ABC Inference for Alpha-Stable Models
A comparative study on Approximate Bayesian Computation (ABC) for parameter estimation of alpha-stable distributions. Given the intractability of the likelihood function, the project implements high-performance simulation kernels and compares various sampling strategies to balance computational cost and posterior accuracy.
Methodology
CMS Generator
Chambers-Mallows-Stuck implementation for exact stable sampling.
Variance Reduction
RQMC
Sobol & Halton sequences to accelerate integration convergence.
Best Performance
SMC-ABC
Geometric epsilon decay for optimal posterior refinement.
Complexity
\(\mathcal{O}(T \cdot \text{cost})\)
Scaling with iterations and summary statistic evaluation.
Algorithm Convergence (L2 Error)
Mathematical Framework & Kernel
To estimate parameters \(\theta = (\alpha, \beta, \gamma, \delta)\), we compare simulated summary statistics \(S(Y_{sim})\) against observed data \(S(Y_{obs})\) using a tolerance threshold \(\epsilon\):
The Sequential Monte Carlo (SMC) approach uses a series of decreasing tolerances \(\epsilon_t\), moving particles toward the true posterior density:
Research Presentation
Detailed technical slides on the ABC implementation and multivariate extensions.