Nonparametric-estimation of Common Regressors for Similar Curve Data

The paper is concerned with data from a collection of different, but related, regression curves (m(j))(j = 1,...,N), N much greater than 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 mj is a linear combination of a small number L much less than N of common functions g(1),...,g(L). For example, if all m(j)'s are straight lines, this holds with L = 2, g(1) = 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),...,g(L) from the data. The procedure combines smoothing techniques with ideas related to principal component analysis. An asymptotic theory is presented which yields detailed insight into properties of the resulting estimators. An application to household expenditure data illustrates the approach.