Locally Private Online Quantile Regression: Estimation and Inference

We study estimation and inference for online quantile regression under a one-report user-level $\eps$-locally differentially private ($\eps$-LDP) protocol.

The main difficulty is that the standard quantile-regression estimating-equation contribution couples covariates with a residual comparison, so a server that receives only privatized reports cannot form the usual online update.

We address this by developing a finite-alphabet channel in which each user computes the contribution locally, applies support-aware stochastic quantization and randomized response to one selected-block category, and sends one report.

A public decoder corrects the randomized-response distortion and reconstructs a server-side estimating-equation input with the correct conditional mean.

These decoded inputs are then used in projected Polyak-Ruppert averaging.

For fixed finite channel designs, we establish local privacy, decoder unbiasedness, consistency, asymptotic normality, and Hessian-free self-normalized inference for prespecified scalar contrasts.

Simulations and a New York City taxi-trip illustration show that the private trajectory approaches the nonprivate online reference as the privacy budget grows and outperforms direct Laplace and face-exponential geometric releases in the reported regimes.