Generation and Parameter Estimation of 2D Stable Distributions

Overview

This project explores the simulation, visualization, and parameter estimation of two-dimensional stable distributions and sub-Gaussian vectors. It includes custom statistical generators and estimators to analyze the impact of the stability index $\alpha$ and spectral measure $\Gamma$ on the behavior of heavy-tailed bivariate data.

Features

• 2D stable vector generator: Simulates symmetric, independent, and dependent asymmetric stable vectors based on defined discrete spectral measures.
• Sub-Gaussian vector generator: Implements a robust generator for 2D sub-Gaussian random vectors.
• Parameter estimation: Estimates the stability index $\alpha$ using a log-log regression tail estimator, and estimates the spectral measure $\Gamma$ by analyzing the directional distribution of tail behavior.
• Characteristic function analysis: Computes and visualizes the empirical characteristic function (real and imaginary parts) and calculates the estimation error against theoretical values.
• Codifference estimation: Calculates the codifference $\tau$ to measure the dependency between vector components.
• Optimal threshold selection: Script designed to dynamically determine the optimal tail cut-off threshold $R$ for spectral measure estimation by minimizing the MSE.

Files & Usage

• main.qmd: The main Quarto document containing all Python/R code, mathematical formulas, and the full analysis.
• To execute the code and generate the PDF report, run the following command in terminal:

  quarto render main.qmd