Stat.Nonparametric.Histogram­Regression

Totalized finite-partition regression histograms

A finite measurable label map represents a partition: its fibers are disjoint and cover the covariate space. Empty and zero-mass fibers are allowed. Samples are tuples of observations, whose covariates and responses are supplied by maps. The estimator clips the empirical response average to [0,1] and uses a fixed default when the observed cell count is zero, including sample size zero.

Reference for the estimator and integrated marginal loss: A. Nobel, Histogram Regression Estimation Using Data-dependent Partitions, Ann. Statist. 24 (1996), 1084–1105, equations (1)–(2), https://nobel.web.unc.edu/wp-content/uploads/sites/13591/2017/01/histreg.pdf. Here partitions are deterministic, and the empty-cell default is arbitrary.

Regression cell means and oscillation bias

The conditional-mean premise is the defining set-integral characterization of E[Y | X] = g(X), together with integrability. It asks only for a population regression identity, never for any sample-risk or rate estimate. A separate bridge accepts Mathlib's conditional expectation on the sigma algebra generated by X. Boundedness of the regression representative is derived almost surely.

Ceiling meshes and optimized regression bandwidth algebra

Use max 1 (ceil (1/b)) bins along each axis. For 0<b≤1 its reciprocal lies between b/2 and b, and its d-dimensional cell count is at most (2/b)^d. The optimized positive-sample bandwidth is m^(-1/(2β+d)). All the real-power and ceiling comparisons needed by the statistical theorem are isolated here and make no probabilistic assumptions.

Euclidean unit-cube partition geometry

The covariate domain is the closed unit cube in Euclidean space, with the Euclidean (L², not sup) metric. Floors at interior grid points choose the cell to their right, and the coordinate value one is assigned to the final cell by clamping the index. This keeps exactly meshCount b ^ d labels, covers the boundary, and gives diameter at most sqrt(d)*b for positive bandwidths at most one. Arbitrary probability distributions on the cube are permitted.

Membership-pattern product moments

A fixed subset of sample indices defines a measurable rectangle: selected coordinates lie in a given cell, and other coordinates lie in its complement. Mathlib's restricted product measure and coordinate-product integral formulas give the factorization directly. The diagonal and centered cross-coordinate moments are isolated before the squared-sum identity. This layer depends only on the estimator definitions, not on binomial counts or regression hypotheses.

Totalized centered means on membership patterns

This layer turns the closed rectangle moments into a reciprocal-count identity on each membership pattern. It also isolates square integrability of the zero-totalized empirical mean, including sample size zero. The global variance formula in Centered can then sum over patterns and invoke the count law, without another combinatorial grouping by cardinality.

Global centered cell-sample variance

Condition on the full subset of observations falling in one cell. Centering annuls cross terms, leaving the cell variance times the reciprocal count. The finite-pattern moments are isolated in PatternMoments, and their totalized-mean consequences in PatternMeans, independently of the binomial identity in Counts; the bounded-response consequence controls the nonempty centered mean. Clipping and the arbitrary empty default are handled separately in Centered. Related infrastructure, without an import dependency on it, is in Causalean.Stat.Sample.Stratified.CenteredNoiseBound and Causalean.Stat.Sample.OccupancyWeightedMean.MomentBounds.

Totalized clipped cell-estimation risk

Combine the closed-form centered-mean variance and empty-cell probability with the deterministic clipping comparison. The fixed default may be any point in the unit interval. No lower cell-mass bound is required.

Binomial cell counts and empty-cell control

This layer transfers iid product sampling to the real binomial weights already provided by Causalean. The empty-cell inequality includes both endpoints of the probability interval and sample size zero. No regression or risk assumption is used here.

Hölder cubical histogram risk and optimized rate

For any observation law with unit-cube covariates and bounded responses whose conditional mean is Hölder, the ceiling-mesh histogram has squared bias of order b^(2β) and variance of order 1/(m*b^d). The optimized bandwidth gives the explicit rate m^(-2β/(2β+d)). Constants depend only on dimension and the Hölder constant/exponent, and all cube boundary points and empty cells are kept.

The integrated marginal loss and zero-empty-cell convention are classical: Andrew Nobel, "Histogram Regression Estimation Using Data-dependent Partitions" (1995), introduction and equation (2), https://nobel.web.unc.edu/wp-content/uploads/sites/13591/2017/01/histreg.pdf. Here the default is generalized to any fixed point of the unit interval; the finite-sample rate is derived from the closed occupancy and mesh lemmas.

Clipping and empty-default comparisons

Clipping an empirical mean to the unit interval contracts squared distance to every unit-interval center. An empty cell contributes at most one when the default lies in the same interval. These deterministic comparisons do not assume bounded responses; they separate estimator algebra from the iid variance analysis. The clipped squared error is integrable under every iid probability law with measurable inputs.

Transferring histogram loss along an iid sample law

The integrated squared histogram loss is a measurable function of the training tuple. Independent observations with a common law therefore have exactly the same expected integrated loss as canonical product samples. This layer does not require bounded responses or a conditional mean, and is independent of the statistical risk inequality in Risk.

Integration over the positive-mass partition

These label-API bridges reuse Mathlib's finite disjoint-union integral theorem, then remove zero-mass fibers. They supply the partition sum for integrated risk and the mass-one simplification for a constant cubical oscillation envelope. Zero-mass cells are retained in the estimator and omitted only from integrals.

Integrated finite-partition histogram risk

The universal bound is twice the mass-weighted squared oscillation plus six times the number of positive-mass cells divided by m+1. Sampling may be presented either as the canonical iid product law or as an independent family on an arbitrary probability space with a common observation law. No density, positive lower cell mass, or nonempty-sample premise is required.