Tail Sets and Level Curves of Bivariate Distributions: Geometry and Estimation

Abstract

By incorporating Gumbel's bivariate exponential (BVE) distribution as the foundation, we aim to minimize losses between two business lines. Motivated by the need for a financial model, we chose Gumbel’s BVE for its thin-tailed property, serving as a great foundation for future extensions to multidimensional risk measures. We derive a closed-form expression for $H(x,y$) using the law of total probability, linking it to the value-at-risk concept represented by set $\mathcal{A}_p$. Recognizing that companies allocate capital near the boundary of $\mathcal{A}_p$, we define set $\mathcal{O}_p$ as the level $p$ curve of optimal values. A convexity analysis via a Hessian matrix and Sylvester’s Criterion provides insight into optimal capital allocation, and we apply Lagrange multipliers to prove a loss-minimization theorem. Graphs and tables illustrate our findings. Overall, this research offers practical insights into resource allocation, boosting company growth and financial stability.