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

Abstract
Data sets that contain an excessive number of zeros appear in several fields of applications. This chapter considers a zero-inflated Lomax distribution as a possible model for these types of data, and presents and analyzes the performance of a Shewhart control chart for process monitoring. Several approaches allow for frequent zero observations, and among them, the most common are zero-inflated models and hurdle models in case of count data, and the use of zero-inflated distributions to model semi-continuous data, that is, data from a continuous distribution with one or more than one point of mass. The chapter presents some motivation for the use of the zero-inflated Lomax distribution together with some properties of this distribution. It proposes a Shewhart-type control chart for monitoring zero-inflated Lomax data, and analyzes its performance under some scenarios.

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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
Background/Objectives: Type 2 diabetes mellitus (T2DM) is a chronic condition marked by hyperglycemia, which can affect metabolic, vascular, and hematological parameters. A low-carbohydrate, high-fat (LCHF) diet has been shown to improve glycemic control and blood pressure regulation. Exercise in hypoxia (EH) enhances insulin sensitivity, erythropoiesis, and angiogenesis. The combination of LCHF and EH may offer a promising strategy for managing T2DM and hypertension (HTN), although evidence remains limited. This study aimed to assess the effects of an eight-week normobaric EH intervention at 3000 m simulated altitude combined with an LCHF diet on hematological and lipid profiles, inflammation, and blood pressure in older patients with T2DM and HTN. Methods: Forty-two diabetic patients with HTN were randomly assigned to three groups: (1) control group (control diet + exercise in normoxia), (2) EH group (control diet + EH), and (3) intervention group (EH+LCHF) Baseline and eight-week measurements included systolic, diastolic, and mean blood pressure (SBP, DBP, MAP), hematological and lipid profiles, and inflammation biomarkers. Results: Blood pressure decreased after the intervention (p < 0.001), with no significant differences between groups (SBP: p = 0.151; DBP: p = 0.124; MAP: p = 0.18). No differences were observed in lipid profile or C-reactive protein levels (p > 0.05). Mean corpuscular hemoglobin (MCH) increased in the EH group (p = 0.027), while it decreased in the EH+LCHF group (p = 0.046). Conclusions: Adding hypoxia or restricting carbohydrates did not provide additional benefits on blood pressure in T2DM patients with HTN. Further elucidation of the mechanisms underlying hematological adaptations is imperative.