## The Annals of Statistics

- Ann. Statist.
- Volume 12, Number 2 (1984), 601-611.

### Regression Models with Infinitely Many Parameters: Consistency of Bounded Linear Functionals

#### 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 $\int^1_0 f^{(m)}(t)^2 dt < \infty$.

#### Article information

**Source**

Ann. Statist., Volume 12, Number 2 (1984), 601-611.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176346508

**Digital Object Identifier**

doi:10.1214/aos/1176346508

**Mathematical Reviews number (MathSciNet)**

MR740914

**Zentralblatt MATH identifier**

0544.62062

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62G05: Estimation

Secondary: 62J05: Linear regression

**Keywords**

Asymptotic consistency consistent region of degree $k$ Fisher's information Hilbert space limiting point of degree $k$ nonparametric regression

#### Citation

Li, Ker-Chau. Regression Models with Infinitely Many Parameters: Consistency of Bounded Linear Functionals. Ann. Statist. 12 (1984), no. 2, 601--611. doi:10.1214/aos/1176346508. https://projecteuclid.org/euclid.aos/1176346508