  It is well-known that observing nonlocal correlations allows us to draw
conclusions about the quantum systems under consideration. In some cases this
yields a characterisation which is essentially complete, a phenomenon known as
self-testing. Self-testing becomes particularly interesting if we can make the
statement robust, so that it can be applied to a real experimental setup. For
the simplest self-testing scenarios the most robust bounds come from the method
based on operator inequalities. In this work we elaborate on this idea and
apply it to the family of tilted CHSH inequalities. These inequalities are
maximally violated by partially entangled two-qubit states and our goal is to
estimate the quality of the state based only on the observed violation. For
these inequalities we have reached a candidate bound and while we have not been
able to prove it analytically, we have gathered convincing numerical evidence
that it holds. Our final contribution is a proof that in the usual formulation,
the CHSH inequality only becomes a self-test when the violation exceeds a
certain threshold. This shows that self-testing scenarios fall into two
distinct classes depending on whether they exhibit such a threshold or not.
