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<!doctype html>
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<head>
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<title>Roadmap — Statlib</title>
<meta name="description" content="Theorems and concepts queued for formalization, by topic.">
<link rel="stylesheet" href="site.css?v=20260703-governance">
</head>
<body class="site-page">
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<nav>
<a href="index.html" class="brand">Statlib</a>
<div class="links">
<a href="index.html">About</a>
<a href="tutorial/index.html">Tutorial</a>
<a href="roadmap.html" class="active">Roadmap</a>
<a href="projects.html">Projects</a>
<a href="contribute.html">Contribute</a>
<a href="governance.html">Governance</a>
</div>
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<main>
<div class="wrap">
<h1>Roadmap</h1>
<p class="intro">We have curated a broad selection of core topics by reviewing foundational textbooks and leading journals in mathematical statistics, prioritizing the bedrock theories that have underpinned research over the last two decades. This list is intentionally non-exhaustive. Our goal is to systematically expand it to encompass modern research programs, such as high-dimensional statistics, random matrix theory, conformal inference, the theory of E-values, and computational-statistical trade-offs.</p>
<p class="intro">Furthermore, we recognize that much of contemporary and classical theoretical statistics relies fundamentally on empirical process theory and concentration inequalities. Given the immense depth of these fields, we have not yet charted specific results. Instead, we aim to populate these areas organically through community engagement, collaborating to discuss and prioritize our formalization efforts.</p>
<div class="topic-toc">
<div class="topic-toc-title">Contents</div>
<div class="topic-toc-part"><a href="#topic-i">Classical Cores (1920's-1990's)</a></div>
<ul class="topic-toc-sub">
<li><a href="#sec-1-1">1.1 Fundamentals of Decision Theory</a></li>
<li><a href="#sec-1-2">1.2 Comparison of Experiments</a></li>
<li><a href="#sec-1-3">1.3 Local Asymptotic Theories</a></li>
<li><a href="#sec-1-4">1.4 Semiparametric Efficiency Theories</a></li>
<li><a href="#sec-1-5">1.5 Nonparametric Function Estimation Theories</a></li>
<li><a href="#sec-1-6">1.6 Minimax Lower Bound Methods</a></li>
<li><a href="#sec-1-7">1.7 Ingster's Nonparametric Detection Theory</a></li>
<li><a href="#sec-1-8">1.8 Donoho-Johnstone Gaussian Sequence Model</a></li>
<li><a href="#sec-1-9">1.9 Function Space Theories and Wavelets</a></li>
<li><a href="#sec-1-10">1.10 Functional Estimation Theories</a></li>
<li><a href="#sec-1-11">1.11 Fundamentals of Bayesian Statistics</a></li>
<li><a href="#sec-1-12">1.12 Causal Identification Theories</a></li>
<li><a href="#sec-1-13">1.13 Core Probabilistic Toolbox</a></li>
</ul>
</div>
<section id="topic-i" class="topic">
<h2>Classical Cores (1920's-1990's)</h2>
<section id="sec-1-1" class="section">
<h3>1.1 Fundamentals of Decision Theory</h3>
<div class="sub-group">Definitions</div>
<ul class="items">
<li><strong>Decision Theoretic Setup</strong></li>
<li><strong>Bayes risk</strong>; <strong>Minimax risk</strong></li>
<li><strong>Admissible decision rules</strong></li>
<li><strong>Complete</strong> and <strong>Essentially complete classes</strong></li>
<li><strong>Invariant / equivariant decision rules under a group action</strong></li>
</ul>
<div class="sub-group">Theorems / methods.</div>
<ul class="items">
<li><strong>Wald's existence theorem</strong></li>
<li><strong>Wald's complete-class theorem</strong></li>
<li><strong>Pointwise characterization of Bayes rules</strong></li>
<li><strong>Minimax-Bayes duality</strong></li>
<li><strong>Stein's phenomenon</strong></li>
<li><strong>James-Stein shrinkage estimation</strong></li>
<li><strong>Hunt-Stein theorem</strong></li>
<li><strong>Pitman estimators under location/scale invariance</strong></li>
<li><strong>Sufficient Statistics, Factorization Theorem</strong></li>
<li><strong>Complete Statistics, Ancillary Statistics, Basu's Theorem</strong></li>
<li><strong>Sufficiency reduction and Rao-Blackwell / Lehmann-Scheffé UMVUE theory</strong></li>
</ul>
</section>
<section id="sec-1-2" class="section">
<h3>1.2 Comparison of Experiments</h3>
<div class="sub-group">Definitions.</div>
<ul class="items">
<li><strong>Statistical Experiment</strong></li>
<li><strong>Markov-kernel / decision-rule formulation of an experiment</strong></li>
<li><strong>Sufficiency of one experiment for another</strong></li>
<li><strong>Le Cam's deficiency and Le Cam pseudo-distance</strong></li>
<li><strong>Equivalent experiments</strong></li>
</ul>
<div class="sub-group">Theorems/methods.</div>
<ul class="items">
<li><strong>Blackwell's theorem</strong></li>
<li><strong>Le Cam's randomization criterion</strong></li>
<li><strong>Bayes-risk characterization of Le Cam's deficiency</strong></li>
<li><strong>Convergence of experiments</strong></li>
<li><strong>Asymptotic equivalence of experiments</strong></li>
</ul>
</section>
<section id="sec-1-3" class="section">
<h3>1.3 Local Asymptotic Theories</h3>
<div class="sub-group">Definitions</div>
<ul class="items">
<li><strong>Contiguity</strong></li>
<li><strong>Local likelihood ratio process</strong></li>
<li><strong>Quadratic Mean Differentiability (QMD), Score, and Fisher Information</strong></li>
<li><strong>Local Asymptotic Normality (LAN)</strong></li>
<li><strong>Local Asymptotic Mixed Normality (LAMN)</strong></li>
<li><strong>Regular estimation</strong></li>
</ul>
<div class="sub-group">Theorems/methods.</div>
<ul class="items">
<li><strong>Le Cam's Lemmas</strong></li>
<li><strong>QMD and LAN Expansion</strong></li>
<li><strong>Hájek's convolution theorem</strong></li>
<li><strong>Hájek's local asymptotic minimax theorem</strong></li>
<li><strong>Properties of MLE and Asymptotic Efficiency</strong></li>
<li><strong>Testing Efficiency: Pitman, Bahadur</strong></li>
</ul>
</section>
<section id="sec-1-4" class="section">
<h3>1.4 Semiparametric Efficiency Theories</h3>
<div class="sub-group">Definitions.</div>
<ul class="items">
<li><strong>Parametric Submodels and Tangent spaces</strong></li>
<li><strong>Efficient score and Information</strong></li>
<li><strong>Efficient influence function</strong></li>
<li><strong>Functionals and Pathwise Differentiability</strong></li>
<li><strong>Canonical gradient</strong></li>
<li><strong>Asymptotically linear estimators</strong></li>
</ul>
<div class="sub-group">Theorems/methods.</div>
<ul class="items">
<li><strong>Nonparametric Information Bound and Local Asymptotic Minimaxity</strong></li>
<li><strong>Nonparametric Convolution theorem</strong></li>
<li><strong>Construction of Efficient estimators and Cross-Fitting</strong></li>
</ul>
</section>
<section id="sec-1-5" class="section">
<h3>1.5 Nonparametric Function Estimation Theories</h3>
<div class="sub-group">Definitions/ objects.</div>
<ul class="items">
<li><strong>Kernel Smoothing</strong></li>
<li><strong>Histogram and partition-based estimators</strong></li>
<li><strong>Series / Projection / orthogonal-series estimators</strong></li>
<li><strong>Sieve estimators</strong></li>
<li><strong>Local polynomial regression</strong></li>
<li><strong>Stone's theorem on universal consistency</strong></li>
<li><strong>Asymptotic Properties of Kernel Smoothing Estimators</strong></li>
<li><strong>Asymptotic Theory of Cross-validation</strong></li>
<li><strong>Pointwise vs. global rates</strong></li>
<li><strong>Curse of dimensionality</strong></li>
<li><strong>Stone's minimax-optimal rate theorem</strong></li>
</ul>
</section>
<section id="sec-1-6" class="section">
<h3>1.6 Minimax Lower Bound Methods</h3>
<div class="sub-group">Definitions.</div>
<ul class="items">
<li><strong>Reduction from estimation to testing</strong></li>
<li><strong>Packing number, Covering number, Metric Entropy</strong></li>
</ul>
<div class="sub-group">Theorems / methods.</div>
<ul class="items">
<li><strong>Le Cam's two-point method</strong></li>
<li><strong>Le Cam's convex-hull method</strong></li>
<li><strong>Assouad's lemma</strong></li>
<li><strong>Birgé's refinement of Assouad</strong></li>
<li><strong>Fano's Lemma</strong></li>
<li><strong>Yang-Barron metric-entropy method</strong></li>
</ul>
</section>
<section id="sec-1-7" class="section">
<h3>1.7 Ingster's Nonparametric Detection Theory</h3>
<div class="sub-group">Definitions</div>
<ul class="items">
<li><strong>Minimax Separation Rates</strong></li>
<li><strong>Detection boundary</strong></li>
</ul>
<div class="sub-group">Theorems/methods.</div>
<ul class="items">
<li><strong>Minimax separation rate over Hölder / Sobolev / Besov classes</strong></li>
<li><strong>Analysis of Chi-square-type test statistics</strong></li>
<li><strong>Wavelet-based and quadratic-form tests</strong></li>
<li><strong>Lower bound via two-point or mixture reduction</strong></li>
<li><strong>Adaptive minimax tests</strong></li>
</ul>
</section>
<section id="sec-1-8" class="section">
<h3>1.8 Donoho-Johnstone Gaussian Sequence Model</h3>
<div class="sub-group">Definitions</div>
<ul class="items">
<li><strong>Gaussian sequence models</strong></li>
<li><strong>Weak and Strong Sparsity classes</strong></li>
<li><strong>Ellipsoids and Besov balls</strong></li>
<li><strong>Soft- and hard-thresholding rules</strong></li>
<li><strong>SureShrink, VisuShrink, block thresholding</strong></li>
</ul>
<div class="sub-group">Theorems / methods.</div>
<ul class="items">
<li><strong>Sharp Minimax risk over $\ell^p$-balls, Ellipsoids, and Pinsker's Theorem</strong></li>
<li><strong>Simultaneous near-minimax over Besov balls and Donoho-Johnstone universal-threshold theorem</strong></li>
<li><strong>Block thresholding and sharp minimax-optimal constants</strong></li>
<li><strong>SURE risk estimator and SureShrink data-driven thresholding</strong></li>
<li><strong>Maxiset theory</strong></li>
<li><strong>Asymptotic equivalence to other nonparametric models</strong></li>
</ul>
</section>
<section id="sec-1-9" class="section">
<h3>1.9 Function Space Theories and Wavelets</h3>
<div class="sub-group">Definitions.</div>
<ul class="items">
<li><strong>Hölder spaces</strong>, <strong>Sobolev spaces</strong>, <strong>Besov spaces</strong>, <strong>Triebel-Lizorkin spaces</strong>, <strong>Nikolskii-Anisotropic Smoothness Spaces</strong></li>
<li><strong>Modulus of continuity characterizations</strong></li>
<li><strong>Orthonormal wavelet basis</strong></li>
<li><strong>Multi-resolution analysis (MRA)</strong></li>
<li><strong>Daubechies compactly-supported wavelets</strong></li>
</ul>
<div class="sub-group">Theorems / methods.</div>
<ul class="items">
<li><strong>Wavelet characterization of Function Spaces</strong></li>
<li><strong>Wavelet Basis Approximation theorems</strong></li>
<li><strong>Jackson and Bernstein inequalities</strong></li>
<li><strong>Embeddings of Function spaces</strong></li>
</ul>
</section>
<section id="sec-1-10" class="section">
<h3>1.10 Functional Estimation Theories</h3>
<div class="sub-group">Definitions</div>
<ul class="items">
<li><strong>Donoho-Liu "geometrizing rates"</strong></li>
</ul>
<div class="sub-group">Theorems / methods.</div>
<ul class="items">
<li><strong>Birgé-Massart oracle inequality for penalized model-selection estimators</strong></li>
<li><strong>Lepski's method and Adaptive minimax rates</strong></li>
<li><strong>Donoho-Liu rates for quadratic functionals</strong></li>
<li><strong>Elbow Effect for Functional Estimation</strong></li>
<li><strong>Modulus-of-continuity / hardest-one-dimensional-subproblems</strong></li>
<li><strong>General Nonlinear Integrated functional Estimation</strong></li>
<li><strong>Higher Order Influence Functions</strong></li>
</ul>
</section>
<section id="sec-1-11" class="section">
<h3>1.11 Fundamentals of Bayesian Statistics</h3>
<div class="sub-group">Definitions</div>
<ul class="items">
<li><strong>Prior on Space of Probability measures</strong></li>
<li><strong>RCP, Posterior distribution and marginal likelihood</strong></li>
<li><strong>Bayes' theorem</strong></li>
<li><strong>Conjugate families of priors</strong></li>
<li><strong>Improper priors and conditions for proper posteriors</strong></li>
<li><strong>Posterior predictive distribution</strong></li>
<li><strong>Bayes factor</strong></li>
<li><strong>Credible intervals / sets</strong></li>
<li><strong>Exchangeable sequences</strong></li>
</ul>
<div class="sub-group">Theorems / methods.</div>
<ul class="items">
<li><strong>de Finetti's theorem</strong></li>
<li><strong>Hewitt-Savage zero-one law for exchangeable events</strong></li>
<li><strong>Doob's consistency theorem</strong></li>
<li><strong>Schwartz's theorem</strong></li>
<li><strong>Bernstein-von Mises theorems</strong></li>
<li><strong>Diaconis-Freedman inconsistency counterexamples</strong></li>
<li><strong>Lindley's paradox</strong></li>
<li><strong>Robbins' empirical Bayes / compound decision theory</strong></li>
<li><strong>Stein's empirical-Bayes interpretation of the James-Stein estimator</strong></li>
</ul>
</section>
<section id="sec-1-12" class="section">
<h3>1.12 Causal Identification Theories</h3>
<div class="sub-group">Definitions</div>
<ul class="items">
<li><strong>Potential outcomes and counterfactuals</strong></li>
<li><strong>Stable Unit Treatment Value Assumption (SUTVA)</strong></li>
<li><strong>Causal estimands</strong></li>
<li><strong>Causal Bayesian network / causal DAG</strong></li>
<li><strong>Structural Causal Model (SCM)</strong></li>
<li><strong>Intervention / do-operator</strong></li>
<li><strong>Interventional distribution</strong></li>
<li><strong>d-separation</strong></li>
<li><strong>Confounding / unmeasured confounding</strong></li>
<li><strong>Back-door path</strong>, <strong>front-door path</strong>, <strong>collider</strong>, <strong>chain</strong>, <strong>fork</strong></li>
<li><strong>Adjustment sets</strong></li>
<li><strong>Propensity score</strong></li>
<li><strong>Instrumental variables (IV)</strong></li>
<li><strong>Compliance types</strong></li>
<li><strong>Time-varying treatments and Confounder</strong></li>
<li><strong>Sequential exchangeability</strong></li>
<li><strong>Marginal Structural Model (MSM)</strong></li>
<li><strong>Structural Nested Mean Model (SNMM) / Structural Nested Model (SNM)</strong></li>
<li><strong>Mediation effects</strong></li>
<li><strong>Partial identification</strong></li>
</ul>
<div class="sub-group">Theorems / methods.</div>
<ul class="items">
<li><strong>Verma-Pearl d-separation theorem</strong></li>
<li><strong>Markov factorization theorem</strong></li>
<li><strong>Back-door criterion</strong></li>
<li><strong>Adjustment formula</strong></li>
<li><strong>Front-door criterion</strong></li>
<li><strong>Do-calculus, soundness theorem</strong></li>
<li><strong>Shpitser-Pearl completeness theorem</strong></li>
<li><strong>ID algorithm</strong></li>
<li><strong>Robins' g-formula</strong></li>
<li><strong>G-computation algorithm</strong></li>
<li><strong>Identification of dynamic treatment regimes</strong></li>
<li><strong>Marginal Structural Model identification</strong></li>
<li><strong>Structural Nested Mean Model identification</strong></li>
<li><strong>Doubly robust identification / estimation</strong></li>
<li><strong>Rosenbaum-Rubin theorem</strong></li>
<li><strong>LATE theorems</strong></li>
<li><strong>Manski bounds</strong></li>
<li><strong>Verma constraints</strong></li>
<li><strong>Mediation identifications</strong></li>
</ul>
</section>
<section id="sec-1-13" class="section">
<h3>1.13 Core Probabilistic Toolbox</h3>
<ul class="items">
<li><strong>Empirical Process Theory</strong></li>
<li><strong>Concentration Inequalities</strong></li>
</ul>
</section>
</section>
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