Alphabeta MathMathematical Statistics
Pathway
Pages are grouped by how many dependency steps into this group they sit. Everything a page needs from this group appears above it.
Level 0
4 pages- Concentration8 results
This page turns moment-generating-function bounds into finite-sample tail guarantees.
1 definition, 2 lemmas, 5 theoremsExamples & counterexamples → - Empirical-Process Complexity15 results
This page defines empirical and Rademacher processes with an explicit countability and outer-probability convention, then proves the independent-copy convex-moment…
5 definitions, 5 lemmas, 5 theoremsExamples & counterexamples → This page defines the causal Bernoulli-shift approximation convention in H=L²([0,1]) and proves the local ingredients needed for its sample-mean central limit theorem.
1 definition, 6 lemmas, 1 theoremExamples & counterexamples →An experiment specifies a measurable observation space and a probability law for each parameter value.
6 definitions, 1 theoremExamples & counterexamples →
Level 1
3 pagesThis page fixes the stochastic-order notation O P(1) and o P(1) using the Euclidean norm for fixed finite-dimensional random vectors.
4 definitions, 4 theoremsExamples & counterexamples →- Domination and Likelihood Ratios7 results
Domination fixes one sigma-finite positive reference measure for every law in an experiment.
3 definitions, 2 lemmas, 2 theoremsExamples & counterexamples → - Estimators and Randomized Procedures4 results
Statistics and estimators are measurable maps from observed data to named target spaces.
3 definitions, 1 lemmaExamples & counterexamples →
Level 2
8 pages- Chaining and Uniform Limits8 results
This page defines Glivenko–Cantelli and Donsker classes with explicit pointwise measurability and outer-probability conventions.
2 definitions, 2 lemmas, 4 theoremsExamples & counterexamples → Exchangeability means invariance of the joint law under coordinate permutations; it is weaker than independent identical sampling.
2 definitions, 2 algorithms, 4 lemmas, 3 theoremsExamples & counterexamples →- Density Estimation9 results
For iid observations with a density on ℝᵈ, this page defines the pointwise density target and the kernel density estimator.
1 definition, 5 lemmas, 3 theoremsExamples & counterexamples → This page develops the finite-dimensional Gaussian route from orthogonal changes of coordinates to independent chi-square components.
2 definitions, 2 lemmas, 4 theoremsExamples & counterexamples →- Loss, Risk and Decision Theory7 results
This page defines nonnegative loss and its possibly infinite sampling risk for deterministic and randomized rules.
3 definitions, 2 lemmas, 2 theoremsExamples & counterexamples → - M- and Z-Estimators4 results
This page defines M-estimators as measurable empirical criterion extrema and Z-estimators as measurable approximate roots.
2 definitions, 2 theoremsExamples & counterexamples → This page builds semiparametric score geometry from common-dominated paths that are differentiable in quadratic mean.
6 definitions, 3 theoremsExamples & counterexamples →This page develops sufficiency through parameter-independent conditional laws and likelihood factorization, then treats minimal sufficiency, complete statistics, exponential families, and Basu’s theorem.
7 definitions, 1 lemma, 9 theoremsExamples & counterexamples →
Level 3
6 pages- Contamination and Robust Risk9 results
This page separates three models: a local gross-error path, an iid Huber mixture, and adversarial replacement after clean iid sampling.
5 definitions, 3 lemmas, 1 theoremExamples & counterexamples → This page proves first- and second-order delta limits for finite-dimensional random vectors and the asymptotic linearity and normality of iid Z-estimators.
2 theoremsExamples & counterexamples →- Finite-Sample Point Estimation7 results
This page defines bias, mean squared error and uniformly minimum-variance unbiased estimation, proves the bias–variance identity, and establishes the Rao–Blackwell and Lehmann–Scheffé theorems with finite-moment hypotheses.
2 definitions, 2 lemmas, 3 theoremsExamples & counterexamples → - Minimax Lower Bounds9 results
This page builds finite-sample minimax lower bounds from KL divergence and the event-supremum total-variation convention defined on the stationary Markov-chain page.
2 definitions, 2 lemmas, 5 theoremsExamples & counterexamples → - Regression Estimation11 results
This page fixes a pointwise version of the regression function before imposing smoothness, then defines local-constant and local-polynomial estimators and pointwise squared-error risk.
4 definitions, 1 lemma, 6 theoremsExamples & counterexamples → - Tests, Neyman–Pearson and Valid P-values11 results
This page defines randomized tests, size, power, level, and validity of p-values.
3 definitions, 6 lemmas, 2 theoremsExamples & counterexamples →
Level 4
8 pagesThis page develops exact finite-sample inference by conditioning and by finite group actions.
4 definitions, 3 lemmas, 4 theoremsExamples & counterexamples →- Confidence Sets and Inversion4 results
This page defines confidence-set procedures through measurable membership events and repeated-sampling coverage.
2 definitions, 2 theoremsExamples & counterexamples → - Decision-Theoretic Upper Bounds13 results
This page studies squared fitted-value loss for fixed-design least squares.
1 definition, 2 algorithms, 4 lemmas, 5 theorems, 1 corollaryExamples & counterexamples → - Robust Means under Heavy Tails20 results
This page separates clean finite-variance estimation from estimation after adversarial sample replacement.
3 algorithms, 10 lemmas, 1 proposition, 3 theorems, 3 counterexamplesExamples & counterexamples → - Scores and Information Bounds4 results
This page defines the score and Fisher information for a dominated parameterized model, then proves the mean-zero score and expected log-likelihood Hessian identities under…
1 definition, 3 theoremsExamples & counterexamples → Develop discrete-time, null-uniform validity for optional monitoring from nonnegative test supermartingales.
3 definitions, 4 theoremsExamples & counterexamples →- Sparse Estimation and Covariance30 results
This page distinguishes sparse fixed-design regression from covariance estimation.
9 definitions, 1 algorithm, 13 lemmas, 6 theorems, 1 counterexampleExamples & counterexamples → - UMP Tests5 results
This page defines monotone likelihood ratio for a scalar statistic and proves the randomized one-sided Karlin–Rubin result.
1 definition, 2 lemmas, 2 theoremsExamples & counterexamples →
Level 5
4 pagesThis page defines pathwise derivatives on the allowed score tangent space, distinguishes parameter influence curves from estimator influence functions, and characterizes all derivative representers.
3 definitions, 2 theoremsExamples & counterexamples →Quadratic-mean differentiability gives an L² notion of smoothness for a dominated model, with a score vector and Fisher information that remain meaningful when pointwise derivatives are unavailable.
2 definitions, 1 lemma, 2 theorems, 1 corollaryExamples & counterexamples →This page separates two observation models that can produce different covariance guarantees.
6 definitions, 1 algorithm, 20 lemmas, 3 theoremsExamples & counterexamples →- Splines and Series Methods11 results
This page develops B-spline spaces and polynomial reproduction, proves the distinct-knot natural cubic interpolation system, and reduces cubic smoothing to a finite-dimensional spline fit.
5 definitions, 5 theorems, 1 corollaryExamples & counterexamples →
Level 6
3 pages- Influence and Breakdown9 results
Define statistical functionals and pointwise gross-error influence, then distinguish their local sensitivity from finite-sample replacement breakdown.
4 definitions, 2 lemmas, 3 theoremsExamples & counterexamples → - LAN, Contiguity and Efficiency18 results
This page defines contiguity and pointwise LAN using mean-one likelihood proxies, then proves the matching-mean criterion for contiguity and Le Cam's third lemma.
3 definitions, 9 lemmas, 6 theoremsExamples & counterexamples → - Nonregular and Misspecified Limits6 results
This page covers two limits that require conditions beyond regular interior likelihood theory.
2 definitions, 1 lemma, 2 theorems, 1 counterexampleExamples & counterexamples →
Level 7
1 page- Semiparametric Efficiency5 results
An asymptotically linear estimator has a Gaussian limit under the iid central limit theorem.
2 lemmas, 3 theoremsExamples & counterexamples →