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Asymptotic Properties of a MiniMax-Optimal Estimator for an Arbitrary Non-Smooth Functional
Subject area: Science,Engineering and Technology · Area of research: Statistics
DOI: 10.64388/IREV10I4-1723739
Abstract
This paper revisits the estimation of the non-smooth functional T(λ)=1/n ∑_(i=1)^n▒|λ_i | from a Gaussian observation Y∼N(λ,I_n ), and studies the finite-sample and asymptotic behaviour of a minimax-optimal estimator constructed by best polynomial approximation and Hermite-polynomial moment estimation. Building on the sharp minimax framework of Cai and Low, we (i) give a self-contained derivation of the general minimax lower bound obtained by testing a pair of composite hypotheses; (ii) construct the estimator and derive its bias and variance with the cutoff-dependence carried through explicitly, showing that the mean squared error attains the minimax rate β_*^2 W^2 (loglogn/logn)^2 over the bounded parameter space Λ_n (W); and (iii) establish asymptotic normality under an undersmoothed cutoff, clarifying the conditions under which interval estimation is valid and the bias correction it requires. We then extend the estimator to finite-population estimation in a model-based survey-sampling setting and compare it, by simulation, with the Nadaraya–Watson and modified Nadaraya–Watson estimators. An independent reproduction across a range of sample sizes exhibits the predicted behaviour and quantifies the severe variance growth of the estimator with the cutoff. The paper is an applied and expository consolidation rather than a new rate result.
Keywords
asymptotic properties; best polynomial approximation; composite hypotheses; Hermite polynomials; minimax lower bound; non-smooth functional
References
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How to cite this paper
@article{1723739,
author = {Mukhwana Moses Kololi},
title = {Asymptotic Properties of a MiniMax-Optimal Estimator for an Arbitrary Non-Smooth Functional},
journal = {Iconic Research And Engineering Journals},
year = {2026},
volume = {10},
number = {4},
pages = {1836-1844},
issn = {2456-8880},
url = {https://www.irejournals.com/formatedpaper/1723739.pdf},
abstract = {This paper revisits the estimation of the non-smooth functional T(λ)=1/n ∑_(i=1)^n▒|λ_i | from a Gaussian observation Y∼N(λ,I_n ), and studies the finite-sample and asymptotic behaviour of a minimax-optimal estimator constructed by best polynomial approximation and Hermite-polynomial moment estimation. Building on the sharp minimax framework of Cai and Low, we (i) give a self-contained derivation of the general minimax lower bound obtained by testing a pair of composite hypotheses; (ii) construct the estimator and derive its bias and variance with the cutoff-dependence carried through explicitly, showing that the mean squared error attains the minimax rate β_*^2 W^2 (loglogn/logn)^2 over the bounded parameter space Λ_n (W); and (iii) establish asymptotic normality under an undersmoothed cutoff, clarifying the conditions under which interval estimation is valid and the bias correction it requires. We then extend the estimator to finite-population estimation in a model-based survey-sampling setting and compare it, by simulation, with the Nadaraya–Watson and modified Nadaraya–Watson estimators. An independent reproduction across a range of sample sizes exhibits the predicted behaviour and quantifies the severe variance growth of the estimator with the cutoff. The paper is an applied and expository consolidation rather than a new rate result.},
keywords = {asymptotic properties; best polynomial approximation; composite hypotheses; Hermite polynomials; minimax lower bound; non-smooth functional},
month = {October},
doi = {https://doi.org/10.64388/IREV10I4-1723739}
}