  This work derives methods for performing nonparametric, nonasymptotic
statistical inference for population means under the constraint of local
differential privacy (LDP). Given bounded observations $(X_1, \dots, X_n)$ with
mean $\mu^\star$ that are privatized into $(Z_1, \dots, Z_n)$, we present
confidence intervals (CI) and time-uniform confidence sequences (CS) for
$\mu^\star$ when only given access to the privatized data. To achieve this, we
study a nonparametric and sequentially interactive generalization of Warner's
famous ``randomized response'' mechanism, satisfying LDP for arbitrary bounded
random variables, and then provide CIs and CSs for their means given access to
the resulting privatized observations. For example, our results yield private
analogues of Hoeffding's inequality in both fixed-time and time-uniform
regimes. We extend these Hoeffding-type CSs to capture time-varying
(non-stationary) means, and conclude by illustrating how these methods can be
used to conduct private online A/B tests.
