Chile are ranked first with TOPSIS, WASPAS, and a combination of them to determine which one has the highest priority. The results show that the Santiago Metropolitan Region has the highest priority for vulnerability in all three methods.

Abstract
Sales forecasting is very important in retail management. It helps with decisions about inventory, staffing, and planning promotions. In this study, we looked at how to balance the accuracy of predictions with how easy it is to understand the machine learning models used in sales forecasting. We used public data from Rossmann stores to study various factors like promotions, holidays, and store features that affect daily sales. We compared a complex, highly accurate model (XGBoost) with simpler, easier-to-understand linear regression models. To find a middle ground, we created a hybrid model called LR XGBoost. This model changes a linear regression model to match the predictions of XGBoost. The hybrid model keeps the strong predictive power of complex models but makes the results easier to understand, which is important for making decisions in retail. Our study shows that our hybrid model offers a good balance, providing reliable sales forecasts with more transparency than standard linear regression. This makes it a valuable tool for retail managers who need accurate forecasts and a clear understanding of what influences sales. The model’s consistent performance across datasets also suggests it can be used in various retail settings to improve efficiency and help with strategic decisions. © 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0)

Abstract
We propose an approach to cluster and classify compositional data. We transform the compositional data into directional data using the square root transformation. To cluster the compositional data, we apply the identification of a mixture of Watson distributions on the hypersphere and to classify the compositional data into predefined groups, we apply Bayes rules based on the Watson distribution to the directional data. We then compare our clustering results with those obtained in hierarchical clustering and in the K-means clustering using the log-ratio transformations of the data and compare our classification results with those obtained in linear discriminant analysis using log-ratio transformations of the data. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.

Abstract
When directional data fall in the positive orthant of the unit hypersphere, a folded directional distribution is preferred over a simple directional distribution for modeling the data. Since directional data, especially axial data, can be modeled using a Watson distribution, this paper considers a folded Watson distribution for such cases. We first address the parameter estimation of this distribution using maximum likelihood, which requires a numerical algorithm to solve the likelihood equations. We use the Expectation-Maximization (EM) algorithm to obtain these estimates and to analyze the properties of the concentration estimator through simulation. Next, we propose the Bayes rule for a folded Watson distribution and evaluate its performance through simulation in various scenarios, comparing it with the Bayes rule for the Watson distribution. Finally, we present examples using both simulated and real data available in the literature.

Abstract