A novel statistical method for confidence interval construction that automatically adapts to model misspecification.
Standard statistical inference faces a fundamental tradeoff:
| Approach | Model Correct | Model Wrong |
|---|---|---|
| Naive OLS | Optimal (narrowest CIs) | Catastrophic (coverage collapses to 35-77%) |
| Robust methods (sandwich SE, bootstrap) | Pays a width penalty | Protected |
Practitioners must choose before seeing the data — but rarely know which regime they're in.
AMRI resolves this tradeoff using a data-driven diagnostic that detects misspecification in real time.
AMRI_v1(X, Y, alpha):
1. Fit OLS → θ̂, residuals e
2. Compute SE_naive and SE_HC3
3. Diagnostic ratio: R = SE_HC3 / SE_naive
4. Adaptive threshold: τ(n) = 1 + 2/√n
5. IF R > τ(n) OR R < 1/τ(n):
SE = SE_HC3 × 1.05 [robust mode]
ELSE:
SE = SE_naive [efficient mode]
6. CI = θ̂ ± t_{n-2, 1-α/2} × SE
AMRI v2 eliminates the hard switch with a continuous blending function, avoiding the pre-test discontinuity that plagues classical procedures:
AMRI_v2(X, Y, alpha, c1=1.0, c2=2.0):
1. Fit OLS → θ̂, residuals e
2. Compute SE_naive and SE_HC3
3. Diagnostic: R = SE_HC3 / SE_naive
4. Blending weight: w = clip((|log R| - c1/√n) / (c2/√n - c1/√n), 0, 1)
5. SE = (1 - w) · SE_naive + w · SE_HC3
6. CI = θ̂ ± t_{n-2, 1-α/2} × SE
Key advantage: The blending weight w transitions smoothly from 0 (fully efficient) to 1 (fully robust), producing a Lipschitz-continuous mapping from DGP parameters to coverage.
AMRI includes a fully interactive web dashboard built with Plotly Dash for exploring simulation results, running custom simulations, and visualizing inference outcomes across all methods.
pip install dash dash-bootstrap-components dash-bootstrap-templates plotly "dash[diskcache]"
cd dashboard && python app.py
# Open http://localhost:8050Configure a DGP, set misspecification severity, and run all 10 inference methods on the same dataset. The forest plot shows each method's confidence interval, point estimate, and whether it covers the true parameter.
Browse 895 pre-computed scenarios with interactive filters by DGP, method, sample size, and delta.
Explore AMRI performance on 61 real-world datasets and see head-to-head comparisons with AKS (2025, Econometrica).
| Metric | AMRI v1 | AMRI v2 | Winner |
|---|---|---|---|
| Mean coverage | 0.9511 | 0.9483 | Both ≥ 0.95 target |
| Coverage std dev | 0.0086 | 0.0073 | v2 (17% less variable) |
| Min coverage | 0.918 | 0.919 | Comparable |
| Narrower CIs | 5% of scenarios | 95% of scenarios | v2 overwhelmingly |
| Metric | AMRI v1 | Best Competitor | Advantage |
|---|---|---|---|
| Overall coverage | 0.949 | HC3: 0.945 | +0.4pp (p=0.003) |
| Coverage at δ=0 (correct model) | 0.948 | Naive: 0.947 | Matches |
| Coverage at δ=1 (severe misspec) | 0.945 | HC3: 0.936 | +0.9pp |
| Width overhead at δ=0 | 1.01× | — | Negligible |
| Degradation rate | +0.007 | Boot-t: −0.003 | Only positive slope |
AMRI is the only method whose coverage improves as misspecification gets worse.
| Dataset | n | Naive Coverage | AMRI Coverage | AMRI Mode |
|---|---|---|---|---|
| California Rooms | 20,640 | 0.355 | 0.965 | Robust |
| Star98 Math | 303 | 0.624 | 1.000 | Robust |
| Fair Affairs | 6,366 | 0.905 | 0.960 | Robust |
| Diabetes BMI | 442 | 0.966 | 0.966 | Efficient |
| Iris Sepal | 150 | 0.966 | 0.966 | Efficient |
Average: Naive 0.853, HC3 0.960, AMRI 0.962
AMRI survived 10/10 deliberately hostile DGPs (coverage ≥ 0.92 in all cases), including threshold-edge attacks, extreme heavy tails (t₂.₅), leverage outliers, and mixture-of-regressions.
Three formal theorems verified both analytically and numerically:
| Theorem | Statement | Verification |
|---|---|---|
| 1. Coverage Continuity | Coverage is Lipschitz-continuous in (δ, n) — no pre-test jumps | Max coverage jump 0.007 < ε=0.01 |
| 2. Asymptotic Validity | Coverage → 1−α for any DGP with finite 4th moments | Coverage 0.9478 ≈ 0.95 at n=5000 |
| 3. Minimax Regret | Max width penalty < always-HC3 | Regret 0.0094 < HC3 regret 0.0652 |
See src/theoretical_guarantees.py for full proof sketches and Monte Carlo verification.
| Test | Result | p-value |
|---|---|---|
| TOST equivalence to 0.95 (ε=0.01) | EQUIVALENT | 0.00008 |
| AMRI > HC3 (paired t-test) | SIGNIFICANT | 0.0029 |
| AMRI > HC3 (permutation, 50K reps) | SIGNIFICANT | 0.0017 |
| Hochberg FWER correction (5 hypotheses) | 4/5 confirmed | — |
| Cohen's d vs Naive OLS | 1.113 (large) | — |
├── README.md # This file
├── HYPOTHESES.md # Complete hypotheses, results, and proofs
├── AMRI_METHOD.md # Standalone AMRI method documentation (v1 + v2)
├── PAPER_DRAFT.md # Full paper draft structure (~19 pages)
├── RESEARCH_PLAN.md # Initial research plan
├── EXPERIMENTAL_DESIGN.md # Experimental design specification
├── Reference/
│ └── REFERENCES.md # Complete bibliography (30+ references)
├── amri/ # Core AMRI package
│ ├── core.py # amri_v1(), amri_v2(), adaptive_ci()
│ ├── estimators.py # OLS, Logistic, Poisson estimators
│ ├── diagnostics.py # SE ratio, blending weight, diagnostics
│ └── tests/ # Unit tests
├── src/
│ ├── simulation.py # Core simulation engine (6 DGPs × 10 methods)
│ ├── competitor_comparison.py # Head-to-head: 10 methods including AKS (2025)
│ ├── formal_theory.py # Theorem 1-3 verification (B=3000)
│ ├── minimax_theory.py # Minimax lower bounds & oracle efficiency
│ ├── real_data_validation.py # 61 real-world datasets (50K bootstrap)
│ ├── statistical_guarantees.py# 9 formal hypothesis tests
│ ├── publication_figures.py # 8 publication-quality figures
│ └── ... # Additional analysis scripts
├── dashboard/ # Interactive Plotly Dash web app
│ ├── app.py # Main entry point (http://localhost:8050)
│ ├── pages/ # 6 pages: overview, simulation, results, etc.
│ ├── components/ # Reusable plots, cards, navbar
│ ├── engine/ # Simulation engine + data loader
│ └── screenshots/ # Dashboard screenshots
├── results/ # Pre-computed CSV results (895+ scenarios)
├── figures/ # Publication-ready figures (PNG + PDF)
└── paper/ # LaTeX manuscript
pip install numpy pandas scipy statsmodels scikit-learn matplotlib seaborn rdatasets# Pilot (fast, ~5 min)
python src/run_fast.py
# Full simulation (1440 scenarios × 2000 reps, 9 methods)
python -u src/run_vectorized_v2.py
# AMRI v2 head-to-head comparison (fast, ~1 min)
python src/run_amri_v2_standalone.py
# Theoretical guarantees verification (3 theorems)
python src/theoretical_guarantees.py# Deep analysis figures
python src/deep_analysis.py
# AMRI comprehensive figure
python src/amri_figure.py
# Formal statistical guarantees (7 tests)
python src/statistical_guarantees.py
# Generalization proof (theory + adversarial + hypothesis testing)
python src/generalization_proof.py
# Real-world data validation (11 datasets, 50K bootstrap)
python src/real_data_validation.py
# One-click reanalysis (after full simulation completes)
python -u src/reanalyze_complete.py| # | Hypothesis | Status | p-value |
|---|---|---|---|
| H1 | Naive coverage degrades monotonically with misspecification | Confirmed | < 10⁻¹⁰ |
| H2 | Sandwich HC3 maintains coverage ≥ 0.93 universally | Confirmed | Simes NS |
| H3 | AMRI achieves best-of-both-worlds | Confirmed | 0.0007 |
| H4 | Degradation rate: Naive >> Bootstrap > Sandwich > AMRI | Confirmed | 0.00008 |
| H5 | Robust methods widen adaptively under misspecification | Confirmed | 0.011 |
| H6 | AMRI v2 soft-thresholding eliminates pre-test discontinuity | Confirmed | Coverage jump < 0.01 |
| H7 | AMRI v2 strictly dominates v1 in width-coverage tradeoff | Mixed | Ceiling effect at high δ |
| Bonus | More data hurts Naive but helps AMRI (sample size paradox) | Confirmed | 0.015 |
- Naive OLS — Model-based standard errors
- Sandwich HC0 — Heteroscedasticity-consistent (White, 1980)
- Sandwich HC3 — Leverage-corrected (MacKinnon & White, 1985)
- Pairs Bootstrap — Resample (Xᵢ, Yᵢ) pairs
- Wild Bootstrap — Rademacher perturbation of residuals
- Bootstrap-t — Studentized bootstrap
- Bayesian — Normal-Gamma posterior with vague priors
- AMRI v1 — Hard-switching adaptive inference (proposed)
- AMRI v2 — Soft-thresholding adaptive inference (proposed, recommended)
| DGP | Misspecification Type | How δ Controls Severity |
|---|---|---|
| 1 | Nonlinearity | Y = X + δ·X² + ε |
| 2 | Heavy-tailed errors | ε ~ (1-δ)·N(0,1) + δ·t(3) |
| 3 | Heteroscedasticity | Var(ε|X) = (1 + δ·X²) |
| 4 | Omitted variable | Y = X + δ·Z + ε, Z⊥X |
| 5 | Clustering | ICC increases with δ |
| 6 | Contaminated normal | ε ~ (1-δ)·N(0,1) + δ·N(0,25) |
If you use this code or method, please cite:
@software{amri2026,
title={AMRI: Adaptive Misspecification-Robust Inference},
author={Anish},
year={2026},
url={https://github.com/ans036/AMRI-Adaptive-Misspecification-Robust-Inference}
}MIT License. See LICENSE for details.