Adaptive spectral density estimation by model selection under local differential privacy

Abstract

We study spectral density estimation under local differential privacy. Anonymization is achieved through truncation followed by Laplace perturbation. We select our estimator from a set of candidate estimators by a penalized contrast criterion. This estimator is shown to attain nearly the same rate of convergence as the best estimator from the candidate set. A key ingredient of the proof are recent results on concentration of quadratic forms in terms of sub-exponential random variables obtained in [GSS19]. We illustrate our findings in a small simulation study.

Estimation of the invariant density for discretely observed diffusion processes: impact of the sampling and of the asynchronicity

“…Even in our context, with the presence of local differential privacy constraints, it seems natural to leverage the minimization of a contrast function technique, incorporating this quantity into the definition of the privacy mechanism employed in our estimation procedure. Furthermore, introducing centered Laplace-distributed noise to bounded random variables is known to ensure α-differential privacy (see [2], [20], [31]). This motivates our choice of the anonymization procedure.…”

“…However, recent advancements in research have introduced mechanisms that extend to a broader array of statistical challenges. These encompass hypothesis testing [8,33], M-estimators [6], robustness [35], change point analysis [9], mean and median estimation [20], and nonparametric estimation [12,11,31], among others. With the escalating importance of data protection, striking the right balance between statistical utility and privacy becomes crucial.…”