Likelihood over Estimation: Robust Quadratic Discriminant Analysis for Heavy-Tailed Distributions with Theory and Evidence

Quadratic Discriminant Analysis (QDA) assumes Gaussian class-conditional distributions, causing systematic misclassification when data exhibit heavy tails. We propose Stable-QDA, which replaces the Gaussian likelihood with a symmetric $\alpha$-stable likelihood that decays polynomially rather than exponentially in Mahalanobis distance. Crucially, we find that correcting likelihood misspecification yields larger gains than robustifying parameter estimation: standard estimators (sample mean, Ledoit--Wolf covariance) often outperform robust alternatives when class heteroscedasticity is discriminative. We provide consistency guarantees under infinite-variance regimes, data-driven diagnostics for estimator selection, and demonstrate 15--53\% error reduction on real-world heavy-tailed benchmarks.