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Statistics Theory

math.ST — Statistics Theory

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3 works

math.ST — Statistics Theory

Nearly Minimax Variance Estimation Under Rough Random Design

Contributed by P. M. Aronow, Patrick Lopatto

We determine, up to a power of $\log n$ , the minimax risk for constant conditional variance estimation under rough random design. The unknown design density is bounded above and away from zero, with no smoothness assumption, and the conditional error laws may depend on the covariates and have uniformly bounded fourth moments. For an $s$ -Hölder regression function with $s>1$ in dimension $d>4s$ , the minimax root-mean-square risk lies, for all sufficiently large $n$ , between $c\Psi_n$ and $C\Psi_n(\log n)^{\Gamma}$ , where $\Psi_n=n^{-2(s+1)/(d+4)}e^{-\kappa\sqrt{\log n}}(\log n)^{(s-1)/(d+4)}$ and the constants $\kappa>0$ and $\Gamma>0$ are explicit. In particular, the minimax exponent is $2(s+1)/(d+4)$ , the minimax risk is smaller than $n^{-2(s+1)/(d+4)}$ by a stretched-exponential factor whose constant $\kappa$ is identified, and the rate proposed by Robins is not uniformly attainable over this model class. For $0<s\le1$ , we show that the exact minimax rate is $n^{-1/2}\vee n^{-4s/(d+4s)}$ ; for $s>1$ and $d\le4s$ , it is $n^{-1/2}$ .

Primarily AI-generated textHuman understanding: some parts

Oct 08, 2026PDF ↗View work →

hexagon:2610.00160v1

stat.TH — Statistics Theory

Binary Regression for BV Class Conditional Probability

Contributed by Rajesh Dachiraju

In this article we solve the problem of binary regression when the class conditional probability is a multivariate function of bounded variation. We study the consistency of binary regression under a nonparametric sampling framework. Sufficient conditions are established for the convergence of the binary regression estimator, and quantitative error estimates are obtained. We further strengthen the analysis by introducing the assumption that the feature vectors possess a point density measure of bounded variation. Under this geometric regularity assumption, the expectation-based error estimate is replaced by a deterministic estimate, yielding deterministic convergence in the $L^{2}$ norm and, consequently, almost sure convergence. The analysis demonstrates that the geometric distribution of the sampling points plays a fundamental role in the approximation properties of the estimator. Although the analysis is carried out on the torus $\mathbb{T}^{m}$ , the results extend to arbitrary bounded Lipschitz domains after affine scaling, embedding into $\mathbb{T}^{m}$ , and zero extension. The point density measure framework provides a deterministic geometric perspective on binary regression and establishes a connection between sampling geometry, bounded variation, and convergence theory.

A mix of human-written and AI-generated textHuman understanding: some parts

Oct 08, 2026PDF ↗View work →

hexagon:2610.00157v1

math.ST — Statistics Theory

Nearly Minimax Rates for Functional Estimation Under Rough Random Design

Contributed by P. M. Aronow, Nathan Kallus, Patrick Lopatto

We establish nearly minimax bounds for missing-at-random means, treatment effects, and expected conditional covariances under rough random design. For two nuisance functions with average H\"older smoothness $s$ in dimension $d$ , the minimax root-mean-square error is $n^{-2s/d+o(1)}$ when $s<d/4$ and of order $n^{-1/2}$ when $s\ge d/4$ . The first rate confirms the rough-design exponent suggested by higher-order influence function theory. In the generic model with an unknown bounded density weight, our upper and lower bounds differ by only polylogarithmic factors and identify a leading correction $e^{-\kappa\sqrt{\log n}}$ when $s<d/4$ , with $\kappa$ explicit in terms of $s/d$ and the weight bounds. For the expected conditional covariance, the same exponent and the same constant $\kappa$ were obtained independently and concurrently by S. Park (arXiv:2610.05006). For models with separately bounded density and propensity, we identify the same polynomial exponent and the explicit leading correction, with an $o(\sqrt{\log n})$ remainder in the logarithm of the risk.

Primarily AI-generated textHuman understanding: some parts

Oct 08, 2026PDF ↗View work →