  The device-independent paradigm has had spectacular successes in randomness
generation, key distribution and self-testing, however most of these results
have been obtained under the assumption that parties hold trusted and private
random seeds. In efforts to relax the assumption of measurement independence,
Hardy's non-locality tests have been proposed as ideal candidates. In this
paper, we introduce a family of tilted Hardy paradoxes that allow to self-test
general pure two-qubit entangled states, as well as certify up to $1$ bit of
local randomness. We then use these tilted Hardy tests to obtain an improvement
in the generation rate in the state-of-the-art randomness amplification
protocols for Santha-Vazirani (SV) sources with arbitrarily limited measurement
independence. Our result shows that device-independent randomness amplification
is possible for arbitrarily biased SV sources and from almost separable states.
Finally, we introduce a family of Hardy tests for maximally entangled states of
local dimension $4, 8$ as the potential candidates for DI randomness extraction
to certify up to the maximum possible $2 \log d$ bits of global randomness.
