  Given a Bell inequality, if its maximal quantum violation can be achieved
only by a single set of measurements for each party or a single quantum state,
up to local unitaries, one refers to such a phenomenon as self-testing. For
instance, the maximal quantum violation of the Clauser-Horne-Shimony-Holt
inequality certifies that the underlying state contains the two-qubit maximally
entangled state and the measurements of one party contains a pair of
anti-commuting qubit observables. As a consequence, the other party
automatically verifies the set of states remotely steered, namely the
"assemblage", is in the eigenstates of a pair of anti-commuting observables. It
is natural to ask if the quantum violation of the Bell inequality is not
maximally achieved, or if one does not care about self-testing the state or
measurements, are we capable of estimating how close the underlying assemblage
is to the reference one? In this work, we provide a systematic
device-independent estimation by proposing a framework called "robust
self-testing of steerable quantum assemblages". In particular, we consider
assemblages violating several paradigmatic Bell inequalities and obtain the
robust self-testing statement for each scenario. Our result is
device-independent (DI), i.e., no assumption is made on the shared state and
the measurement devices involved. Our work thus not only paves a way for
exploring the connection between the boundary of quantum set of correlations
and steerable assemblages, but also provides a useful tool in the areas of DI
quantum certification. As two explicit applications, we show 1) that it can be
used for an alternative proof of the protocol of DI certification of all
entangled two-qubit states proposed by Bowles et al., and 2) that it can be
used to verify all non-entanglement-breaking qubit channels with fewer
assumptions compared with the work of Rosset et al.
