. 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 AND PHRASES. frequency estimation, nonparametric regression, semiparametric estimation, Nadaraya-Watson estimator, MACHO project, variable star data. SHORT TITLE. Estimation of a periodic function 1 Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 02...