Stat.Sample.Stratified.Treatment­Regression

Categorical within-cell regression statistics

Counts, empirical treatment fractions, the residualized treatment Gram sum, and the real-outcome regression estimator for an arbitrary finite cell type. Empty cells contribute zero and the estimator is zero when the Gram sum is zero. These definitions are independent of any paper or causal model.

Finite conditional treatment moments

Explicit enumeration of independent Boolean assignments at fixed categorical labels. The expected cell Gram is (N-1)_+ times pi(1-pi), including empty and singleton cells. Bernoulli design moments in Causalean provide the independent one- and two-coordinate expectations; these statements expose the required count and denominator algebra.

Categorical conditioning, including null cells

An arbitrary joint law of a finite cell and a Boolean treatment is disintegrated by explicit finite sums. Treatment propensity is guarded at zero cell mass; zero-mass label vectors have zero product weight. The fixed-label Gram identity is consequently a genuine conditional expectation statement on every occupied label fiber, without requiring overlap in null cells.

Lower-tail bound for within-cell treatment variation

In a regression of an outcome on a binary treatment with cell fixed effects, the denominator of the treatment coefficient is the within-cell treatment variation ∑ₖ tₖ (mₖ − tₖ) / mₖ, where cell k holds mₖ observations of which tₖ are treated. For n i.i.d. draws of (cell, treatment) with at most n cells and propensity in [ε, 1 − ε] in every cell of positive probability (0 < ε ≤ 1/2), this denominator has mean at least ε(1 − ε) n / 4 when n ≥ 2, and falls below ε(1 − ε) n / 8 with probability at most exp(−2 (ε(1 − ε)/8)² n), hence at most C(ε)/n with C(ε) = 1 / (2 (ε(1 − ε)/8)²) for every n ≥ 1. The bounds are uniform over all such laws; the tail bound is McDiarmid's inequality, since replacing one observation changes the denominator by at most one.

Main definitions and results

* gramThreshold, gramTailConstant — the constants ε(1 − ε)/8 and C(ε). * integral_gram_lower — the lower bound on the expected denominator (n ≥ 2). * gram_lower_tail_exp — the exponential lower-tail bound (n ≥ 2). * gram_lower_tail — the C(ε)/n bound for every n ≥ 1; at n = 1 it holds because C(ε) ≥ 1. * observed_gram_lower_tail — the same bound for the design observed from any i.i.d. sample with a measurable cell label and Boolean treatment; no overlap condition is imposed on null cells.

McDiarmid's inequality is taken from Causalean.Stat.Concentration.TailBounds.McDiarmid (formalized in FoML, github.com/auto-res/lean-rademacher).

Conditional regression residual second moments

Arm-cell residuals need only be centered and square integrable with their conditional second moments bounded by M². Arbitrary nonnegative functions of the finite observed design can weight the residual regression square. Cross-coordinate terms vanish, and the sum of squared residualized treatment weights equals the Gram, leaving the guarded reciprocal Gram envelope. Conditional moments are stated as normalized positive-mass design-fiber integrals, rather than choosing a conditional-expectation version on null fibers.

Squared-error bounds for interval clipping

Clipping a real estimate to a symmetric interval contracts its squared error at every target in that interval and bounds the error by four times the squared interval radius. These deterministic facts require no sampling law.

Algebra and coordinate oscillation of the categorical Gram denominator

The Gram is both a sum of squared residualized treatment weights and the treatment-weighted residualized numerator. Adding a single observation to a cell changes its Gram by a number in [0,1]; deleting and adding an observation therefore gives a coordinate-replacement oscillation bound of one.

Almost-sure support of observed categorical designs

A finite categorical sample almost surely visits only positive-mass cells. This supports occupied-cell assumptions in both conditional residual bounds and risk calculations without imposing assumptions on null cells.

Uniform repeat-count expectation

Under an arbitrary categorical iid law, the expected repeat count is n minus the expected number of occupied cells. Convexity of the empty-cell probability shows that the uniform law minimizes repetitions among laws with at most n available cells. For n at least two this gives n/4; at n=1 repetitions are identically zero. Null cells are included throughout.

Scalar occupancy inequalities

The empty-cell polynomial for any probability vector with at most n entries has at least n/4 expected repetitions for n at least two. This module isolates the analytic inequalities from the finite-product probability calculations.

Arbitrary finite-design residual coefficients

This module isolates the square-integrability and variance calculation for a sum of arm-cell residuals with arbitrary coefficients depending on the entire observed categorical design. It imposes no bounded-outcome condition. The regression application takes each coefficient to be the residualized treatment weight times the guarded inverse Gram. No categorical conditioning or Gram tail theorem is needed in this lower dependency layer.

Finite-sample MSE of clipped within-cell regression

The good-design conditional second moment costs M²/(c n). On the bad event, interval clipping bounds squared error by four times the squared interval radius. Combining the derived Gram tail and residual theorem yields a uniform 1/n risk bound, including D=0, n=1, arbitrary masses, and null cells.

The general theorem allows a clipping radius R distinct from the residual moment envelope M. With R=M and epsilon=1/4 its constant is 33152/9, less than the reference constant 72+4(1296+729/64).