Nonparametric spectral density estimation under local differential privacy

We consider nonparametric estimation of the spectral density of a stationary time series under local differential privacy. In this framework only an appropriately anonymised version of a finite snippet from the time series can be observed and used for inference. The anonymisation procedure can be chosen in advance among all mechanisms satisfying the condition of local differential privacy, and we propose truncation followed by Laplace perturbation for this purpose. Afterwards, the anonymised time series snippet is used to define a surrogate of the classical periodogram and our estimator is obtained by projection of this privatised periodogram to a model given by a finite dimensional subspace of \(L^2([-\pi ,\pi ])\) $L^2([-\pi ,\pi ])$ . This estimator attains nearly the same convergence rate as in the case where the original time series can be observed. However, a reduction of the effective sample size in contrast to the non-privacy framework is shown to be unavoidable. We also consider adaptive estimation and suggest to select an estimator from a set of candidate estimators by means of a penalised contrast criterion. We derive an oracle inequality which shows that the adaptive estimator attains nearly the same rate of convergence as the best estimator from the candidate set. Concentration inequalities for quadratic forms in terms of sub-exponential random variables, which have been recently derived in Götze et al. (Electron J Probab 26:1–22, 2021), turn out to be essential for our proof. Finally, we illustrate our findings in a small simulation study.