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Biometrika 2000 87(3):545-557; doi:10.1093/biomet/87.3.545
© 2000 by Biometrika Trust
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Nonparametric estimation of a periodic function

P HallZ, J ReimannZZ and J RiceY

Z Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia E-mail: halpstat@pretty.anu.edu.au ZZ Biostatistics Department, Genentech, Inc., South San Francisco, CA 94080, USA E-mail: reimann@gene.com Y Department of Statistics, University of California, Berkeley, CA 94720, USA E-mail: rice@stat.berkeley.edu

Motivated by applications to brightness data on periodic variable stars, we study nonparametric methods for estimating both the period and the amplitude function from noisy observations of a periodic function made at irregularly spaced times. It is shown that nonparametric estimators of period converge at parametric rates and attain a semiparametric lower bound which is the same if the shape of the periodic function is unknown as if it were known. Also, first-order properties of nonparametric estimators of the amplitude function are identical to those that would obtain if the period were known. Numerical simulations and applications to real data show the method to work well in practice.

Key Words: frequency estimation; MACHO project; Nadaraya-Watson estimator; nonparametric regression; semiparametric estimation; variable star data


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