Inference for Many Quantiles under Local Differential Privacy

In this paper, we study the problem of simultaneous inference for multiple quantiles under local differential privacy (LDP). We develop a general online framework that reduces the multiple-quantile problem to a categorical one-hot estimation problem. A projected Robbins-Monro recursion guarantees noncrossing at every iteration, while Polyak-Ruppert averaging together with multivariate self-normalization yields a joint asymptotically pivotal ellipsoidal confidence region without requiring estimation of nuisance densities or asymptotic covariance matrices. We also present several important examples in detail, including direct $k+1$-ary randomized response (kRR), optimized unary encoding (OUE), optimized local hashing (OLH), and Hadamard response (HR). We evaluate our methodology through numerical experiments based on these examples, and the results provide positive support for the proposed framework.