Privacy-preserving estimation for non-randomly distributed data

This paper investigates data distributed across various machines in a non-random manner. We introduce two innovative distributed estimators, tailored to accommodate varying levels of communication cost and data privacy protection. The proposed estimators adeptly addresses the challenges associated with the non-random distribution of data. Both methods are communication-efficient, necessitating only two rounds of communication between the Master and worker machines, and safeguard data privacy by solely sharing summary statistics. Under mild conditions, we establish the ${\ell }_2$-error bound and the asymptotic distribution of the estimators. Theoretical analysis confirms that the proposed estimators are statistically efficient. Additionally, numerical simulations and two real-world applications demonstrate the good performance of the proposed methods.