Nonasymptotic anytime-valid inference for quantiles under local differential privacy

We propose a framework for sequential inference on population quantiles under local differential privacy (LDP). Specifically, we construct a nonasymptotic, anytime-valid confidence sequence by sequentially testing candidate quantiles with betting $e$-processes and permanently eliminating those rejected at level $\alpha$. A ternary randomized response mechanism based on adaptively chosen query intervals provides the LDP guarantee. The proposed algorithm updates the confidence sequence in a single pass by taking the closed convex hull of the surviving candidates, while a nested-lattice implementation reduces memory requirements. We establish distribution-free, nonasymptotic coverage of at least $1-\alpha$ uniformly over time, ensuring valid inference at data-dependent stopping times. Under suitable regularity conditions, the interval width contracts at the iterated-logarithm rate $\mathcal{O}(\sqrt{\log \log t/t})$. Numerical experiments demonstrate the finite-sample validity and practical effectiveness of our algorithm.