2023/01/25 by Lukas Steinberger, Steinberger, Lukas · 4 citations
Economics, Econometrics and Finance · Social Sciences · #Corruption and Economic Development #Economic Policies and Impacts #FOS: Mathematics #Local Government Finance and Decentralization #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2301.10600
openalex publication_date 2023/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop a theory of asymptotic efficiency in regular parametric models when data confidentiality is ensured by local differential privacy (LDP). Even though efficient parameter estimation is a classical and well-studied problem in mathematical statistics, it leads to several non-trivial obstacles that need to be tackled when dealing with the LDP case. Starting from a standard parametric model \mathcal P=(Pθ)θ∈Θ, Θ⊆\mathbb Rp, for the iid unobserved sensitive data X1,…, Xn, we establish local asymptotic mixed normality (along subsequences) of the model Q(n)\mathcal P=(Q(n)Pθn)θ∈Θ generating the sanitized observations Z1,…, Zn, where Q(n) is an arbitrary sequence of sequentially interactive privacy mechanisms. This result readily implies convolution and local asymptotic minimax theorems. In case p=1, the optimal asymptotic variance is found to be the inverse of the supremal Fisher-Information supQ∈\mathcal Qα Iθ(Q\mathcal P)∈\mathbb R, where the supremum runs over all α-differentially private (marginal) Markov kernels. We present an algorithm for finding a (nearly) optimal privacy mechanism Q and an estimator θn(Z1,…, Zn) based on the corresponding sanitized data that achieves this asymptotically optimal variance.