@techreport{causalsmith_stat_mar_rareq_logfrontier_v1,
  title       = {Treatment effect estimation with missing outcomes and many covariate categories},
  author      = {{CausalSmith}},
  year        = {2026},
  month       = oct,
  institution = {CausalSmith Working Papers},
  number      = {CSWP-2026-034},
  url         = {https://causalsmith.org/papers/stat_mar_rareq_logfrontier_v1/},
  note        = {Version 5, 2 Oct 2026. Machine-generated; formal statements verified in Lean 4 at commit 15d86b3}
}

Introduction

The precision of a randomized trial at a prespecified analysis horizon depends on both the primary outcomes that have arrived and the information recorded for the full sample. Baseline categories, treatment assignments, and short-term measurements can remain available while primary outcomes are being collected. When outcome arrival varies across these observed characteristics, estimating a population treatment effect requires combining population cell weights with outcome information within cells. This paper characterizes the resulting statistical precision in a balanced randomized experiment with binary primary outcomes, a binary surrogate, and a finite baseline alphabet.

The model imposes independent and identically distributed sampling, treatment assignment independent of baseline characteristics and potential outcomes and surrogates, assignment probability one half, and consistency. Outcome arrival is missing at random conditional on treatment, baseline category, and realized surrogate. Each occupied cell has an unknown arrival probability at least a supplied value $q$, where $0<q\le 1$. This lower bound indexes the model and is used to tune the estimator and interval; individual cell probabilities remain unknown. Baseline probabilities may vary arbitrarily, including zero probabilities. The surrogate supplies observed conditioning information for ascertainment, and the target is the population average treatment effect on the primary outcome. Under these conditions, Lemma 3 expresses the effect as a signed sum of cell membership probabilities multiplied by outcome means among arrivals. Membership information from every sampled individual enters this representation.

Our first result is a constructive squared-error guarantee across the entire positive-arrival domain. Write $n$ for the sample size, $d$ for the number of baseline categories, and $N=nq$ for the effective sample-size scale. The mixed-count estimator in Algorithm 1 satisfies a uniform risk bound of order $$r_{n,d,q}=\min \left\{1,\frac{1}{N}+{\left[\frac{d}{N\log (e+N)}\right]}^2\right\},$$ where $r_{n,d,q}$ is the comparison rate, with a universal comparison constant, as established in Theorem 6. The construction separates membership, pilot, and outcome estimation into three auxiliary streams. Membership counts estimate cell weights; the pilot selects between empirical arrived-outcome means and a Chebyshev polynomial statistic built from marked factorial counts. Exact conditional averaging over the auxiliary randomization produces a deterministic estimator.

Under the rare-arrival restriction $$q{\log }^2(e+nq)\le \frac{1}{64},$$ Theorem 4 establishes matching minimax squared-error order $r_{n,d,q}$, with universal constants. The decision class includes all bounded parameter-independent randomized estimators. Its minimax value agrees with that for deterministic estimators by Lemma 1. The lower bound retains the full observed experiment, including memberships whose outcomes have yet to arrive. A one-category family supplies the reciprocal effective-sample-size contribution, while an activated moment-matching construction supplies the category contribution. Along sequences satisfying the rare-arrival restriction and $N\to \infty$, Proposition 5 consequently gives consistency exactly when $d=o\{N\log (e+N)\}$, and squared-error order $N^{-1}$ exactly when $d=O\{\sqrt{N}\log (e+N)\}$.

The same rare-arrival experiment admits a matching uncertainty-quantification comparison. For each fixed miscoverage level $\alpha \in (0,1)$, Theorem 7 establishes optimal maximal expected length of order $\sqrt{r_{n,d,q}}$ among connected intervals with uniform coverage at least $1-\alpha$. The comparison constants may depend on $\alpha$. Coverage and expected length integrate jointly over the sample and any endpoint randomization. A deterministic squared-error envelope supplies an attaining interval, and the converse covers the full randomized endpoint decision class.

Near complete ascertainment, the remaining deficit supplies a complementary precision scale. Throughout the arrival-floor model, Theorem 1 gives $$\frac{1}{256n}\le {\mathfrak{R}}_{n,d,q}^{+}\le \min \left\{1,\overline{C}{r}_{n,d,q},\frac{4}{n}+(1-q{)}^2\right\},$$ where ${\mathfrak{R}}_{n,d,q}^{+}$ is the minimax squared-error risk and $\overline{C}$ is a universal constant. The observed-outcome arm contrast attains the last upper bound. For $d\ge 1024n^2$, Theorem 2 establishes matching order $n^{-1}+(1-q{)}^2$ uniformly in $q$. Hence, along arrival-floor sequences, an eventual $n^{-1}$ risk guarantee holding simultaneously over every alphabet size is equivalent to $1-q_n=O(n^{-1/2})$, by Proposition 1. Complete arrival gives $n^{-1}$ order uniformly over arbitrary alphabet growth.

The construction also connects randomized outcome arrival to observational adjustment. An independent fair assignment retains an observational outcome when the synthetic assignment agrees with the observed treatment. Under a fixed strict-interior treatment-positivity margin, this transformation preserves the baseline-weighted regression contrast and supplies a randomized missing-at-random completion. Averaging the mixed-count estimator over synthetic assignments yields the deterministic observational procedure in Theorem 8. At each fixed margin strictly between zero and one half, its minimax squared-error order is $$\min \left\{1,\frac{1}{n}+{\left[\frac{d}{n\log (e+n)}\right]}^2\right\},$$ with margin-dependent constants. Consistency and conditional exchangeability give the observational contrast its population causal interpretation. The same experiment comparison supplies the matched randomized point-risk order at every fixed $0<q<1/2$ in Theorem 5, with constants depending on that floor.

The statistical mechanism is a weighted-cell-mean problem with many scarce arrived counts. Ordinary cell ratios accumulate a category-dependent bias; on the difficult large-alphabet branch, a Chebyshev correction reduces the aggregate bias to the scale $d/\{N\log (e+N)\}$. The converse uses moment-matched reciprocal priors whose activated observation laws remain close even though all cell memberships are retained. Each augmented record carries an independent activation flag with probability $qH$, where $H$ is the upper endpoint of the latent prior support and has squared-logarithmic order. This explains the rare-arrival condition ensuring a valid same-experiment construction. Section 5 and Theorem 12 present these two sides.

These results connect three strands of existing work. Surrogate-assisted analysis explains how early measurements inform treatment-effect estimation and efficiency under appropriate observation and nuisance conditions (Kallus et al., 2025). Decaying-labelling analysis establishes asymptotic inference under effective-sample-size and nuisance-rate conditions (Zhang et al., 2025). Our comparisons characterize uniform squared error and honest expected interval length as category frequencies, alphabet size, and ascertainment vary over the specified finite models. For discrete observational adjustment, Zeng et al. (2026) provide the closest rate benchmark. The synthetic-assignment construction supplies an attaining polynomial estimator, while their lower-bound conclusion supports the fixed-positivity comparisons. The polynomial and moment-matching methods build on discrete functional estimation (Jiao et al., 2015; Wu et al., 2016; Wu et al., 2019); the expected-length criterion connects to honest confidence-interval theory (Donoho, 1994; Cai et al., 2004; Armstrong et al., 2020).

The exposition proceeds through related work, the experiment and identification conditions, and the point-risk comparisons across ascertainment regimes. It then develops the constructive estimator, honest intervals under rare arrival, and the observational extension through synthetic assignment. Discussion and open questions follow. The appendices collect identification and decision-theory preliminaries, risk certificates, full-experiment lower bounds, interval and observational arguments, the verification note, and proofs of the main results.

Related work

The availability of short-term measurements before primary outcomes motivates both surrogate analysis and methods for incomplete trial data. Prentice (1989) formulates criteria for surrogate endpoints, while Fleming et al. (1996) emphasizes the substantive evidence required to connect surrogate responses to clinical effects. Short-term outcomes can also inform treatment-effect estimation while primary outcomes are still being collected, as in Van Lancker et al. (2020). Here, the surrogate enters the conditioning information for outcome arrival, and the target remains the population average treatment effect. Identification rests on randomized assignment, consistency, and outcome missingness at random conditional on baseline category, treatment, and surrogate, as specified in Assumptions 2, 4, 5, and 6.

The closest comparison for estimation rates is Zeng et al. (2026). Their observational experiment comprises independent observations of discrete covariates, treatment, and outcome, with treatment propensities bounded away from zero and one by a fixed positivity margin. Writing $n$ for the sample size and $d$ for the number of covariate categories, their plug-in analysis gives a squared-error upper bound of order $n^{-1}+d^2/n^2$. Their minimax lower bound is of order $n^{-1}+d^2/\{n^2{\log }^{2}n\}$ in the stated regime $d\lesssim n\log n$, with constants depending on positivity. These conclusions give a logarithmic comparison between the plug-in guarantee and the minimax benchmark. Appendix C.5 develops the moment-matching argument underlying that benchmark. For the corresponding observational regression contrast under fixed strict-interior positivity on occupied categories, Theorem 8 supplies a constructive polynomial-estimation guarantee through synthetic assignment. Its positivity margin, $ϵ$, is held fixed in this comparison.

An earlier conference announcement by Kennedy (2019) considers an average causal effect with arbitrarily high-dimensional discrete confounders. It reports plug-in risk bounds, a minimax lower bound, and a variation with propensity-score positivity deteriorating with sample size, thereby establishing earlier consideration of the shrinking-positivity observational problem. The theorem-level comparison permitted by the available abstract is recorded in the limitations section.

The randomized analysis addresses a different observation mechanism. In the randomized fixed-horizon trial with a finite baseline alphabet, treatment is assigned with known probability one half, and primary outcomes arrive with unknown probabilities conditional on the observed cells. The arrival floor $q$ is imposed on occupied cells in Assumption 7. Under the rare-arrival restriction in Assumption 8, Theorem 4 matches upper and lower squared-error bounds at the rate defined in Definition 17, $$\min \left\{1,\frac{1}{nq}+{\left[\frac{d}{nq\log (e+nq)}\right]}^2\right\}.$$ This comparison separates fixed treatment positivity in the observational experiment from outcome ascertainment in a balanced randomized experiment. The latter analysis quantifies the joint contribution of the effective sample size and the number of categories, while the synthetic-assignment construction connects the two experiments.

Surrogate-assisted efficiency provides a complementary perspective. Kallus et al. (2025) characterize efficiency bounds with different patterns of surrogate and outcome observation, under treatment unconfoundedness, outcome missingness at random, and the overlap conditions appropriate to each observation pattern. Their labelled-sample asymptotics allow the labelled fraction to vanish while the expected number of labelled observations diverges, with fixed conditional distributions within the labelled and unlabelled populations. Attainment of the efficiency bound requires their nuisance convergence conditions: relevant products of regression and propensity or density-ratio errors must be negligible relative to the inverse square root of the expected labelled sample size. Treatment propensity estimates satisfy the stated boundedness condition, and estimated labelling propensities remain positive. Their moment requirements include uniformly bounded moments of order greater than two for the outcome residual, density ratio, and specified regression terms (Kallus et al., 2025). These results explain how predictive surrogates improve asymptotic precision under those conditions. The finite-alphabet minimax analysis here characterizes uniform risk as category frequencies and the alphabet size vary over the stated model class.

Decaying outcome observation is also central to Zhang et al. (2025). Their triangular-array observational framework combines treatment overlap with positive, potentially vanishing labelling probabilities conditional on treatment and covariates. Theorem 2.2 establishes asymptotic normality when both nuisance models are correctly specified, the inverse-propensity effective sample size diverges, and the product-rate condition in equation (2.3) holds. Its conditions also include finite inverse-propensity expectations, bounded outcome variance, conditional residual variance bounded above and away from zero, and the stated weighted nuisance and tail requirements. Their bias-reduced estimator in Theorem 4.1 accommodates nuisance approximation error under its product restriction and alternative sparsity conditions, together with sub-Gaussian design and conditional noise assumptions, covariance lower bounds, propensity moment control, and bounded offset signals (Zhang et al., 2025). These conditions support asymptotic inference with estimated nuisance functions. Our randomized rare-arrival comparison instead measures worst-case squared error over a finite categorical model, and the interval analysis imposes uniform coverage over that same experiment.

The approximation methods have established antecedents in the estimation of discrete-distribution functionals. Polynomial estimators and moment-matched lower bounds are developed by Jiao et al. (2015), Wu et al. (2016), and Wu et al. (2019). Their pairing reflects the dual relationship between polynomial approximation and separation of distributions with matching moments. Classical approximation theory supplies the reciprocal-function approximation used in the converse argument (Meinardus, 1967); explicit reciprocal approximation formulas are also presented in Plonka et al. (2026). The application here combines these tools with observed cell memberships and selectively observed outcomes. The roles of reciprocal approximation and moment-matched priors are stated in Lemmas 6 and 7, and the resulting comparison concerns the full observed experiment in Theorem 12.

Bias correction for regression functionals supplies another comparison. Higher-order influence functions use higher-order U-statistics to obtain rate-optimal point and interval procedures for nonlinear functionals under smoothness and nuisance-estimation conditions (Robins et al., 2008; Robins et al., 2009). Augmented minimax linear estimation corrects a regression plug-in estimator by a minimax linear estimate of its error and obtains semiparametric efficiency under its function-class conditions (Hirshberg et al., 2021). Weak-overlap analysis instead thresholds doubly robust scores and derives asymptotic normality and Wald inference under propensity-tail, outcome-smoothness, and nuisance-rate restrictions (Dorn, 2025). The present model replaces smooth nuisance classes by arbitrary masses and heterogeneous arrival probabilities on a finite, growing alphabet. Its guarantees are nonasymptotic worst-case squared-error and honest-length comparisons, and its logarithmic improvement comes from polynomial correction of scarce arrived-cell counts rather than smooth regression approximation.

The interval criterion connects the analysis to uncertainty quantification through maximal expected length subject to uniform coverage. Donoho (1994) relates statistical estimation to optimal recovery, Cai et al. (2004) studies expected-length comparisons and adaptation for confidence intervals, and Armstrong et al. (2020) develops honest intervals that account for worst-case bias in nonparametric regression. For the present missing-outcome experiment, Definition 29 specifies honesty for connected intervals, and Theorem 7 gives the corresponding length comparison under rare outcome arrival. This places estimation error and honest interval length within a common finite-alphabet decision problem.

At complete outcome arrival, Sudijono et al. (2026) provides a useful comparison from randomized-experiment theory. They jointly optimize the assignment design and estimator for the sample average treatment effect in a finite population with binary potential outcomes, obtaining a sharp minimax analysis that includes a second-order risk expansion. Our complete-arrival result in Theorem 3 concerns a population average treatment effect under independent and identically distributed sampling and a fixed balanced assignment design. The common first-order sampling scale therefore arises in two distinct decision problems: joint design and estimation for a finite-population sample effect, and estimation of a population effect under a specified randomized sampling experiment.

Experiment, identification, and risk

Throughout, $n\ge 1$ is the sample size, $d\ge 1$ is the number of baseline categories, and $q\in (0,1]$ is a supplied lower bound on occupied-cell outcome arrival. Generic positive constants $c$ and $C$ are independent of these parameters unless dependence is stated explicitly. We write $u\lesssim v$ for $u\le Cv$ and $u\asymp v$ for two such inequalities, uniformly over the stated parameter domain. Scalar errors are measured by absolute value. The arm index $a$ and surrogate index $s$ range over $\{0,1\}$, while the baseline index $x$ ranges over $\{0,\dots ,d-1\}$. We use $N$, the effective sample-size scale, for $N=nq$; it combines the sample size with the guaranteed within-cell arrival probability.

Records and cells

The experiment has a finite baseline category $X$, a binary treatment assignment $A$, binary potential surrogates $S(a)$, binary potential outcomes $Y(a)$, and an outcome-arrival indicator $R$. The following definitions specify the record space and the coordinates selected by treatment assignment.

Treatment assignment selects the realized surrogate $S$ from the two potential surrogate coordinates.

The assigned-arm outcome $Y$ is selected in the same way, and the finite record space carries its discrete measurable structure.

A full-data probability law $P$, defined in Definition 4, specifies the joint distribution of these coordinates.

Outcome ascertainment is described within treatment–baseline–surrogate cells. The cell-index set ${\mathcal{J}}_d$ contains a cell $j$ for each possible triple of these observed coordinates.

Two probabilities distinguish membership from outcome arrival: $p_{axs}(P)$ is the full-membership cell probability, and ${\rho }_{axs}(P)$ is the conditional arrival probability in an occupied cell.

Thus an occupied cell has positive membership probability, and its arrived-cell probability is the product of its membership and conditional arrival probabilities. For the treatment contrast, the treatment sign $g(a)$ assigns opposite signs to the two arms.

The observed record $O$ is $(X,A,S,R,RY)$. Its sampled copies $O_i$ retain cell membership and arrival status for every individual; when $R=1$, the final coordinate records the outcome. The canonical observed sample law ${\mathbf{O}}_n$, defined in Definition 11, is the corresponding product distribution.

Sampling, randomization, and outcome arrival

Outcome consistency completes the link between observed and potential outcomes.

Independent and identically distributed sampling is imposed within each triangular-array row, allowing the underlying law and alphabet size to vary across experiments (Zeng et al., 2026). Outcome arrival is governed by the following conditional independence condition.

For later aggregation, we index treatment–baseline–surrogate cells explicitly.

The next definition records each cell’s population mass and its conditional arrival probability.

The supplied arrival floor is imposed only on occupied cells.

The preceding randomization, consistency, and arrival restrictions define the full positive-arrival model.

Outcome arrival missing at random given treatment, baseline, and surrogate equates the outcome distribution among arrivals with the full-cell outcome distribution whenever arrivals have positive probability (Rubin, 1976; Robins et al., 1994; Kallus et al., 2025). The surrogate therefore serves as an observed conditioning variable for ascertainment: its realized value can capture variation in arrival associated with the assigned treatment and the outcome. Positive ascertainment within occupied cells is imposed next.

The occupied-cell arrival floor is specific to this analysis. It guarantees an outcome-arrival probability of at least the supplied value $q$ in every cell represented in the population, while permitting both cell membership probabilities and arrival probabilities to vary across cells. This condition supplies positive arrived-cell mass wherever the population cell contribution can be nonzero. A prospective analysis must justify this lower bound; the stated tuning and guarantees apply when it is valid.

Together, the randomization, consistency, and arrival restrictions define the arrival-floor model ${\mathcal{M}}_{n,d,q}^{+}$ in Definition 7.

This family covers the full range of positive arrival floors, including complete outcome arrival at $q=1$. Sampling determines the product experiment induced by each member of the family.

Identification and squared-error risk

The causal target is the population average treatment effect $\tau (P)$ in Definition 8.

Binary potential outcomes place this target in $[-1,1]$. Its observed-data representation combines full-membership probabilities with outcome means among arrivals. The total signed cell functional $\Phi (P)$, defined in Definition 10, uses an arrived-cell outcome mean ${\mu }_{axs}(P)$ and a membership-weighted contribution $c_{axs}(P)$, with explicit conventions for cells having zero arrived mass.

To distinguish the two roles in subsequent formulas, we write ${\mathbf{O}}_n=(O_1,\dots ,O_n)$ for the random observed-record tuple and ${\mathcal{L}}_P({\mathbf{O}}_n)$ for its product law. The canonical-law display above specifies the latter distribution.

For every law in Definition 7, Lemma 3 establishes $\tau (P)=\Phi (P)$; its proof appears in the appendix. The representation assigns zero contribution to null cells and uses observable arrived-cell means on occupied cells. Full cell membership remains observed even when the outcome is missing, so the representation separates population cell weights from within-cell outcome ascertainment. Balanced assignment supplies the factor of two in the signed contrast.

We evaluate estimation over the entire arrival-floor model through the minimax squared-error risk ${\mathfrak{R}}_{n,d,q}^{+}$ in Definition 12. Its decision class ${\mathcal{T}}_n$ contains all parameter-independent bounded statistical decision rules, including a possibly randomized estimator $\mathsf{T}$.

The inner integral averages over the procedure’s conditional output distribution given the sample, and the outer integral averages over sampling. Parameter independence means that the procedure is common to all laws in the specified model. Under squared loss, replacing a randomized output by its conditional mean given the sample weakly reduces risk for each law, as stated in Lemma 1. This comparison makes the all-procedure criterion equivalent to its deterministic counterpart.

The rare-arrival experiment and comparison rate

A second model family restricts the arrival floor relative to the effective sample-size scale. Its defining condition uses the logarithmic scale $\ell$, which equals $\log (e+N)$ on the statistical parameter domain fixed above.

The rare-arrival regime is selected by the following restriction on $q$ and this logarithmic scale.

The rare-arrival restriction is specific to this analysis and selects parameter triples for which the arrival floor is small relative to the squared logarithmic scale. It restricts the guaranteed floor; individual occupied cells may have larger arrival probabilities. The rare-arrival model ${\mathcal{M}}_{n,d,q}$ in Definition 14 combines this parameter restriction with the preceding statistical conditions.

At parameter triples satisfying Assumption 8, this model uses the same full-data restrictions as Definition 7, together with the stated sampling condition. The observed experiment ${\mathcal{E}}_{n,d,q}$ in Definition 15 is explicitly indexed by this rare-arrival model.

The decision problem retains every sampled cell membership, including those whose outcomes have yet to arrive. Its minimax squared-error risk ${\mathfrak{R}}_{n,d,q}$, defined in Definition 16, uses the same decision class and joint sample-and-procedure expectation as the unrestricted criterion.

The two risk definitions distinguish the parameter domains over which the subsequent comparisons are stated. For the rare-arrival analysis, the comparison-rate expression $r_{n,d,q}$ in Definition 17 combines the effective sample-size scale with the number of baseline categories.

In terms of the setup notation, the expression is $\min \{1,N^{-1}+[d/(N\ell ){]}^2\}$. Its first term records the reciprocal effective sample-size scale, while its second term expresses the joint dependence on alphabet size and outcome ascertainment. The truncation at one matches the uniform squared-error bound of the estimator that always reports zero for a target in $[-1,1]$. The estimation results that follow compare the defined risks with these scales under their stated ascertainment regimes.

Point estimation across ascertainment regimes

The squared-error comparisons depend jointly on outcome ascertainment and the number of baseline categories. For every arrival floor $0<q\le 1$, the unrestricted minimax risk ${\mathfrak{R}}_{n,d,q}^{+}$ of Definition 12 admits an upper envelope combining the mixed-count rate in Definition 17 with a dimension-free observed-outcome bound. The latter separates a sampling contribution of order $n^{-1}$ from the squared ascertainment deficit $(1-q{)}^2$. We first specify the estimators supporting this envelope, then give matched comparisons for large alphabets, rare arrival, and a fixed arrival floor $0<q<1/2$.

An all-arrival envelope and its estimators

The clipped observed-outcome arm contrast ${\hat{\tau }}_n^{\mathrm{HT}}$ uses balanced assignment to form a signed average of arrived outcomes. Its projection ${\Pi }_{[-1,1]}$ keeps the estimate in the range of the population treatment effect. The following definition also fixes the clipping convention used by the mixed-count estimator.

The mixed-count construction combines cell membership information with cell-specific arrived-outcome statistics. It uses the effective sample size $N=nq$ and the logarithmic scale $\ell =\log (e+N)$ in Definition 13. Its polynomial degree $k$ grows with this logarithmic scale, according to the following total tuning rule.

On the polynomial branch, the first-kind Chebyshev polynomial $T_k$ determines the needle polynomial $Q_k(t)$. Here $t$ denotes an arrived-count intensity, and the threshold $B$ sets the scale of the construction. The next two definitions specify the polynomial and its coefficient sequence $a_v$, where $v$ indexes the degree.

The estimator assigns observations to membership, pilot, and estimation streams. The pilot arrived count $C_j^{\prime }$ selects the cell statistic, while the estimation arrived count $C_j$ and arrived-one count $U_j$ supply its outcome information. Their definitions retain the observed cell memberships and distinguish the two uses of arrived outcomes.

The selected cell statistic $G_j$ uses a polynomial expression when the pilot count is small and an empirical arrived-outcome mean on the remaining cells. The associated factorial statistic $H_j$ records the polynomial expression separately, and the elementary ratio $D_j$ specifies the empirical-mean branch, including its value when the arrived count is zero.

Cell weights use the membership count $M_j$, which includes observations with either arrival status. The weight $W_j$ divides this count by the stream intensity $m$. The following definitions therefore separate estimation of cell mass from estimation of its outcome mean.

These ingredients define the mixed-count estimator ${\hat{\tau }}_{n,d,q}^{\mathrm{mix}}$. A Poisson prefix length $K_0$ and independent stream assignments produce an auxiliary estimate; averaging over this randomization yields a deterministic function of the observed sample.

The resulting deterministic risk envelope ${\mathfrak{U}}_{n,d,q}$ is specified branch by branch in the next result. Its certificate uses the effective stream size $N_0$, equal to $mq$. Together with the observed-outcome contrast, it provides two upper bounds applicable throughout the unrestricted arrival domain.

The universal comparison constant $\overline{C}$ is independent of sample size, alphabet size, and arrival floor. The two upper bounds express different ways to control squared error: the mixed-count bound accounts for category complexity through $r_{n,d,q}$, whereas the observed-outcome bound accounts for missing outcomes through the ascertainment deficit. As $q$ approaches one, the latter supplies a dimension-free guarantee. Conditional averaging in Algorithm 1 preserves the mixed-count guarantee while eliminating the auxiliary randomization from the reported estimate.

Large alphabets and the uniform near-complete criterion

For alphabets at least quadratic in sample size, the sampling and ascertainment-deficit terms also give a matching lower bound. The next result establishes their joint order with universal constants throughout the stated large-alphabet region.

The comparison identifies two sources of difficulty on the same scale: ordinary sampling variation and the remaining ascertainment deficit. When $(1-q{)}^2$ exceeds $n^{-1}$, it determines the risk order in this region; when it is at most of order $n^{-1}$, sampling variation determines that order. Thus the dimension-free observed-outcome guarantee in Theorem 1 is sharp up to universal constants for $d\ge 1024n^2$.

A guarantee uniform over every alphabet size must cover this large-alphabet region. Combining its lower bound with the all-arrival upper envelope yields an exact sequence-level criterion for parametric squared-error risk.

The quantifier over all $d$ is central to this criterion. A deficit of order $n^{-1/2}$ is necessary and sufficient for an eventual $n^{-1}$ upper bound holding simultaneously over every alphabet size. The explicit sequence $d_n=1024n^2$ witnesses the converse when the rescaled deficit is unbounded. For any fixed deficit multiplier, the sufficient window also gives the following numerical comparison along arbitrary alphabet-size sequences.

At complete ascertainment, the deficit is zero. The following endpoint specialization records the observed-outcome contrast as a direct witness to the dimension-free parametric benchmark.

Complete arrival makes the observed-outcome arm contrast sufficient to attain the minimax order for every alphabet size. The sampling lower bound remains of the same order, so the endpoint comparison is uniform even when the alphabet grows arbitrarily quickly. Its sequence formulation follows directly.

Rare arrival and effective sample size

Under Assumption 8, the rate in Definition 17 gives a matched comparison for the rare-arrival risk of Definition 16. This comparison retains the full observed experiment, including memberships whose outcomes have yet to arrive, and applies to the entire decision class in that risk definition.

The universal lower constant $\underline{c}$ makes the two components of the rate jointly informative about minimax difficulty. The reciprocal effective sample size captures the sampling scale, while the squared category term captures the cost of estimating the treatment contrast across many cells: $$r_{n,d,q}=\min \left\{1,\frac{1}{N}+{\left(\frac{d}{N\ell }\right)}^2\right\}.$$ At fixed alphabet size, the category term becomes asymptotically negligible as $N$ grows. The next result also establishes this effective-sample-size order within the submodel having a constant surrogate in both treatment arms.

Allowing the alphabet to grow separates consistency from effective-sample-size precision. The category term vanishes when $d$ is small relative to $N\log (e+N)$, and it remains at most of order $N^{-1}$ when $d$ is at most of order $\sqrt{N}\log (e+N)$. The matched comparison turns these rate calculations into exact equivalences for the minimax risk.

These thresholds quantify how many baseline categories can be accommodated at a given outcome-maturity floor. They apply along sequences satisfying the rare-arrival restriction and growing effective sample size. In particular, they distinguish the broader category-growth range permitting consistency from the smaller range preserving squared-error risk of order $N_t^{-1}$.

A fixed strict-interior arrival floor

A separate matched comparison applies to the unrestricted model when the arrival floor is fixed at $0<q<1/2$. Its lower comparison uses the discrete observational lower bound of Zeng et al. (2026), including the restriction setting the control regression to zero. The resulting comparison holds for every $n,d\ge 1$, with constants depending on the fixed arrival floor.

For a fixed arrival floor, effective sample size is proportional to sample size, so the comparison can be written using $n$ and $\log (e+n)$. The lower constant $\underline{c}(q)$ and upper constant $\overline{C}(q)$ may depend on that floor. This is a second logarithmic matched regime: Theorem 4 gives universal constants under the rare-arrival restriction, while the present result gives a comparison over the unrestricted model at each fixed $q\in (0,1/2)$.

The fixed-floor comparison yields corresponding sequence-level thresholds. The next statement records both the exact consistency condition and the necessary and sufficient category-growth order for parametric squared-error risk.

The fixed-floor thresholds express consistency on the $n\log (e+n)$ category scale and parametric precision on the $\sqrt{n}\log (e+n)$ scale. For a prescribed category-growth multiplier, the eventual upper constant may depend on that multiplier and on $q$. This dependence differs from the universal constants in the rare-arrival comparison and from the explicit deficit-dependent constant in the near-complete window.

The following table collects the comparisons while keeping the universal upper envelope, the matched large-alphabet region, and the two logarithmic matched regimes separate. Here $N=nq$, $\ell =\log (e+N)$, and $r_{n,d,q}$ is the rate in Definition 17.

Squared-error comparisons across ascertainment regimes

Domain	Risk comparison	Constants	Supporting results
$n,d\ge 1,0<q\le 1$	$\frac{1}{256n}\le {\mathfrak{R}}_{n,d,q}^{+}\le \min \{1,\overline{C}{r}_{n,d,q},4/n+(1-q{)}^2\}$	Universal	Theorem 1
$d\ge 1024n^2,0<q\le 1$	${\mathfrak{R}}_{n,d,q}^{+}\asymp n^{-1}+(1-q{)}^2$	Universal	Theorem 2
$1-q_n\le M/\sqrt{n}$ eventually, all $d\ge 1$	$\frac{1}{256n}\le {\mathfrak{R}}_{n,d,q_n}^{+}\le \frac{4+M^2}{n}$; uniform $n^{-1}$ order iff $1-q_n=O(n^{-1/2})$	Explicit dependence on $M$	Propositions 1 and 2
$q=1,d\ge 1$	${\mathfrak{R}}_{n,d,1}^{+}\asymp n^{-1}$, uniformly in $d$	Universal	Theorem 3 and Proposition 3
Rare-arrival restriction	${\mathfrak{R}}_{n,d,q}\asymp \min \{1,N^{-1}+[d/(N\ell ){]}^2\}$	Universal	Theorem 4 and Proposition 5
Fixed $0<q<1/2$	${\mathfrak{R}}_{n,d,q}^{+}\asymp \min \{1,n^{-1}+[d/\{n\log (e+n)\}{]}^2\}$	Depend on fixed $q$	Theorem 5 and Proposition 6

A constructive estimator

The risk comparisons admit an explicit statistical construction. The mixed-count estimator in Algorithm 1 combines information about cell membership with information about arrived outcomes, assigning these two tasks to separate parts of each cell contribution. Its elementary branch uses empirical cell means; its polynomial branch modifies the contribution of cells with few arrivals. We collect the ingredients by their statistical role before describing the resulting estimate and its risk guarantee.

Mechanism of the mixed-count correction

The canonical formulas are collected once in Algorithm 1 and its immediate ingredients Definitions 13, 19, 27, 23, 28, 26, 21, 25, 24, 22, and 20. Their statistical roles are simple. Membership-labelled records estimate the population cell weights, while pilot and estimation labels separate branch selection from estimation of each arrived-cell mean. An ordinary ratio assigns zero to a cell with no arrived outcome. Across many rare cells, those zero-count events create the category-dependent bias that appears in the elementary certificate.

On the large-alphabet branch, the polynomial extension of $\{1-Q_k(t)\}/t$ supplies the reciprocal correction implicit in a cell mean. Marked factorial counts estimate its monomials without discarding the association between an outcome and its cell: for cell intensity $t_j$ and arrived-cell mean ${\mu }_j$, the factorial statistic satisfies $E(H_j)={\mu }_j\{1-Q_k(t_j)\}$. Thus $Q_k$ controls the residual bias, and the bound on $t_j|Q_k(t_j)|$ controls its membership-weighted aggregate. With degree $k$ proportional to $\log (e+N)$, this changes the aggregate bias scale from the elementary scarce-cell scale to $d/\{N\log (e+N)\}$, while a pilot count sends well-observed cells back to the stable empirical ratio. Independent membership, pilot, and estimation streams justify the comparison calculation; the capped-prefix transfer and conditional averaging turn that comparison into the deterministic fixed-sample estimator.

The construction is explicit on every parameter triple. Its polynomial branch activates at $\log (e+nq)\ge 128$, equivalently $nq\ge e^{128}-e$. Exact evaluation averages $3^{k_0}$ assignments at each retained prefix length $k_0$; the limitations section discusses this computational scale separately from the attainment guarantee.

Risk control and the deterministic envelope

The construction supplies a squared-error guarantee throughout the unrestricted arrival class in Definition 7. Its deterministic envelope ${\mathfrak{U}}_{n,d,q}$ depends on the sample size, alphabet size, and arrival floor. The following result gives the envelope explicitly, compares it with the rate in Definition 17, and records the risk improvement from conditional averaging.

The bound separates effective-sample-size error from category complexity through $$r_{n,d,q}=\min \left\{1,\frac{1}{N}+{\left(\frac{d}{N\ell }\right)}^2\right\},$$ as defined in Definition 17. The polynomial certificate reflects the mechanism of the construction: its leading squared contribution involves $dB/(N_0{k}^2)$, which scales as $d/(N\ell )$ on the polynomial branch. Membership weights and arrived-outcome statistics thus enter a common aggregate guarantee even when many cells have few arrived observations. The final inequality in Theorem 6 also shows why the conditional average is the reported estimator: averaging the bounded auxiliary outputs preserves their risk guarantee and can reduce squared-error risk.

The detailed cellwise moment bounds and their aggregation are collected in Theorem 9. For the subsequent interval construction, the essential feature is that ${\mathfrak{U}}_{n,d,q}$ is a known, deterministic upper bound on squared-error risk, uniform over the stated model class. It supplies the radius $\sqrt{{\mathfrak{U}}_{n,d,q}/\alpha }$, where $\alpha \in (0,1)$ is the miscoverage level, used by the clipped interval in Definition 30.

Honest intervals under rare outcome arrival

The deterministic squared-error envelope in Theorem 6 also supports uncertainty quantification for the average treatment effect. The relevant criterion combines uniform coverage over the rare-arrival model with maximal expected interval length. Fix a miscoverage level $\alpha \in (0,1)$, so that the required coverage is $1-\alpha$. To include procedures with auxiliary randomization, both coverage and expected length are evaluated under the joint distribution of the observed sample and the reported endpoints. The following definition specifies this decision problem.

Uniform honesty requires the coverage inequality separately for every law in Definition 14. For a randomized endpoint procedure $\mathsf{I}$, this requirement averages over its auxiliary draw as well as sampling variation. The same averaging governs length. Thus the admissible class ${\mathcal{C}}_{n,d,q,\alpha }$ treats deterministic and randomized procedures within one criterion, and the minimax length ${\mathfrak{L}}_{n,d,q,\alpha }$ measures the smallest attainable worst-case expected width at the specified coverage.

A concrete honest interval uses the mixed-count estimator ${\hat{\tau }}_{n,d,q}^{\mathrm{mix}}$ from Algorithm 1 and its deterministic risk envelope ${\mathfrak{U}}_{n,d,q}$ in Theorem 6. Both are calibrated to the supplied arrival floor $q$. The envelope determines a radius before observing the sample; intersecting the resulting interval with the target range gives the following construction. Its coverage guarantee therefore requires the supplied floor to be no larger than every occupied-cell arrival probability.

The interval $I_{\alpha }^{\mathrm{mix}}$ is deterministic conditional on the observed sample and therefore enters Definition 29 through a Dirac endpoint rule. Its radius converts uniform squared-error control into coverage at least $1-\alpha$. The next result compares its maximal expected length with the best uniformly honest connected interval, using the comparison rate $r_{n,d,q}$ defined in Definition 17.

The upper bound reflects the square-root conversion from squared-error risk to an interval radius: the deterministic envelope controls coverage, while clipping bounds the realized length by the width of the untruncated interval. The converse connects statistical indistinguishability with the geometry of a connected interval. Coverage transfers between competing laws in the joint sample-and-endpoint distribution, and a realized interval that meets several separated target neighborhoods must span their separation. Working with this joint distribution retains the auxiliary endpoint draw throughout both the coverage comparison and the length calculation. The parameter-independent decision-rule comparison in Lemma 2 supplies the corresponding control when an endpoint procedure is appended to the observed experiment.

For each fixed coverage level $1-\alpha \in (0,1)$, Theorem 7 establishes expected-length order $\sqrt{r_{n,d,q}}$, with constants that may depend on $\alpha$, uniformly over the sample sizes, dimensions, and arrival floors satisfying its conditions. Alongside the squared-error order in Theorem 4, this gives an uncertainty-quantification interpretation of the same comparison rate: its square root is the attainable and necessary order of maximal expected width for uniformly honest connected intervals. In particular, along sequences satisfying Assumption 8, the optimal expected length tends to zero exactly when the rate in Definition 17 tends to zero.

An observational extension through synthetic assignment

The discrete observational experiment connects adjustment for many covariate categories to the randomized experiment with missing outcomes. Its target is a baseline-weighted regression contrast under a fixed strict positivity margin. An independent synthetic assignment converts observational records into randomized records whose outcomes arrive when the synthetic and observational treatments agree. This construction preserves the contrast and transfers the mixed-count estimator to observational data.

We begin by specifying the observational law and its occupied-category positivity condition.

For a law $P_Z$ in the class ${\mathcal{D}}_{d,ϵ}$, both values of the observational treatment $B^{\mathrm{obs}}$ have positive conditional probability in every occupied category. The margin bounds these probabilities away from zero while allowing arbitrary baseline probabilities, including null categories. Consequently, both conditional outcome means are defined wherever the baseline distribution assigns positive mass.

The estimand aggregates the within-category regression contrasts using that baseline distribution.

The contrast ${\theta }_Z(P_Z)$ is a functional of the observational law. Under outcome consistency, $Y=Y(B^{\mathrm{obs}})$, and conditional exchangeability, $$\{Y(0),Y(1)\}\perp \perp B^{\mathrm{obs}}\mid X,$$ positivity identifies each occupied-category regression mean with the corresponding conditional potential-outcome mean. The contrast then equals the population average treatment effect $E[Y(1)-Y(0)]$, giving the usual causal interpretation of covariate adjustment (Rosenbaum et al., 1983).

Let ${\mathbf{Z}}_n$ denote the sample of $n$ independent observational records. The decision problem allows estimation procedures to use independent auxiliary randomness as well as the sample.

The observational minimax risk ${\mathfrak{R}}_{n,d,ϵ}^Z$ therefore measures the smallest worst-case squared error over the full decision class in Definition 33. To connect this problem to the randomized construction, we present together the transformation in Lemma 5 and the estimation procedure in Theorem 8.

Synthetic assignment and observational estimation.

For an observational record $(x,b,y)$, draw a fair coin $U$, independently of the record, and define $$o(x,b,y;u):=(x-1,u,0,1\{b=u\},1\{b=u\}y),u\in \{0,1\}.$$ The five coordinates are the randomized baseline, assignment, surrogate, arrival indicator, and arrived outcome. For a sample realization ${\mathbf{Z}}_n=((x_i,b_i,y_i){)}_{i=1}^n$, use independent fair coins $U_1,\dots ,U_n$ and set $$o_i(u_i):=o(x_i,b_i,y_i;u_i).$$ The transformation uses the observed record and a known coin distribution, so its rule is independent of the unknown observational law.

The induced randomized full-data law $P_Z^{\ast }$ in Lemma 5 can be specified through a latent representation. Draw the baseline with probabilities $p_x$. Conditional on an occupied category $x$, draw $B^{\mathrm{obs}}$ with success probability ${\pi }_x$, and draw independent binary potential outcomes with means ${\mu }_{0x}$ and ${\mu }_{1x}$, independently of $B^{\mathrm{obs}}$. Draw an independent fair assignment $A$, retain $X-1$ as the randomized baseline, and set $$S(0)=S(1)=0,Y=Y(A),R=1\{B^{\mathrm{obs}}=A\}.$$ This representation reproduces the synthetic observed-record distribution. Its arm-specific arrival probabilities are $$P_Z^{\ast }(R=1\mid X-1=x-1,A=1,S=0)={\pi }_x,P_Z^{\ast }(R=1\mid X-1=x-1,A=0,S=0)=1-{\pi }_x,$$ and its target satisfies $$\tau (P_Z^{\ast })={\theta }_Z(P_Z).$$ Both arrival probabilities are at least $ϵ$. Independence of the latent observational treatment and potential outcomes conditional on the baseline supplies the missing-at-random condition in this constructed law. The causal interpretation of the original observational contrast continues to rest on consistency and conditional exchangeability.

Apply the mixed-count estimator ${\hat{\tau }}_{n,d,ϵ}^{\mathrm{mix}}$ from Algorithm 1 to the synthetic sample. Averaging over all fair-coin assignments defines the deterministic observational estimator ${\hat{\theta }}_{n,d,ϵ}^{\mathrm{poly}}$: $${\hat{\theta }}_{n,d,ϵ}^{\mathrm{poly}}({\mathbf{Z}}_n):=\frac{1}{2^n}\sum _{(u_1,\dots ,u_n)\in \{0,1{\}}^n}{\hat{\tau }}_{n,d,ϵ}^{\mathrm{mix}}(o_1(u_1),\dots ,o_n(u_n)).$$ This finite average defines the procedure for every observational sample. Conditional averaging weakly reduces squared-error risk relative to retaining the auxiliary coin draws.

The resulting risk bound is uniform over sample size and alphabet size. At each fixed strict positivity margin, it also yields a matching all-estimator comparison.

The statistical mechanism is the conversion of observational treatment imbalance into selective outcome arrival under independent assignment. Positivity supplies a common arrival floor, and the target-preserving transformation makes the randomized MAR risk certificate applicable. Averaging over the synthetic assignments then removes their auxiliary variation. For fixed $ϵ$, the two terms in the comparison describe sampling error and the cost of adjusting for many categories; the truncation captures the bounded scale of the contrast.

Under consistency and conditional exchangeability, Theorem 8 therefore characterizes minimax squared-error risk for the observational average treatment effect. As $n$ tends to infinity at a fixed strict positivity margin, uniform mean-square consistency is attainable exactly when $$d=o\{n\log (e+n)\}.$$ The constants in the two-sided comparison may depend on that margin, while the constructive bound retains the dependence on $ϵ$ through the rate in Definition 17.

The observational lower-bound input comes from Zeng et al. (2026). Their argument sets the control regression to zero on occupied categories and supplies the lower order $$\frac{1}{n}+{\left(\frac{d}{n\log n}\right)}^2$$ for fixed strict positivity, sufficiently large $n$, and $1\le d\le c_{0}n\log n$, where $c_0>0$ is a constant from that result. The comparison in Theorem 8 extends the risk characterization to every pair $n,d\ge 1$, with a total estimator and the bounded-risk truncation.

Finally, the same construction fixes the direction of the experiment comparison supporting the fixed-arrival analysis. At the fixed arrival floor $q=ϵ\in (0,1/2)$, the induced laws belong to the randomized class in Definition 7. Any estimator for their randomized observed samples can be applied to observational records by appending independent synthetic assignments, with the same target and risk. The observational sample is therefore at least as informative as its synthetic image. Taking worst-case risk over the containing randomized class gives a risk at least as large as the observational minimax risk, so the published observational lower bound transfers in this direction to the fixed-arrival lower comparison in Theorem 5. The formal target-preserving map is given in Lemma 5.

Discussion and extensions

The precision comparisons distinguish three features of the observed experiment: the availability of primary outcomes, the number of covariate categories across which ascertainment must be accommodated, and the remaining ascertainment deficit. Their relative importance changes with the arrival regime. The resulting interpretation is a comparison of attainable precision over the stated model classes, with each regime retaining its own conditions and quantifiers.

Informative outcomes and adjustment complexity

The effective sample size $N=nq$ measures outcome availability at the guaranteed occupied-cell arrival floor. Because individual cells may have arrival probabilities exceeding that floor, it is a conservative scale for outcome information. Under Assumption 8, the matched comparison in Theorem 4 separates the sampling contribution $N^{-1}$ from the category contribution $${\left\{\frac{d}{N\log (e+N)}\right\}}^2,$$ with their sum truncated at a constant. Thus increasing outcome availability improves both components, whereas increasing the alphabet size raises the cost of estimating the aggregate contrast across sparsely observed cells. The comparison incorporates the memberships observed for individuals whose primary outcomes have yet to arrive.

Along sequences satisfying the rare-arrival restriction and $N\to \infty$, Proposition 5 gives two distinct measures of manageable adjustment complexity. Consistency corresponds to $d=o\{N\log (e+N)\}$, while squared-error risk of order $N^{-1}$ corresponds to $d=O\{\sqrt{N}\log (e+N)\}$. These thresholds distinguish category growth compatible with learning the treatment effect from category growth compatible with preserving the effective-sample-size precision benchmark. At each fixed coverage level, Theorem 7 gives the corresponding interpretation for uniformly honest connected intervals: optimal maximal expected length has the order of the square root of the matched squared-error rate.

Near complete ascertainment, the deficit $1-q$ provides a complementary measure of difficulty. The observed-outcome contrast attains the upper bound $4/n+(1-q{)}^2$ throughout the unrestricted model in Theorem 1. For $d\ge 1024n^2$, Theorem 2 establishes the matching order $n^{-1}+(1-q{)}^2$. At complete arrival, Theorem 3 gives squared-error order $n^{-1}$ uniformly over every alphabet size. Together, these results explain how sufficiently complete ascertainment supports precision governed by sampling variation even under arbitrary category growth.

The quantifiers distinguish a uniform guarantee from a guarantee along a specified dimension sequence. In Proposition 1, a single risk constant and an eventual sample-size threshold apply simultaneously to every $d\ge 1$. Such an $n^{-1}$ guarantee is equivalent to $1-q_n=O(n^{-1/2})$. A specified sequence can satisfy a different sufficient condition through its category growth: at any fixed $q\in (0,1/2)$, Proposition 6 characterizes $n^{-1}$ risk by $d_n=O\{\sqrt{n}\log (e+n)\}$. The comparison constants may depend on the fixed floor and the category-growth multiplier. The near-complete criterion therefore describes precision uniformly over arbitrary alphabets, while the fixed-floor criterion describes precision along sequences with controlled adjustment complexity.

Surrogates and prospective trial interpretations

The surrogate serves as an observed variable for conditioning outcome arrival. Under Assumption 6, arrivals represent the outcome distribution within treatment–baseline–surrogate cells; Assumption 7 supplies positive ascertainment in every occupied cell. Its realized value may depend on assignment through Assumption 4. The target remains the population average treatment effect on the primary outcome in Definition 8, with the cell aggregation specified in Definition 10.

The constant-surrogate specialization provides a useful benchmark for this interpretation. When both potential surrogates equal zero, the arrival condition reduces to conditioning on treatment and baseline category. For fixed alphabet size, sequences in the rare-arrival slice with diverging effective sample size retain minimax squared-error order $N^{-1}$ within this specialization, as established by Proposition 4. That benchmark isolates the precision scale associated with outcome availability in a model where the surrogate supplies a single category.

Clinical-trial applications can be interpreted prospectively through these conditions. At a prespecified analysis horizon, a binary primary endpoint would supply the outcome, an available short-term measurement would supply the binary surrogate, and recorded baseline categories would determine the adjustment alphabet. Applying the comparisons would require independent sampling, balanced randomized assignment, consistency, and the stated conditional arrival and occupied-cell positivity conditions. The delayed binary endpoints considered by Marschner et al. (2001) provide motivation for this interpretation. The use of short-term outcomes and baseline covariates for interim decisions in Van Lancker et al. (2020) supplies related motivation. Here, these examples connect the statistical observation scheme to prospective trial settings, while the precision conclusions retain the conditions of the cited internal results.

Further questions concerning scope and research priorities are collected in Section 9.

Limitations and open questions

The risk comparisons leave several questions about joint changes in outcome arrival and alphabet size. The unrestricted upper bound in Theorem 6 provides an envelope across arrival regimes, while Theorems 4 and 5 give matching comparisons under their respective conditions. Matching the unrestricted envelope throughout intermediate moving-arrival and alphabet regimes remains open. This requires lower bounds that track simultaneous changes in the arrival floor and the number of baseline categories.

A related question concerns the distinction between a uniform near-complete requirement and a requirement for a particular alphabet sequence. Proposition 1 characterizes the uniform comparison, and Theorem 2 supplies the large-alphabet obstruction. These results leave open the precise near-complete requirement along individual smaller-alphabet sequences. A sequence-specific characterization would determine how the necessary degree of ascertainment depends on the growth of the alphabet.

For uncertainty quantification, Theorem 7 compares uniformly honest connected intervals under the rare-arrival restriction in Assumption 8. Extending matching expected-length comparisons beyond that restriction is another open problem. Such an extension would require interval lower bounds appropriate to the additional arrival regimes, together with procedures attaining the corresponding lengths under uniform coverage.

The observational comparison in Theorem 8 concerns a fixed positivity margin strictly between zero and one half, as specified in Definition 31. The endpoint at which every occupied category has treatment propensity one half calls for a separate risk characterization. Determining how this endpoint relates to the strict-interior comparisons would clarify the role of treatment-propensity variation in the observational problem.

The conditional arrival assumption in Assumption 6 also defines an important boundary of the model. Extending the analysis beyond missing at random requires specifying how arrival may depend on the outcome after conditioning on the recorded covariates, assignment, and surrogate. Alternative identifying restrictions or sensitivity parameters would then determine the target and the precision comparison. The present risk results do not supply those extensions.

Statistical attainment and practical evaluation are separate questions. The procedure in Algorithm 1 averages $3^{k_0}$ stream assignments at each retained prefix length $k_0$, and the observational procedure in Theorem 8 additionally averages over $2^n$ synthetic assignments. Their risk guarantees concern those exact procedures. Moreover, the polynomial branch begins only when $\log (e+nq)\ge 128$, or $nq\ge e^{128}-e$. These scales make the displayed formulas mathematical attainment constructions rather than evidence of numerical usefulness at ordinary sample sizes. Computational evaluation should seek more efficient representations and quantify additional error from approximations. Prospective clinical evaluation must also justify the supplied arrival floor used for tuning and coverage, examine the choice of baseline categories and the conditional arrival assumption, and compare with established trial analyses.

Kennedy (2019) earlier announced average-causal-effect estimation with arbitrarily high-dimensional discrete confounders, including a variation with sample-size-dependent weakening of propensity-score positivity. The available abstract does not state its precise model class, rate, loss, or attaining construction. It therefore supports priority for studying that shrinking-positivity observational setting but cannot settle theorem-level overlap with the fixed-margin observational result here. A comparison with a complete theorem statement or manuscript remains open.

Appendices

Identification and decision-theory preliminaries

This appendix collects the decision-theoretic comparisons, cell-identification identities, and synthetic-assignment map used in the risk analysis. The generic experiment and testing notation introduced below is local to this appendix.

Finite decision experiments

For the decision experiment in Lemma 1, let $\Lambda$ denote a parameter set, $\mathcal{Z}$ a finite observation space with measurable singletons, $Q_{\lambda }$ the observation law at parameter $\lambda$, and $\vartheta (\lambda )\in [-1,1]$ the scalar target. A randomized procedure is a parameter-independent Markov kernel $\kappa$: conditional on observation $o$, it draws an action from $\kappa (o,\cdot )$. Its squared risk averages the squared estimation error over both the observation and the action. A deterministic procedure is represented by a kernel concentrated at one action for each observation. For actions in $[-1,1]$, the barycenter is the conditional mean of the action. The following comparison establishes the risk effect of replacing a randomized action by this mean, together with the corresponding clipping and uniform-seed representations.

The comparison holds at every parameter, so conditional averaging also preserves a uniform squared-risk bound. It supplies the decision-theoretic justification for the auxiliary average in Algorithm 1 and the synthetic-assignment average in Theorem 8. Clipping weakly reduces squared loss for targets in the stated interval; on the real line, this clipping map coincides with Euclidean projection onto that interval. The uniform-seed representation places bounded randomized procedures and procedures implemented with independent auxiliary randomization in the same decision class.

For the testing comparison in Lemma 2, let $Q_0,Q_1$ denote two laws on $\mathcal{Z}$, and let $\mathcal{H}$ be a measurable action space. The notation $Q_i⋉\kappa$ denotes the joint observation–action law, whereas $Q_i\kappa$ denotes its action marginal. Total variation, written $\mathrm{TV}$, is the supremum of absolute probability differences over measurable events. For binary actions, a randomized test has conditional probability $\psi (o)=\kappa (o,\{1\})$ of selecting alternative $1$. The next comparison applies both to events that retain the observation and to events determined by the action alone.

Retaining the observation alongside the randomized action preserves total variation, while taking the action marginal weakly reduces it. When the event represents selection of alternative $1$, the displayed probability sum is the sum of the two testing errors. These comparisons allow the lower-bound analysis in Theorem 12 and the interval analysis in Theorem 7 to cover parameter-independent randomized procedures using the same observation-law comparison.

Identification through observed cells

The cell functional in Definition 10 expresses the target through membership probabilities and arrived-outcome means, with zero contributions assigned to null cells. Its connection to the average treatment effect uses randomized independence, balanced assignment, consistency, conditional missing at random, and positive arrival in occupied cells, as specified in Assumptions 2, 3, 4, 5, 6, and 7. The first identity applies to the unrestricted-arrival class.

The identity makes the observed-cell functional an exact population target throughout the unrestricted-arrival class. The surrogate enters through the cells on which arrival is conditioned, and the signed sum aggregates their contributions into the treatment contrast. This is the identification relation used by the unrestricted risk bound in Theorem 6.

For the rare-arrival class, the same relation is recorded under the model conditions governing its minimax analysis.

The rare-arrival restriction in Assumption 8 specifies the regime for the risk comparison, while this identity fixes its estimand. In particular, the null-cell convention gives a total cell functional across laws with different occupied categories. The mixed-count analysis underlying Theorem 4 therefore targets the same average treatment effect throughout the rare-arrival model.

Synthetic assignment and experiment comparison

The observational class in Definition 31 supplies an experiment that can be mapped into the unrestricted randomized model. The map assigns an independent fair binary arm to each observational record, records outcome arrival when that arm matches the observational treatment, and sets the surrogate to zero. The following statement establishes a parameter-independent observation map together with a full-data completion preserving the target.

The map preserves both the product observation law and the target at the corresponding full-data completion. Its parameter independence permits composition with any randomized estimator in Definition 12. This fixes the direction of the experiment comparison used in Theorem 5: observational minimax lower bounds transfer to the unrestricted randomized model at the same arrival floor.

For construction, the same map supplies the synthetic records used by the observational procedure in Theorem 8. The target equality connects its squared loss to that of the randomized estimator, and Lemma 1 supplies the risk comparison for averaging over the synthetic assignments. Thus the observation map supports both the fixed-arrival lower-bound comparison and the constructive observational extension.

Risk certificates for the constructive estimator

The risk analysis separates estimation of cell membership probabilities from estimation of arrived-cell outcome means. Independence holds in the uncapped Poisson comparison experiment used for this analysis. The reported finite-sample estimator instead averages assignments of a capped prefix, and the prefix-transfer argument below connects its risk to that independent-stream comparison. The polynomial branch controls the contribution of sparsely observed cells. The resulting certificate bounds squared error for the observed-cell functional in Definition 10; identification then connects that certificate to the causal upper bound in Theorem 6.

For the deterministic mixed-count estimator ${\hat{\tau }}_{n,d,q}^{\mathrm{mix}}$ of Algorithm 1, write $N=nq$ for the effective sample size, $m=n/6$ for the intensity of each auxiliary stream, and $N_0=mq=N/6$ for the effective stream size. The logarithmic scale $\ell$, degree $k$, and threshold $B$ are specified in Definitions 13, 19, and 20. Define the prefix-tail constant by $${\gamma }_0=\log 2-\frac{1}{2}.$$ On the polynomial branch, where $N>1$, $\ell \ge 128$, and $d\ge \sqrt{N}$, introduce the certificate quantities $$b_{n,d,q}=2\left\{\frac{4dB}{N_0{k}^2}+3e^{-16\ell }\right\}$$ and $$v_{n,d,q}=8\left(\frac{4}{N_0}+\frac{1}{m}\right)+64d(e^{\ell /8}+1)\left(\frac{B^2}{N_0^2}+\frac{B}{m{N}_0}\right)+32e^{-32\ell }\left(1+\frac{1}{m}\right).$$ Here $b_{n,d,q}$ is the bias allowance and $v_{n,d,q}$ is the additive second-moment allowance. These quantities summarize the polynomial approximation, marked-count fluctuations, and estimation of membership weights.

The deterministic squared-error envelope ${\mathfrak{U}}_{n,d,q}$ combines these allowances with the elementary branch and the prefix-overflow contribution. The following statement gives its full formula and compares it with the rate $r_{n,d,q}$ of Definition 17 throughout the unrestricted-arrival class ${\mathcal{M}}_{n,d,q}^{+}$ of Definition 7.

The certificate distinguishes information from arrived outcomes, measured by $N_0$, from information about cell membership, measured by $m$. On the polynomial branch, the squared bias allowance captures the aggregate effect of estimating means in sparsely observed cells, while the additive allowance accounts for fluctuations in both components of the cell contribution. The elementary branch supplies the corresponding bound in its stated parameter range. Together, the branches give a deterministic precision benchmark depending on $n,d,q$, with a universal comparison constant.

Polynomial bias and marked-count second moments

The polynomial statistic $H_j$ and the pilot-selected statistic $G_j$ are specified in Definitions 25 and 24. Their mathematical role is to replace the elementary cell statistic where the pilot count indicates sparse outcome arrival. The degree and threshold jointly determine the approximation contribution $4dB/(N_0{k}^2)$ in the bias allowance. The exponentially small term in that allowance accounts for branch control. The certificate combines these contributions before squaring, preserving the distinction between bias and second-moment terms.

Marked counts retain the association between an arrived outcome and its cell. Consequently, the second-moment analysis concerns the marked polynomial statistic, together with the pilot-selected branch, rather than an unmarked count alone. The expression $v_{n,d,q}$ records the resulting allowance: its terms involve the effective arrived-outcome intensity, the membership-stream intensity, the alphabet size, and the polynomial tuning parameters. This decomposition explains why the degree enters the bias allowance while the coefficient growth associated with the polynomial construction enters the additive allowance.

Membership weights and branch control

The membership weight $W_j$ in Definition 28 is formed from the membership-labelled records. In the uncapped Poisson comparison experiment, that stream is independent of the pilot and estimation streams. The finite-sample estimator uses the capped prefix and conditional averaging in Algorithm 1; the overflow allowance and prefix-transfer comparison connect it to the independent-stream analysis. The weight supplies the population mass attached to each arrived-cell mean while retaining information from all observed memberships.

The two sample scales appear explicitly in the certificate through $1/N_0$, $1/m$, and the mixed term $B/(m{N}_0)$. These terms retain the cost of estimating membership weights alongside the cost of estimating arrived-cell means. The pilot-selected statistic in Definition 24 organizes the latter task within the polynomial branch, whereas the elementary ratio in Definition 26 supplies the estimator on the elementary branch. The branch conditions in Theorem 9 specify where each deterministic expression applies.

Transfer to the fixed observed sample and causal target

The auxiliary construction in Algorithm 1 uses an independent Poisson prefix and uniform stream assignments. It assigns zero output when the prefix exceeds the available $n$ records and clips the output to $[-1,1]$. The term $4e^{-n{\gamma }_0}$ in Theorem 9 is the prefix-overflow allowance. Its dependence on the full record count $n$ reflects the supply of records for the auxiliary construction, while $N=nq$ governs the effective outcome information. Clipping supplies the bounded-loss allowance represented by the minimum with $4$; the zero estimator on the small-effective-sample branch supplies the bound $1$.

Conditional averaging over the auxiliary randomization produces the deterministic estimator in Algorithm 1. The squared-loss comparison for such averaging is supplied by Lemma 1. Thus the certificate concerns the estimator evaluated on the fixed observed sample, with the auxiliary construction serving its risk analysis.

For the causal interpretation, Lemma 3 identifies the observed-cell functional with the average treatment effect under the unrestricted model conditions. Substituting that identity into Theorem 9 gives the causal risk control used in Theorem 6. The same identification step applies throughout the rare-arrival class ${\mathcal{M}}_{n,d,q}$ of Definition 14. The following specialization records the resulting envelope for that class.

The rare-arrival restriction places the model within the domain of the unrestricted certificate, and Lemma 4 fixes the causal target there. Accordingly, the specialization retains the same numerical envelope. Its polynomial bias, second-moment, and prefix-transfer components are those of Theorem 9; model inclusion and identification supply the specialization step.

The next statement records this causal upper bound together with the squared-loss comparison between the deterministic estimator and its auxiliary randomized output. It makes explicit the role of conditional averaging in the final estimator.

Averaging over the prefix and stream assignments removes auxiliary variation under squared loss. The first inequality in Theorem 11 is the certificate of Theorem 10, and the final comparison is the conditional-averaging relation of Lemma 1 applied to the specified auxiliary output. These references preserve the same construction and envelope across the two statements. For a prospective choice of $n,d,q$, the envelope therefore supplies a deterministic causal risk benchmark that incorporates cell complexity, outcome arrival, and the fixed-sample implementation.

Lower bounds with full observed memberships

The lower comparisons concern the observed experiment in Definition 15, which retains baseline category, assignment, surrogate, arrival status, and arrived outcome for every sampled individual. Two constructions account for the two components of the rate in Definition 17: a one-category family isolates the effective sample-size contribution, while an activated many-category experiment supplies the category-complexity comparison. We first introduce the activated moment-matching construction, then state its approximation inputs and the one-category bounds. A separate comparison addresses the ascertainment deficit in the unrestricted large-alphabet regime.

The activated comparison experiment

The construction in Algorithm 2 augments each observed record with an activation flag whose distribution is common to the two hypotheses. Conditional on a latent vector drawn once for the sample, it generates exactly $n$ independent records with balanced treatment assignment. The following algorithm specifies both augmented laws and the conditions on their finite priors.

The augmented pair ${\mathfrak{H}}_{\mathrm{act}}$ in Algorithm 2 preserves all original observations. Its revealed activation flag ${\mathsf{B}}_{\mathrm{act}}$ has a distribution depending on the experiment’s parameters and independent of the latent coordinates and prior index. Projection onto the original record coordinates is therefore a common, parameter-independent observation map. The total-variation comparison in Lemma 2 supplies the decision-theoretic connection between an observed-law comparison and any common randomized procedure.

The tuning constant $\eta$, moment-matching degree $K$, support endpoint $H$, nominal rare-cell mass $b_0$, and rare-cell count $J$ are specified in Algorithm 2. The mass cap reserves at least half of the baseline probability for the reservoir. Each rare cell has a single latent coordinate $Z_x$, shared by its two treatment arms and all its sampled records. In the table, this coordinate governs arrival, while its reciprocal governs the treated arrived-outcome mean; the control outcome is zero. This arrangement places reciprocal separation and polynomial moment matching in the same experiment.

Reciprocal approximation and finite priors

The approximation inputs concern the interval and degree used by Algorithm 2. Best uniform approximation quantifies the discrepancy between the reciprocal function and polynomials of the matching degree. The first input gives that discrepancy exactly.

Lemma 6 identifies the approximation scale underlying the prior construction. Its mathematical role is to distinguish the reciprocal functional from the polynomial moments used to compare the two experiments. The next input supplies finite probability laws with exactly matching moments and a fixed reciprocal-mean separation.

The finite priors ${\Pi }_0,{\Pi }_1$ supplied by Lemma 7 satisfy the prior requirements of Algorithm 2. Matching includes the zeroth moment, and reciprocal separation remains bounded below by an absolute constant. Finite support makes the augmented laws finite mixtures of experiments generated conditional on a fixed latent vector.

A one-category family

For the effective sample-size component, the local family in Lemma 8 places the entire baseline mass on category zero. Both potential surrogates and the control potential outcome are zero; the treated potential outcome is Bernoulli with mean $u\in [1/4,3/4]$. Treatment is an independent fair coin, and arrival is an independent Bernoulli variable with probability $q$. Thus $P_u$, the full-data law indexed by the treated outcome mean and defined through Lemma 8, has average treatment effect $u$. The following statement records both squared-error and honest-interval comparisons for this family.

The squared-error comparison already applies when all individuals share a baseline category and both potential surrogates are constant. It isolates the information supplied by arrived treated outcomes, whose frequency is governed by balanced assignment and the arrival probability. The same family supports the interval comparison: the family is common across coverage levels, while the finite grid and interval lower constant may depend on the chosen coverage level. Both conclusions retain the randomized decision classes of Definitions 16 and 29.

The many-category comparison in the full experiment

The many-category component uses the augmented laws of Algorithm 2, with the finite priors supplied by Lemma 7. Their membership distribution, treatment allocation, and activation distribution are common to the two hypotheses. The comparison therefore concerns the full fixed-sample experiment, including the information in repeated memberships and in both randomized arms.

Two mathematical inputs have distinct roles in this comparison. Moment matching governs comparison of the integrated observation laws, while reciprocal-mean separation governs separation of the treatment-effect functional across the prior constructions. The common activation mechanism connects these roles while preserving the original record coordinates. Together with the one-category component in Lemma 8, these inputs support the following lower bound for the entire decision class.

The bound combines effective sample-size uncertainty with uncertainty across many categories. The one-category family accounts for the reciprocal effective sample-size term, and the activated comparison accounts for the squared category term in Definition 17. Full membership observations remain available throughout the experiment. Consequently, the lower bound describes the difficulty of estimating the treatment contrast with all recorded baseline, assignment, surrogate, and arrival information under the stated rare-arrival conditions.

Large alphabets and the ascertainment deficit

The separate unrestricted comparison is Theorem 2. Its domain is $n\ge 1$, $d\ge 1024n^2$, and $0<q\le 1$, with the model and risk defined in Definitions 7 and 12. The observed-law comparison concerns all $n$ original records, including their category memberships. Target separation supplies the ascertainment-deficit component, quantified by the lower bound $(1-q{)}^2/1024$; the sampling component is bounded below by $1/(256n)$.

These components have complementary interpretations. The sampling term remains present at complete ascertainment. The squared deficit measures the additional difficulty captured by the large-alphabet comparison when the arrival floor is below one. The matching upper bound $4/n+(1-q{)}^2$ in Theorem 2 comes from the observed-outcome envelope of Theorem 1. Thus the large-alphabet result supplies a comparison with universal constants for the joint sampling-and-deficit scale.

Squared-error comparisons and their consequences

The lower bound in Theorem 12 and the constructive upper comparison in Theorem 6 provide the two sides of Theorem 4. Under Assumption 8, the minimax squared-error risk is therefore comparable, with universal constants, to the rate in Definition 17. Conditional averaging preserves the upper guarantee by Lemma 1, while the lower comparison covers the full class of possibly randomized estimators.

For a fixed alphabet and growing effective sample size, Proposition 4 records risk of order $N^{-1}$, including within the constant-surrogate submodel. Along growing-alphabet sequences satisfying the rare-arrival restriction and $N\to \infty$, Proposition 5 gives consistency exactly when $d=o\{N\log (e+N)\}$, and risk of order $N^{-1}$ exactly when $d=O\{\sqrt{N}\log (e+N)\}$. These consequences distinguish the category-growth range permitting consistency from the range preserving effective sample-size precision.

For the unrestricted model, Proposition 1 combines the large-alphabet lower comparison with the dimension-free upper envelope. An eventual risk bound of order $n^{-1}$, simultaneous over every alphabet size, is equivalent to an ascertainment deficit $1-q_n=O(n^{-1/2})$. The numerical comparison along arbitrary alphabet-size sequences is given by Proposition 2. At complete arrival, Theorem 3 and Proposition 3 establish the same sampling order uniformly over category growth.

Finally, the fixed-floor comparison in Theorem 5 concerns each fixed $q\in (0,1/2)$, with constants depending on that floor. Its lower comparison invokes the discrete observational result of Zeng et al. (2026), including the submodel with zero control regression. The resulting category-growth consequences are stated in Proposition 6: consistency corresponds to $d=o\{n\log (e+n)\}$, and parametric squared-error order corresponds to $d=O\{\sqrt{n}\log (e+n)\}$. These comparisons preserve the distinct parameter domains and constant dependencies of the rare-arrival, fixed-floor, and near-complete ascertainment results.

Interval and observational proofs

The interval and observational comparisons use two features of the constructive analysis: a deterministic bound on squared-error risk and a parameter-independent transformation preserving the target. The decision-theoretic results in Lemmas 1 and 2 supply the corresponding comparisons for procedures using auxiliary randomization.

Coverage and expected interval length

Fix $\alpha \in (0,1)$, and suppose that $n,d\ge 1$, $0<q\le 1$, and Assumption 8 holds. The deterministic envelope ${\mathfrak{U}}_{n,d,q}$ in Theorem 6 controls squared-error risk uniformly over the model in Definition 14. Its role in Definition 30 is to determine the radius before observing the sample. Markov’s inequality converts this risk control into coverage at least $1-\alpha$. Because the target belongs to $[-1,1]$, intersecting the interval with that range preserves coverage.

The mixed-count estimator ${\hat{\tau }}_{n,d,q}^{\mathrm{mix}}$ is defined in Algorithm 1. The resulting interval $I_{\alpha }^{\mathrm{mix}}$ has length bounded by $$\mathrm{length}(I_{\alpha }^{\mathrm{mix}})\le \min \left\{2,2\sqrt{\frac{{\mathfrak{U}}_{n,d,q}}{\alpha }}\right\}.$$ Thus the envelope comparison with the rate $r_{n,d,q}$ in Definition 17 supplies the upper expected-length comparison in Theorem 7. The upper constant ${\overline{C}}_{\alpha }$ may depend on the fixed miscoverage level and is uniform over the sample sizes, dimensions, and arrival floors satisfying these conditions.

The lower comparison concerns every endpoint procedure admitted by Definition 29. Its sampling component is supplied directly by Lemma 8. For each fixed $\alpha$, that result supplies a finite grid of distinct target values within the same one-category family and bounds maximal expected length below at order $$\min \{1,(nq{)}^{-1/2}\}.$$ The underlying family is common across coverage levels; the grid and its comparison constant may depend on $\alpha$. This finite-family formulation supports every coverage level $1-\alpha \in (0,1)$.

The many-category component of Theorem 7 uses the separation and observation-law comparison associated with Algorithm 2 and Lemma 7. Moment matching controls statistical distinguishability, while reciprocal separation determines the target scale. Connectedness supplies the geometric link to expected length: an interval containing separated target values spans their separation. Finite testing families at the specified coverage level retain this link throughout the coverage range in the theorem.

Auxiliary endpoint randomization enters through the joint observation–action laws defined in Lemma 2. Apply that result with the action space equal to the ordered endpoint pairs and the common decision rule equal to the interval procedure. The joint-law total variation equals the total variation of the observation laws. Consequently, coverage events can be compared while retaining both the observed sample and the reported endpoints. Expected length is evaluated under this same joint law, exactly as required by Definition 29.

Together, these components give the lower constant ${\underline{c}}_{\alpha }$ in Theorem 7. The minimax expected length ${\mathfrak{L}}_{n,d,q,\alpha }$ is therefore comparable to $\sqrt{r_{n,d,q}}$, with constants depending on $\alpha$. The square-root scale reflects both sources of uncertainty: effective sample size and the number of covariate categories. Along sequences satisfying the rare-arrival restriction, at fixed $\alpha$, optimal expected length tends to zero exactly when $r_{n,d,q}$ tends to zero.

Synthetic assignment and observational risk

Fix a strict positivity margin $0<ϵ<1/2$. The observational class ${\mathcal{D}}_{d,ϵ}$ is defined in Definition 31, and its target is the contrast in Definition 32. The transformation in Lemma 5, with arrival floor $q=ϵ$, assigns an independent fair arm to each observational record and retains the outcome when that arm agrees with the observational treatment. Positivity supplies the arrival floor in both arms.

The induced full-data law $P_Z^{\ast }$, defined by Lemma 5, belongs to the unrestricted randomized class and preserves the target: $$\tau (P_Z^{\ast })={\theta }_Z(P_Z).$$ The same statement identifies the synthetic sample law with the product observed-record law of this completion. These two properties make the unrestricted risk certificate in Theorem 6 applicable to the mixed-count estimator evaluated on the synthetic records.

The observational estimator ${\hat{\theta }}_{n,d,ϵ}^{\mathrm{poly}}$ in Theorem 8 averages over the independent synthetic assignments conditional on the observational sample. The barycenter comparison in Lemma 1 weakly reduces squared-error risk under this averaging. Accordingly, the constructive bound has a universal upper constant and retains the positivity dependence explicitly through $$r_{n,d,ϵ}=\min \left\{1,\frac{1}{nϵ}+{\left[\frac{d}{nϵ\log (e+nϵ)}\right]}^2\right\}.$$ Synthetic assignment converts observational treatment imbalance into selective outcome arrival; conditional averaging then removes the variation introduced by that assignment. Target preservation connects both operations to the same observational contrast.

The matching lower comparison invokes Zeng et al. (2026). The relevant published conclusion concerns fixed strict positivity, sufficiently large $n$, and $1\le d\le c_{0}n\log n$, where $c_0>0$ is a constant in that conclusion. Its proof uses the submodel with zero control regression on occupied categories and supplies the lower order $$\frac{1}{n}+{\left(\frac{d}{n\log n}\right)}^2.$$ This conclusion supplies the observational lower-bound input. The matched comparison in Theorem 8 covers every $n,d\ge 1$, incorporates bounded-risk truncation, and uses $\log (e+n)$.

For each fixed $ϵ$, the observational minimax risk ${\mathfrak{R}}_{n,d,ϵ}^Z$ consequently has the two-sided comparison stated in Theorem 8, with positive finite constants $c_{ϵ}$ and $C_{ϵ}$ depending on that margin. This dependence is distinct from the universal constant in the constructive bound expressed through $r_{n,d,ϵ}$.

Fixed arrival and category-growth consequences

The same transformation determines the direction of the fixed-arrival experiment comparison. Fix $q\in (0,1/2)$. For every observational law in ${\mathcal{D}}_{d,q}$, Lemma 5 supplies a target-preserving completion in the unrestricted class ${\mathcal{M}}_{n,d,q}^{+}$ of Definition 7. Composing any randomized estimation procedure with the synthetic assignment gives an observational procedure with the same risk at the corresponding completion. Hence the unrestricted randomized minimax risk ${\mathfrak{R}}_{n,d,q}^{+}$ dominates the observational minimax risk at positivity margin $q$. The published observational lower comparison therefore transfers to this randomized class.

Combined with the constructive upper comparison, this transfer supports Theorem 5. Its constants $\underline{c}(q)$ and $\overline{C}(q)$ depend only on the fixed arrival floor and apply uniformly over $n,d\ge 1$. The resulting category-growth conclusions are recorded in Proposition 6: consistency is equivalent to $$d_n=o\{n\log (e+n)\},$$ and parametric squared-error order is equivalent to $$d_n=O\{\sqrt{n}\log (e+n)\}.$$ For an eventual bound $d_n\le M\sqrt{n}\log (e+n)$, the lower risk constant may depend on $q$, while the upper constant may depend on both $q$ and $M$. The eventual sample-size threshold may depend on the category sequence. These dependencies specify the fixed-arrival scope of the matched comparison and its sequence-level consequences.

Verification note

The theorem and lemma statements and proofs attached to the frozen results are machine-checked in Lean 4 under their stated hypotheses and cited dependencies. The checked conclusions cover the identification identities in Lemmas 3 and 4, the decision-rule comparisons in Lemmas 1 and 2, and the synthetic-assignment construction in Lemma 5. They also cover the estimator risk certificates in Theorems 9, 10, and 11, the approximation and testing conclusions in Lemmas 6, 7, and 8, and the lower bound in Theorem 12. The point-risk comparisons and their stated phase-boundary consequences, the connected-interval comparison in Theorem 7, and the observational comparison in Theorem 8 are checked with the conditions and constant dependencies recorded in their statements.

The model restrictions enter as assumptions. These include independent sampling, randomized assignment and balance, surrogate and outcome consistency, conditional independence of outcome arrival, and the occupied-cell arrival floor in Assumptions 1, 2, 3, 4, 5, 6, and 7. Results for rare arrival additionally use Assumption 8; the observational comparison uses the positivity class specified in Definition 31. Presentation definitions specify the laws, targets, decision classes, and procedures to which the mathematical assertions apply.

Published mathematical conclusions used through theorem-local cited dependencies remain part of the assumptions of the corresponding checks. The generated local verification disclosures identify those conclusions, preserve their source locators, and distinguish them from proofs formalized in Lean. Together, the stated hypotheses and those disclosures specify the scope of the machine-checked conditional conclusions.

Proofs of the main results

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