Conical Extremes of a Multivariate Sample

Area of Science:

• Statistics
• Probability Theory
• Extreme Value Theory

Background:

• Multivariate extreme value theory is crucial for understanding rare events in high-dimensional data.
• Existing models often assume specific dependence structures, limiting their applicability.

Purpose of the Study:

• To extend the theory of multivariate extremes to directional extremes within a cone.
• To establish convergence results for the k-th order statistics in this directional setting.

Main Methods:

• Developing a framework for analyzing extremes in a specific direction (cone).
• Utilizing properties of asymptotically independent radial and spherical components.
• Applying techniques for regularly varying tails in probability distributions.

Main Results:

• Established convergence results for the number of k-th extremes in the specified directional context.
• Demonstrated the applicability to distributions with asymptotically independent radial and spherical components.
• Characterized the behavior of extremes under these specific distributional assumptions.

Conclusions:

• The study provides a theoretical foundation for analyzing directional multivariate extremes.
• The findings are relevant for statistical modeling of rare events in complex systems.
• Offers new insights into the limiting behavior of order statistics in high dimensions.