Adaptive Robust Confidence Intervals for High-Dimensional & Heavy-Tailed Data

Constructing reliable confidence intervals breaks down when data exhibits heavy tails, extreme outliers, or time-varying volatility. Learn how adaptive robust confidence intervals dynamically adjust tuning parameters and M-estimators to guarantee exact coverage probabilities without making rigid distributional assumptions.

Generate sample data with heavy-tailed Cauchy noise (outliers)

np.random.seed(42) clean_data = np.random.normal(loc=10.0, scale=2.0, size=200) outliers = np.random.cauchy(loc=10.0, scale=50.0, size=20) y = np.concatenate([clean_data, outliers])

1. Adaptive robust center estimation using Huber M-estimator

Threshold epsilon adapts to sample scale via MAD (Median Absolute Deviation)

mad = np.median(np.abs(y - np.median(y))) huber = HuberRegressor(epsilon=1.35, alpha=0.0) huber.fit(np.ones((len(y), 1)), y) robust_mean = huber.intercept_

2. Non-parametric bootstrapped Adaptive Confidence Interval

boot_means = [] for _ in range(1000): boot_sample = np.random.choice(y, size=len(y), replace=True) h_boot = HuberRegressor(epsilon=1.35, alpha=0.0) h_boot.fit(np.ones((len(boot_sample), 1)), boot_sample) boot_means.append(h_boot.intercept_)

ci_lower, ci_upper = np.percentile(boot_means, [2.5, 97.5])

print(f"Robust Point Estimate: {robust_mean:.3f}") print(f"95% Adaptive Robust CI: [{ci_lower:.3f}, {ci_upper:.3f}]")Key Takeaway: Adaptive robust confidence intervals provide valid statistical coverage and high efficiency by dynamically tuning truncation parameters according to the empirical distribution, offering essential protection against heavy tails and extreme data corruption.