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June, 1984 Regression Models with Infinitely Many Parameters: Consistency of Bounded Linear Functionals
Ker-Chau Li
Ann. Statist. 12(2): 601-611 (June, 1984). DOI: 10.1214/aos/1176346508

Abstract

Consider a linear model with infinitely many parameters given by $y = \sum^\infty_{i = 1} x_i\theta_i + \varepsilon$ where $\mathbf{x} = (x_1, x_2, \cdots)'$ and $\theta = (\theta_1, \theta_2, \cdots)'$ are infinite dimensional vectors such that $\sum^\infty_{i = 1}x^2_i < \infty$ and $\sum^\infty_{i = 1} \theta^2_i < \infty$. Suppose independent observations $y_1, y_2, \cdots, y_n$ are observed at levels $\mathbf{x}_1, \mathbf{x}_2, \cdots, \mathbf{x}_n$. Under suitable conditions about the error distribution, the set of all bounded linear functionals $T(\theta)$ for which there exists an estimate $\hat{T}_n$ such that $\hat{T}_n \rightarrow T(\theta)$ in probability will be characterized. An application will be extended to the nonparametric regression problem where the response curve $f$ is smooth on the interval [0, 1] in the sense that $f$ has an $(m - 1)$th derivative that is absolutely continuous and $\int^1_0 f^{(m)}(t)^2 dt < \infty$.

Citation

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Ker-Chau Li. "Regression Models with Infinitely Many Parameters: Consistency of Bounded Linear Functionals." Ann. Statist. 12 (2) 601 - 611, June, 1984. https://doi.org/10.1214/aos/1176346508

Information

Published: June, 1984
First available in Project Euclid: 12 April 2007

zbMATH: 0544.62062
MathSciNet: MR740914
Digital Object Identifier: 10.1214/aos/1176346508

Subjects:
Primary: 62G05
Secondary: 62J05

Rights: Copyright © 1984 Institute of Mathematical Statistics

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Vol.12 • No. 2 • June, 1984
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