Exact Conditional Distributions of Chi-Square-Family Statistics for Two-Way Contingency Tables, by Cell-Separable Dynamic Programming

William J. Dwyer

Home/Archive/arXiv

arXiv · 2608.08576

Exact Conditional Distributions of Chi-Square-Family Statistics for Two-Way Contingency Tables, by Cell-Separable Dynamic Programming

William J. Dwyer

View original source ↗Read source document ↗Export metadata ↓

Abstract

Every chi-square-family statistic for a two-way contingency table (Pearson's X^2, the power-divergence members, the variance-stabilized T_root) is referred to an approximate null distribution (chi-square, moment-matched chi-square, saddlepoint, or bootstrap) that miscalibrates on sparse or heterogeneous tables, where the exact distribution is a lattice of atoms rather than a continuous curve. The exact reference conditions on both margins (the multivariate Fisher noncentral hypergeometric law) but is usually treated as uncomputable, because the number of tables sharing a margin is astronomical. We show that the exact conditional moments and the exact conditional distribution of any chi-square-family statistic are computable without enumerating tables, because the statistic is additive over cells and the margin-conditional law factorizes cell by cell: a dynamic program walks the table one cell at a time, carrying per row-capacity state either a few moment accumulators, a map from statistic value to probability, or one complex number (the characteristic function), at a cost set by the number of margin states rather than the number of tables. The moment engine computes the exact conditional moments of a 5x5 table at three per cell (about 79 billion tables on the margin fibre) in three seconds; the distribution engine returns the exact tail, verified against enumeration to machine precision, where a three-moment chi-square fit misses it by up to 0.28. The engine furnishes Monte-Carlo-free ground truth against which any approximate reference can be measured, extends to the non-null alternative, and is the computational substrate beneath the exact conditional interval and reporting tools of the companion papers.

Explore related subjects

stat.ME stat.AP stat.CO

Keep this discovery

BibTeX RIS

William J. Dwyer. 2026-08-09. Exact Conditional Distributions of Chi-Square-Family Statistics for Two-Way Contingency Tables, by Cell-Separable Dynamic Programming. https://arxiv.org/abs/2608.08576

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Surprise Reduction and Nullification in Bayesian and Inverse Bayesian Inference under Ambiguous Prediction-Error Attribution

In non-stationary environments, prediction errors may signal environmental change or transient outliers, and adaptive systems must track such changes without overreacting to outliers. We distinguish surprise reduction, which updates beliefs to fit observations, from surprise nullification, which weakens constraints imposed by the predictive structure, and formalize both within Bayesian and inverse Bayesian (BIB) inference. Belief and likelihood updates are derived from variational objectives sharing a nullification strength, determined endogenously by minimizing surprise under the candidate post-update predictive distribution. In the Gaussian case, nullification expands belief and likelihood variances by a common factor relative to standard Bayesian updating, leaving the ratio unchanged. BIB thus defers attribution of the prediction error, committing to neither latent-state change nor observation-process uncertainty. The nullification strength is carried over as a candidate and is maintained or released according to the predictive surprise of the next observation. In a mean estimation task with outliers and changepoints, no scanned parameter setting of a Sage-Husa-type adaptive Kalman filter, fixed-strength BIB variant, or belief-forgetting-only variant outperforms BIB in both changepoint tracking and post-outlier stability. An oracle-informed reduced Bayesian model tracks changepoints better but is less stable after outliers. Although BIB maintains no explicit hypotheses about changepoints or outliers, it generates event-dependent dynamics. The learning rate increases after changepoints, whereas after outliers, nullification is released, and this increase is suppressed. Deferring attribution and letting subsequent observations differentiate the responses may constitute a principle of adaptive inference in non-stationary environments.

stat.ME↗

arXiv2026

Generalized Ridge Refitting for the Lasso and Prediction Improvement Bounds

We study a class of Lasso based estimators obtained by applying a quadratic correction on the Lasso equicorrelation set. The penalty matrix determines both the magnitude and geometry of the correction and contains, among other cases, the isotropic Lasso--Ridge correction, least squares refitting, Gram proportional interpolation between the Lasso and least squares, and coordinate specific penalties. We first derive a closed form representation and isolate the positive gain component of the resulting prediction improvement. We then control the remaining stochastic linear term in expectation by localizing the random signed equicorrelation model around a deterministic reference support. This yields a finite sample expectation bound that explicitly accounts for the randomness induced by Lasso model selection. The resulting decomposition provides a unified framework for understanding when Lasso based quadratic corrections can improve prediction.

stat.ME↗

arXiv2026

Discretization in covariate-adaptive randomization: gains and losses

Covariate-adaptive randomization(CAR) is widely implemented in clinical trials to balance prognostic covariates across treatment arms. Continuous covariates are often discretized into strata in practice, yet their consequences are not clearly understood. This paper provides a comprehensive study of the impact of discretization on both the CAR design process and the inferential results thereafter. We establish the asymptotic properties of both imbalance measures and treatment effect estimators under discretized and non-discretized settings. Practical recommendations are given on when and how discretization should be employed. We show that discretization in design is generally recommended, as it enhances robustness against model misspecification. However, if the true model is known, the most efficient strategy is to balance covariates according to that model in the design. The theoretical results are corroborated by extensive simulation studies and an empirical application to a diabetes trial dataset. Together, the results clarify the gains and losses of discretization in CAR and pave the way for learning impact of discretization to other designs and beyond.

stat.ME↗