Optimal statistical analyses of Bell experiments

We show how both smaller and more reliable p-values can be computed in Bell-type experiments by using statistical deviations from no-signalling equalities to reduce statistical noise in the estimation of Bell's S or Eberhard's J. Further improvement is obtained by using Wilks' likelihood ratio test based on the four tetranomially distributed vectors of counts of the four different outcome combinations, one 4-vector for each of the four setting combinations. The methodology is illustrated by application to the loophole-free Bell experiments of 2015 and 2016 performed in Delft and Munich, at NIST, and in Vienna respectively; and also to the earlier Innsbruck experiment of Weihs et al. (1998) and the recent Munich experiment of Zhang et al. (2022), which investigates use of a loophole-free Bell experiment as part of a protocol for Device Independent Quantum Key Distribution, DIQKD.