Nonparametric Estimation of Common Regressors for Similar Curve Data

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.