Hermite regression estimation in noisy convolution model

IF 0.8 4区数学 Q3 STATISTICS & PROBABILITY Journal of Statistical Planning and Inference Pub Date : 2024-03-26 DOI:10.1016/j.jspi.2024.106168

Ousmane Sacko

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引用次数: 0

Abstract

In this paper, we consider the following regression model: $y (k T / n) = f ⋆ g (k T / n) + ɛ_{k}, k = - n, \dots, n - 1$ , $T$ fixed, where $g$ is known and $f$ is the unknown function to be estimated. The errors ${(ɛ_{k})}_{- n \leq k \leq n - 1}$ are independent and identically distributed centered with finite known variance. Two adaptive estimation methods for $f$ are considered by exploiting the properties of the Hermite basis. We study the quadratic risk of each estimator. If $f$ belongs to Sobolev regularity spaces, we derive rates of convergence. Adaptive procedures to select the relevant parameter inspired by the Goldenshluger and Lepski method are proposed and we prove that the resulting estimators satisfy oracle inequalities for sub-Gaussian $ɛ$ ’s. Finally, we illustrate numerically these approaches.

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噪声卷积模型中的赫米特回归估计

本文考虑以下回归模型：y(kT/n)=f⋆g(kT/n)+ɛk,k=-n,...,n-1, T 固定，其中 g 为已知函数，f 为待估计的未知函数。误差 (ɛk)-n≤k≤n-1 是独立且同分布的中心误差，具有有限的已知方差。利用赫米特基的特性，我们考虑了 f 的两种自适应估计方法。我们研究了每种估计方法的二次风险。如果 f 属于 Sobolev 正则空间，我们将得出收敛率。受 Goldenshluger 和 Lepski 方法的启发，我们提出了选择相关参数的自适应程序，并证明所得到的估计器满足亚高斯ɛ的oracle 不等式。最后，我们用数字说明了这些方法。

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来源期刊

Journal of Statistical Planning and Inference 数学-统计学与概率论

CiteScore

2.10

自引率

11.10%

发文量

审稿时长

3-6 weeks

期刊介绍： The Journal of Statistical Planning and Inference offers itself as a multifaceted and all-inclusive bridge between classical aspects of statistics and probability, and the emerging interdisciplinary aspects that have a potential of revolutionizing the subject. While we maintain our traditional strength in statistical inference, design, classical probability, and large sample methods, we also have a far more inclusive and broadened scope to keep up with the new problems that confront us as statisticians, mathematicians, and scientists. We publish high quality articles in all branches of statistics, probability, discrete mathematics, machine learning, and bioinformatics. We also especially welcome well written and up to date review articles on fundamental themes of statistics, probability, machine learning, and general biostatistics. Thoughtful letters to the editors, interesting problems in need of a solution, and short notes carrying an element of elegance or beauty are equally welcome.

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