  A stylized feature of high-dimensional data is that many variables have heavy
tails, and robust statistical inference is critical for valid large-scale
statistical inference. Yet, the existing developments such as Winsorization,
Huberization and median of means require the bounded second moments and involve
variable-dependent tuning parameters, which hamper their fidelity in
applications to large-scale problems. To liberate these constraints, this paper
revisits the celebrated Hodges-Lehmann (HL) estimator for estimating location
parameters in both the one- and two-sample problems, from a non-asymptotic
perspective. Our study develops Berry-Esseen inequality and Cram\'{e}r type
moderate deviation for the HL estimator based on newly developed non-asymptotic
Bahadur representation, and builds data-driven confidence intervals via a
weighted bootstrap approach. These results allow us to extend the HL estimator
to large-scale studies and propose \emph{tuning-free} and \emph{moment-free}
high-dimensional inference procedures for testing global null and for
large-scale multiple testing with false discovery proportion control. It is
convincingly shown that the resulting tuning-free and moment-free methods
control false discovery proportion at a prescribed level. The simulation
studies lend further support to our developed theory.
