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2020 The limiting behavior of isotonic and convex regression estimators when the model is misspecified
Eunji Lim
Electron. J. Statist. 14(1): 2053-2097 (2020). DOI: 10.1214/20-EJS1714

Abstract

We study the asymptotic behavior of the least squares estimators when the model is possibly misspecified. We consider the setting where we wish to estimate an unknown function $f_{*}:(0,1)^{d}\rightarrow \mathbb{R}$ from observations $(X,Y),(X_{1},Y_{1}),\cdots ,(X_{n},Y_{n})$; our estimator $\hat{g}_{n}$ is the minimizer of $\sum _{i=1}^{n}(Y_{i}-g(X_{i}))^{2}/n$ over $g\in \mathcal{G}$ for some set of functions $\mathcal{G}$. We provide sufficient conditions on the metric entropy of $\mathcal{G}$, under which $\hat{g}_{n}$ converges to $g_{*}$ as $n\rightarrow \infty $, where $g_{*}$ is the minimizer of $\|g-f_{*}\|\triangleq \mathbb{E}(g(X)-f_{*}(X))^{2}$ over $g\in \mathcal{G}$. As corollaries of our theorem, we establish $\|\hat{g}_{n}-g_{*}\|\rightarrow 0$ as $n\rightarrow \infty $ when $\mathcal{G}$ is the set of monotone functions or the set of convex functions. We also make a connection between the convergence rate of $\|\hat{g}_{n}-g_{*}\|$ and the metric entropy of $\mathcal{G}$. As special cases of our finding, we compute the convergence rate of $\|\hat{g}_{n}-g_{*}\|^{2}$ when $\mathcal{G}$ is the set of bounded monotone functions or the set of bounded convex functions.

Citation

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Eunji Lim. "The limiting behavior of isotonic and convex regression estimators when the model is misspecified." Electron. J. Statist. 14 (1) 2053 - 2097, 2020. https://doi.org/10.1214/20-EJS1714

Information

Received: 1 July 2019; Published: 2020
First available in Project Euclid: 6 May 2020

zbMATH: 07210995
MathSciNet: MR4094468
Digital Object Identifier: 10.1214/20-EJS1714

Subjects:
Primary: 62G08, 62G20
Secondary: 62G10

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Vol.14 • No. 1 • 2020
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