Literature

establishes $M^{-1}$ as the natural D sensitivity metric and provides the archetype for a global optimality certificate.

sequentially add points chosen by sensitivity; convergence and generalized-variance bounds.

extends generation ideas to $D_s$, selected parameter subsets, and discrete designs.

general concave criterion/equivalence viewpoint; consequences for iterative construction and transformations including $D_s$-related ideas.

broad $\Phi$-optimal theory including D, L, E and other criteria.

prior-art frame for DS15: determinant criteria reward extremes and tolerate singular information at optima, in contrast to distortion criteria (topic 4), which penalize both. The binning problem sits in the intersection: mass dilution ($W_b\mu_b\mu_b^\top/W_b$-structure) restores the distortion side's (M2) while keeping the design side's (M3) failure.

prior-art frame for DS15: determinant criteria reward extremes and tolerate singular information at optima, in contrast to distortion criteria (topic 4), which penalize both. The binning problem sits in the intersection: mass dilution ($W_b\mu_b\mu_b^\top/W_b$-structure) restores the distortion side's (M2) while keeping the design side's (M3) failure.

$D_s$ equivalence including singular cases; simplified sufficient conditions.

the comparison template for DS19's tilt certificate — a dual multiplier certifies optimality.

prior-art frame for DS15: determinant criteria reward extremes and tolerate singular information at optima, in contrast to distortion criteria (topic 4), which penalize both. The binning problem sits in the intersection: mass dilution ($W_b\mu_b\mu_b^\top/W_b$-structure) restores the distortion side's (M2) while keeping the design side's (M3) failure.

optimizes the worst scalar asymptotic relative efficiency across parameter values using score-function threshold quantizers and an iterative design.

high-resolution asymptotics, optimal scalar point density, FI loss versus bit depth.

multivariate conditional-score geometry and trace-FI bounds under finite-bit quantization; special geometric optimality results.

quantization-induced identifiability/FIM singularity limitations.

carries the quantized-Fisher geometry into minimax lower bounds for interactive communication protocols.

for exponential families, an optimal deterministic K-level trace-FIM quantizer can be chosen with convex-polytopal cells in sufficient-statistic space.

classical affine-invariant determinant grouping criteria.

likelihood/determinant clustering under multivariate models.

single-point exchange for classical partition criteria including determinant-type criteria.

determinant exchange routines and matrix/scatter update machinery in executable form.

computational study of determinant-style clustering and first-improvement versus steepest-ascent moves.

the classical anchors for DS8 (scalar efficient-score DP) and the $\mathrm{btw}\le\hat v_K$ leg of DS15's Proposition 4; the DS15 audit verified contiguity against full partition lattices, including duplicate values and unequal weights.

DS15's assumption (S) — uniqueness of the optimal efficient-score interval quantizer — holds for Gaussian efficient scores by any of these; with Pollard (1981) this makes the population→empirical transfer of the scalar optimum fully rigorous in one dimension.

closest structural precedent for the DS14 bridge shape — set convergence of empirical fixed-point (locally certified, not globally optimal) quantizers to the population fixed-point set, for convex additive distortions.

CVT theory, Lloyd algorithm, geometric quantization.

the published record of the finite-sample singleton phenomenon at exact optima — the qualitative shape of the $N\le18$ evidence (N-DS-BRIDGE-TREND). Hennig also supplies the normative point that one-point clusters are a modeling choice, not an invalidity.

nearest published instance of a uniform LLN over Voronoi-type partition classes (DS14 Step 3 companion to C1) and of existence/stationarity machinery for population quantizers. Thm 4.1/4.2 (secondary-verified via Levrard and Liu–Pagès): every cell of an optimal quantizer carries positive mass with null overlaps — the population (M2) analogue on the distortion side, and the deletion-vs-$W_{K-1}$ mechanism reused in DS15's rigidity lemma. Thm 5.1: scalar log-concave uniqueness (the book anchor for DS15's assumption (S)).

the published record of the finite-sample singleton phenomenon at exact optima — the qualitative shape of the $N\le18$ evidence (N-DS-BRIDGE-TREND). Hennig also supplies the normative point that one-point clusters are a modeling choice, not an invalidity.

DS15's assumption (S) — uniqueness of the optimal efficient-score interval quantizer — holds for Gaussian efficient scores by any of these; with Pollard (1981) this makes the population→empirical transfer of the scalar optimum fully rigorous in one dimension.

direct prior art for the compact-codebook rigidity and uniform-law core of DS16.1. What does not transfer: their feasible objects are nearest-center codebooks, whereas DS16 starts from arbitrary sample groupings; they also do not handle unbounded finite-second-moment sources, signed nuisance moments, profiled Fisher information, or a nuisance-margin price.

conceptual template for the DS14 slab-margin assumption (M4) and for a future fast-rate refinement of the bridge. Levrard's $p_{\min}$ (mass) and $B$ (separation) are the published codification of what (M2)/(M5) buy; their positivity at population optima is inherited from Graf–Luschgy, never assumed of empirical optima.

adjacent common/anisotropic metric Voronoi geometry.

forces any polynomial exact algorithm for the tilted ScoreQuant dual to prove structure specific to the interval-segmentation DAG and its quadratic segment costs.

DS15's assumption (S) — uniqueness of the optimal efficient-score interval quantizer — holds for Gaussian efficient scores by any of these; with Pollard (1981) this makes the population→empirical transfer of the scalar optimum fully rigorous in one dimension.

the closest published structure to DS15's Proposition 6 — constraints imposed at asymptotically negligible cost — and the reference to engage before any publication of the steering construction.

practical adjacent categorization method; not exact hard D partition theory.

defines the event-by-event sensitivity $\gamma_i=(1/w_i)\,\partial w_i/\partial\theta$ — the single-parameter score — shows the binned-fit information is $I_\theta=\sum_k s_k\phi_k^2$ with $\phi_k=\langle\gamma\rangle_k$ the cell-mean sensitivity, and argues the optimal partitioning variable is $\gamma$ itself. Defines FIP$_3=I_\theta/I_\theta^{(\rm ideal)}\in[0,1]$ and factors it as FIP$_{\rm efS}\times$FIP$_{\rm shS}\times$FIP$_{\rm shB}$ (efficiency $\times$ signal sharpness $\times$ background sharpness).

$(F^{-1})_{r_sr_s}$ is the inverse Schur complement, so the SANNT objective is the profiled criterion of DS-SCHUR with $s=1$ — the production-scale realisation of the $D_s$ theme, arrived at independently.

projects the score into a locally/asymptotically Fisher-preserving summary for parameters of interest using the nuisance Schur geometry.

normal-theory exact distribution of the sample correlation ratio $\eta^2$ (between-group over total centred sum of squares).

Chapter 28, functions of sample moments: a smooth function of sample moments is asymptotically normal with variance from the first-order expansion, under the moment conditions that make the underlying sample moments asymptotically normal.

§26.21–26.22 define the population correlation ratio (26.40), (26.47) and the sample version (26.48) with $0\le r^2\le e^2\le1$ (26.49); §26.23–26.24 give the decomposition (26.50)–(26.51) and the normal-theory $F$ tests (26.52)–(26.55) for $\eta^2=0$, $\rho=0$ and linearity, attributed to Fisher, via Cochran's theorem with the arrays fixed. No large-sample variance of the correlation ratio and no moment-based non-normal form are given.

the non-null distribution of the LR (determinant-ratio) criterion for the multivariate linear hypothesis, expanded to order $N^{-2}$ under a fixed alternative with no rank assumption on the noncentrality matrix; normal theory.

the formal Edgeworth expansion of $\sqrt n[H(\bar Z)-H(\mu)]$ is valid for smooth $H$ of a sample-mean vector under moment and Cramér-type conditions; the class covers all smooth functions of sample moments.

the symmetric, linearly invariant measures of association between two sets of variables are all functions of the canonical correlations, generalize $R^2$, and are strictly ordered for any two sets.

delta-method variance of the ANOVA intraclass-correlation estimator under the one-way random-effects model.

for populations with finite fourth moments, the sample canonical correlations and their test statistics are asymptotically normal with a covariance involving fourth-order cumulants; elliptical simplification and a corrected chi-squared test for zero coefficients.

Chapter 3 (transformations of given statistics), Section 3.3 Theorem B: the multivariate delta method for functions of asymptotically normal vectors.

closed-form Hampel influence functions for the generalized variance, the regression-coefficient matrix, the MANOVA noncentrality matrix $\Sigma^{-1}\delta$ and its eigenvalues, canonical correlations, principal components and the parameters behind Pillai's statistic, Hotelling's $T_0^2$ and Wilks' $\Lambda$; $r^2$, $R^2$ and Mahalanobis $D^2$ as special cases.

limit laws of smooth functions of the eigenvalues of the standard covariance/MANOVA/canonical-correlation random matrices under nonnormal sampling (fourth-cumulant covariance).

for a statistical functional $T$ at $F$, the asymptotic variance is $V(T,F)=\int \mathrm{IF}(x;T,F)^2\,dF(x)$.

Edgeworth expansions for the smooth function model; the studentized statistic's skewness term differs in sign and size from the non-studentized one, the standard explanation of small-$n$ under-coverage of plug-in Wald intervals.

perturbation theory of the generalized eigenproblem yields the influence function of each squared canonical correlation and canonical vector pair; three sample versions; the squared multiple correlation and correspondence-analysis eigenvalues as special cases.

null distributions of the dimensionality (rank) tests in CCA and MANOVA under nonnormal populations.

Theorem 3.1 (delta method): if $r_n(T_n-\theta)\Rightarrow T$ and $\phi$ is differentiable at $\theta$, then $r_n(\phi(T_n)-\phi(\theta))\Rightarrow\phi'_\theta(T)$. Chapter 3 works the sample variance under $E X^4<\infty$; Chapter 5 (Theorem 5.23) gives the M-estimator sandwich and the plug-in variance principle.

Part Three, Chapter 8 "Some important differentials": §8.3 "The differential of a determinant" (p. 149, $d|X|=|X|\operatorname{tr}X^{-1}dX$), §8.4 "The differential of an inverse" (p. 151); Chapter 9 §9.10 "Scalar functions of a matrix, II: determinant" (p. 178).

rank tests from the smallest estimated characteristic roots of a $\sqrt n$-consistent, asymptotically normal matrix estimate; under rank $r$ the $n$-scaled smallest roots converge to a weighted sum of independent $\chi^2$ variables.

§7.5 "The generalized variance"; §8.3–8.5 likelihood-ratio criterion for linear hypotheses, its distribution and asymptotic expansion; §8.9 "Multivariate analysis of variance"; §12.2–12.4 canonical correlations, estimation and inference; §13.5–13.6 asymptotic distributions of roots (one and two Wishart matrices). Normal theory throughout these sections.

influence functions of canonical correlations and vectors for general scatter-matrix functionals; for the sample covariance the squared correlation has $\operatorname{IF}(\rho_i^2)=2\rho_iu_iv_i-\rho_i^2(u_i^2+v_i^2)$ in the canonical variates $u_i,v_i$.

noncentral-$F$/$t$/$\chi^2$ confidence intervals for $\eta^2$, $R^2$ and Cohen's $d$ under normal-theory ANOVA (MBESS).