Before revision:  We report (i) evidence of strong upward biases on the estimation of n when using power law memory kernels in the presence of a few outliers, (ii) strong effects on nresulting from the form of the regularization part of the power law kernel, (iii) strong edge effects on the estimated n when using power law kernels, and (iv) the need for an exhaustive search of the absolute maximum of the log-likelihood function due to its complicated shape. Moreover, we demonstrate that the calibration of the Hawkes process on mixtures of pure Poisson process with changes of regime leads to completely spurious apparent critical values for the branching ratio (n = 1) while the true value is actually n=0.
After revision:  We report (i) evidence of strong upward biases on the estimation of n when using power law memory kernels in the presence of a few outliers, (ii) strong effects on nresulting from the form of the regularization part of the power law kernel, (iii) strong edge effects on the estimated n when using power law memory kernels. We demonstrate that the calibration of the Hawkes process on mixtures of pure Poisson process with changes of regime leads to completely spurious apparent critical values for the branching ratio (n = 1) while the true value is actually n=0.