  In a local realist world view, physical properties are defined prior to and
independent of measurement, and no physical influence can propagate faster than
the speed of light. Proper experimental violation of a Bell inequality would
show that the world cannot be described within local realism. Such experiments
usually require additional assumptions that make them vulnerable to a number of
"loopholes." A recent experiment [Giustina et al, Nature, 2013] violated a Bell
inequality without being vulnerable to the detection (or fair-sampling)
loophole, therefore not needing the fair-sampling assumption. Here we analyze
the more subtle coincidence-time loophole, and propose and prove the validity
of two different methods of data analysis that avoid it. Both methods are
general and can be used both for pulsed and continuous-wave experiments. We
apply them to demonstrate that the experiment mentioned above violates local
realism without being vulnerable to the coincidence-time loophole, therefore
not needing the corresponding fair-coincidence assumption.
