Alphabeta MathStatistical Models and Experiments
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
21 pages- BH independent FDR4 results
This page defines false discovery proportion, false discovery rate, and family-wise error rate for a fixed finite testing family, then specifies the Benjamini–Hochberg step-up rule.
1 definition, 1 algorithm, 1 lemma, 1 theoremExamples & counterexamples → On a finite horizon, the page defines a simple unit-jump counting process, predictable at-risk and intensity processes, and a continuous compensator A t=∫ 0ᵗY sλ s ds.
4 definitions, 7 lemmas, 3 theoremsExamples & counterexamples →- Detection and quantification limits4 results
This page separates the blank decision threshold L C, the concentration L D meeting a specified false-negative target, and the concentration L Q meeting a declared precision criterion.
1 definition, 2 lemmas, 1 theoremExamples & counterexamples → This page proves a finite-horizon maximal bound for a nonnegative discrete-time test supermartingale, then passes to the infinite-horizon crossing and supremum bounds by…
1 lemma, 1 theoremExamples & counterexamples →- Full factorials3 results
For k two-level factors, a full factorial contains every one of the 2ᵏ coded runs.
1 definition, 2 theoremsExamples & counterexamples → - Group-sequential alpha spending3 results
This page defines a prescheduled group-sequential design under a stipulated canonical Gaussian simple-null law and proves the recursive boundaries exist uniquely.
1 definition, 1 algorithm, 1 theoremExamples & counterexamples → This page proves finite-dimensional terminal normal limits from a separately established subject-level influence expansion.
2 theoremsExamples & counterexamples →- IV and LATE9 results
For a binary instrument Z, binary treatment receipt A, and outcome Y, this page fixes unit-level potential treatments and outcomes, consistency, exclusion, joint instrument independence, relevance and monotonicity.
3 definitions, 4 lemmas, 2 theoremsExamples & counterexamples → This page distinguishes a latent value from its measured value and fixes the direction of the classical and Berkson additive error equations.
2 definitions, 1 proposition, 2 theoremsExamples & counterexamples →- Method agreement2 results
Method agreement compares paired measurements of the same target under a declared population of measurement conditions.
1 definition, 1 theoremExamples & counterexamples → - Missingness and nonidentification5 results
Missing-data analysis starts with a declared full-data target and an observation-pattern law.
2 definitions, 1 lemma, 2 theoremsExamples & counterexamples → - Poisson and Cox point processes13 results
This page defines locally finite counting measures with multiplicities, Poisson and Cox processes, and the scalar count law.
3 definitions, 7 lemmas, 3 theoremsExamples & counterexamples → - Polyhedral selective inference8 results
Condition on one fixed finite polyhedral selection event in a known-variance Gaussian location model.
3 definitions, 2 lemmas, 2 theorems, 1 corollaryExamples & counterexamples → This page fixes a target population and treatment versions before defining potential outcomes, consistency, the ATE/CATE, conditional exchangeability, and positivity.
4 definitions, 1 lemma, 1 theoremExamples & counterexamples →- Power and sample size11 results
A planning record fixes the statistical model, null and target alternative, size and power requirements, allocation, sidedness, integer candidate sizes, and any precision target.
1 definition, 2 algorithms, 4 lemmas, 1 proposition, 3 theoremsExamples & counterexamples → This page states the National Academies' convention for computational reproducibility, study replicability, and generalizability, and defines transparent reporting as an…
1 definitionExamples & counterexamples →This page defines the record needed to interpret a scientific statistical analysis: its observation law, units and indices, target, observation mechanism, support…
1 definitionExamples & counterexamples →- Sequential likelihood-ratio tests4 results
For two specified laws P 0 and P 1 whose restrictions to each finite-time sigma-algebra are equivalent, this page defines a two-boundary sequential likelihood-ratio rule.
1 algorithm, 1 lemma, 2 theoremsExamples & counterexamples → For an integer-indexed real process, strict stationarity concerns all finite-dimensional laws, while weak stationarity requires finite second moments, a constant mean and lag-dependent covariance.
2 definitions, 1 theorem, 1 corollaryExamples & counterexamples →For one fixed null law and interim sigma-algebra, the page proves that an adapted design keeps its null rejection probability within the original conditional-error budget…
1 lemma, 1 theorem, 1 corollaryExamples & counterexamples →This page defines a fixed finite family of null hypotheses, null-valid p-values, the true-null set, FWER, and FDR.
1 definition, 2 theoremsExamples & counterexamples →
Level 1
21 pagesThis page treats the finite assignment law as the source of probability while experimental units and their potential-outcome schedule remain fixed.
3 definitions, 1 lemma, 1 theoremExamples & counterexamples →- BY arbitrary dependence and BH PRDS7 results
The Benjamini–Yekutieli (BY) step-up rule uses critical values kq/(mH m), where H m=∑ j=1ᵐ1/j.
1 definition, 1 algorithm, 3 lemmas, 2 theoremsExamples & counterexamples → Specify the measurand and measurement model, carry correction and reference standard uncertainty through the full input covariance matrix, and distinguish Type A from Type B evaluation by how the information is evaluated.
1 definition, 1 theoremExamples & counterexamples →This page distinguishes censoring after sampling from truncation before sample inclusion.
1 definition, 1 theoremExamples & counterexamples →- Finite-history linear prediction7 results
For a weakly stationary real process, the finite-history best linear forecast is the L² projection onto the affine span of the observed history.
1 definition, 1 algorithm, 1 lemma, 4 theoremsExamples & counterexamples → - Finite-window spatial likelihood12 results
For a bounded Borel observation window W⊂ℝᵈ, this page fixes the reference measure and boundary convention, then derives the spatial Poisson density relative to unit-rate Poisson measure and its log likelihood.
4 definitions, 6 lemmas, 1 proposition, 1 theoremExamples & counterexamples → This page develops least squares for a fixed matrix X∈ℝ^n× p and response vector Y∈ℝⁿ.
3 definitions, 1 lemma, 4 theoremsExamples & counterexamples →Regular two-level fractions are selected from the coded {-1,+1}ᵏ run cube by independent generator equations.
1 definition, 3 theoremsExamples & counterexamples →This page specifies finite independent response vectors under an explicit product sampling law, then derives the generalized linear model's mean, variance, and likelihood derivatives.
3 definitions, 1 algorithm, 4 lemmas, 4 theoremsExamples & counterexamples →This page defines Holm's step-down procedure for a finite hypothesis family fixed before observing the data, verifies that its rejection events are measurable, and proves strong familywise error control.
1 algorithm, 1 lemma, 1 theoremExamples & counterexamples →- Missingness weights and sensitivity4 results
This page defines the response-probability weighted sample mean, proves the population inverse-weighting identity under the explicit MAR conditional independence condition…
1 definition, 1 algorithm, 1 lemma, 1 theoremExamples & counterexamples → This page fixes the vector sampling unit, the n-1 sample covariance convention, and the Wishart law as a Gaussian outer-product sum, including singular-support cases.
2 definitions, 2 lemmas, 4 theorems, 1 corollaryExamples & counterexamples →- PCA and eigengap stability7 results
For a specified parametric model and finite-second-moment vector, covariance PCA orders the eigenspaces of the centered covariance matrix; correlation PCA first divides each centered coordinate by its positive standard deviation.
1 definition, 1 lemma, 4 theorems, 1 corollaryExamples & counterexamples → - Random fields and variograms5 results
A random field assigns a real random variable to each spatial location on one probability space.
1 definition, 1 lemma, 3 theoremsExamples & counterexamples → - SCM, DAGs, adjustment14 results
This page specifies finite recursive structural causal models on full DAGs, including latent vertices, their topological probability-kernel law, and the global Markov…
4 definitions, 7 lemmas, 3 theoremsExamples & counterexamples → - Spatial summaries and Palm laws18 results
For a stationary locally finite point process on ℝᵈ with finite positive intensity, this page fixes the recentering convention and constructs the full Palm law from the refined Campbell identity.
5 definitions, 1 algorithm, 3 lemmas, 9 theoremsExamples & counterexamples → For a real weakly stationary sequence on ℤ, the covariance sequence is positive definite.
1 definition, 1 lemma, 3 theoremsExamples & counterexamples →This page defines the population standardization and inverse-probability-weighted functionals for a fixed discrete or atomic treatment value.
2 algorithms, 1 lemma, 2 theoremsExamples & counterexamples →- State-space model laws5 results
For a finite horizon T, a state-space model specifies a joint law for a latent state path X 0,…,X T and observations Y 1,…,Y T on finite products of standard-Borel spaces.
2 definitions, 1 lemma, 2 theoremsExamples & counterexamples → This page derives a time-uniform confidence sequence for a scalar mean from a discrete-time conditional moment-generating-function bound.
2 definitions, 3 lemmas, 2 theoremsExamples & counterexamples →- Temporal intensity and likelihood4 results
This page treats simple, locally finite event sequences on a finite observation window when a conditional waiting-time law and its predictable hazard are supplied, together…
1 definition, 2 lemmas, 1 theoremExamples & counterexamples →
Level 2
14 pages- ARMA roots, spectra and inference7 results
This page fixes the lag-polynomial convention, defines white noise by uncorrelatedness rather than independence, and uses coprime normalized polynomial pairs.
1 definition, 1 lemma, 4 theorems, 1 corollaryExamples & counterexamples → - Blocking, pairs, and Latin squares7 results
This page separates three sources of structure. A randomized complete-block design randomizes a treatment permutation independently inside each complete block. With fixed…
2 definitions, 2 lemmas, 1 proposition, 2 theoremsExamples & counterexamples → - Cox PH and partial likelihood7 results
This page specifies the Cox conditional event-intensity model, risk-set partial likelihood, and hazard-ratio interpretation.
1 definition, 4 lemmas, 1 theorem, 1 corollaryExamples & counterexamples → Derive exact finite-sample coefficient, nested-model, mean-response, and prediction inference for a fixed full-column-rank Gaussian linear model.
1 lemma, 2 theoremsExamples & counterexamples →This page treats a finite population's unit values as fixed and places the probability law on sampled subsets.
1 definition, 1 lemma, 2 theoremsExamples & counterexamples →- Interference and exposure mappings10 results
Model fixed finite-population outcomes that may respond to assignments inside a group while stating partial interference across groups as a separate design assumption.
4 definitions, 3 lemmas, 3 theoremsExamples & counterexamples → - Kriging3 results
Kriging chooses a linear predictor from finitely many field observations.
1 definition, 2 theoremsExamples & counterexamples → - Logrank and weighted logrank8 results
This page defines the two-group risk-set logrank score, its conventional hypergeometric plug-in variance, weighted scores, and the normal-calibrated statistic.
1 definition, 4 lemmas, 2 theorems, 1 corollaryExamples & counterexamples → This page defines controlled and natural direct and indirect effects before stating their identification assumptions.
4 definitions, 1 lemma, 1 theoremExamples & counterexamples →- Mixed effects and longitudinal models18 results
Develop the finite-dimensional linear mixed model and its longitudinal interpretation with explicit subject, visit, and observation-pattern roles.
6 definitions, 7 lemmas, 5 theoremsExamples & counterexamples → This page defines the pooled risk-set estimators with a fixed event and censoring endpoint convention.
1 definition, 2 lemmas, 2 theoremsExamples & counterexamples →- Parametric survival and reliability5 results
This page defines survival, reliability, density hazards, cumulative hazards, and exponential and Weibull lifetime families.
1 definition, 4 theoremsExamples & counterexamples → - Response surfaces and optimal design6 results
Response-surface models are local polynomial approximations on a declared factor region.
3 definitions, 1 lemma, 2 theoremsExamples & counterexamples → - Spatial summary estimators13 results
The Van Hove window and spatial-average claims assume Countable Choice AC ω, which makes every Borel window Lebesgue measurable; the boundary excesses are required to be Lebesgue measurable as well.
4 definitions, 4 lemmas, 4 theorems, 1 corollaryExamples & counterexamples →
Level 3
5 pagesThis page defines fixed process-condition, nested batch/day, and repeat indices, then distinguishes repeatability within stated short-term conditions, reproducibility across…
1 definition, 1 theoremExamples & counterexamples →- Biological and technical replication2 results
This page distinguishes biological, assignment, and technical measurement units, then proves the mean and variance of a balanced average under a two-level random-intercept model.
1 definition, 1 theoremExamples & counterexamples → - Cluster and two-stage sampling5 results
Cluster sampling selects whole primary sampling units (PSUs) and observes their secondary sampling units (SSUs).
2 definitions, 3 theoremsExamples & counterexamples → For finitely many causes, this page distinguishes the latent cause-specific times from the observed first time and cause.
1 definition, 1 algorithm, 1 lemma, 1 theoremExamples & counterexamples →- HT variance and SRSWOR5 results
This page develops design-based variance for fixed finite populations.
2 definitions, 3 theoremsExamples & counterexamples →
Level 4
1 page- Stratification and allocation4 results
This page treats the population values as fixed and the sample selections as random under the design.
1 definition, 3 theoremsExamples & counterexamples →