Locally Private Online Quantile Regression: Estimation and Inference

• URL: http://arxiv.org/abs/2607.05312v1
• Date: Mon, 06 Jul 2026 16:52:54 GMT
• Title: Locally Private Online Quantile Regression: Estimation and Inference
• Abstract summary: We study estimation and inference for online quantile regression under a one-report user-level $eps$-locally differentially private protocol.<n>For fixed finite channel designs, we establish local privacy, decoder unbiasedness, consistency, inference normality, and Hessian-free self-normalized scalars.
• Score: 6.828516229529806
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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