  Self-testing refers to the phenomenon that certain extremal quantum
correlations (almost) uniquely identify the quantum system under consideration.
For instance observing the maximal violation of the CHSH inequality certifies
that the two parties share a singlet. While self-testing results are known for
several classes of states, in many cases they are only applicable if the
observed statistics are almost perfect, which makes them unsuitable for
practical applications. Practically relevant self-testing bounds are much less
common and moreover they all result from a single numerical method (with one
exception which we discuss in detail). In this work we present a new technique
for proving analytic self-testing bounds of practically relevant robustness. We
obtain improved bounds for the case of self-testing the singlet using the CHSH
inequality (in particular we show that non-trivial fidelity with the singlet
can be achieved as long as the violation exceeds $\beta^{*} = (16 + 14
\sqrt{2})/17 \approx 2.11$). In case of self-testing the tripartite GHZ state
using the Mermin inequality we derive a bound which not only improves on
previously known results but turns out to be tight. We discuss other scenarios
to which our technique can be immediately applied.
