Using the periodogram to estimate period in nonparametric regression
Centre for Mathematics and its Applications, Australian National University, Canberra ACT 0200, Australia. peter.hall{at}anu.edu.au, ming.li{at}maths.anu.edu.au
Properties of the periodogram are seldom studied in the setting of nonparametric regression, although that is the context in which the periodogram is widely applied in astronomy. There it is a competitor with more recent least-squares methods. The periodogram has the advantage of providing significant graphical insight into statistical and numerical aspects of the problem. However, as we show in the present paper, it also has drawbacks. The estimator that it produces has somewhat higher variance than its least-squares counterpart, and a periodogram-based approach is more prone to suffer difficulties caused by periodicity of the observation schedule. While the periodogram remains a very attractive tool, the information provided in this paper allows users to assess more readily the extent to which it can be relied upon in a nonparametric setting. This aspect of our contributions is discussed theoretically and illustrated by numerical studies involving a real dataset.
Key Words: Aliasing; Astronomy; Central limit theorem; Convergence rate; Design density; Frequency; Least-squares; Nonparametric regression; Number theory; Regression-mean; Relatively rational; Wavelength.
Received May 2005. Revised November 2005.