## The Annals of Statistics

### Nonparametric Estimation of Common Regressors for Similar Curve Data

Alois Kneip

#### Abstract

The paper is concerned with data from a collection of different, but related, regression curves $(m_j0_{j = 1}, \ldots, N, N \gg 1$. In statistical practice, analysis of such data is most frequently based on low-dimensional linear models. It is then assumed that each regression curve $m_j$ is a linear combination of a small number $L \ll N$ of common functions $g_1, \ldots, g_L$k. For example, if all $m_j$'s are straight lines, this holds with $L = 2, g_1 \equiv 1$ and $g_2(x) = x$. In this paper the assumption of a prespecified model is dropped. A nonparametric method is presented which allows estimation of the smallest $L$ and corresponding functions $g_1, \ldots, g_L$ from the data. The procedure combines smoothing techniques with ideas related to principal component analysis. An asymptotic theory is presented with yields detailed insight into properties of the resulting estimators. An application to household expenditure data illustrates the approach.

#### Article information

Source
Ann. Statist., Volume 22, Number 3 (1994), 1386-1427.

Dates
First available in Project Euclid: 11 April 2007

https://projecteuclid.org/euclid.aos/1176325634

Digital Object Identifier
doi:10.1214/aos/1176325634

Mathematical Reviews number (MathSciNet)
MR1311981

Zentralblatt MATH identifier
0817.62029

JSTOR