Testing for periodicity at an unknown frequency under cyclic long memory, with applications to respiratory muscle training

A frequent problem in applied time series analysis is the identification of dominating periodic components. A particularly difficult task is to distinguish deterministic periodic signals from periodic long memory. In this paper, a family of test statistics based on Whittle’s Gaussian log-likelihood approximation is proposed. Asymptotic critical regions and bounds for the asymptotic power are derived. In cases where a deterministic periodic signal and periodic long memory share the same frequency, consistency and rates of type II error probabilities depend on the long-memory parameter. Simulations and an application to respiratory muscle training data illustrate the results.