tokens: we also define . We chose three different statistics: (). This statistic picks up the maximum deviation from the continuum spectrum for each simulated PDS. The observed value is then compared with the simulated distribution and the significance is evaluated directly. By construction, it implicitly accounts for the multitrial search performed all over the frequencies. is the Anderson–Darling (AD) statistic ( Anderson Darling 1952 ) obtained for the th set of compared with a distribution. Analogously, KS is the Kolmogorov–Smirnov (KS) statistic obtained for the th set of compared with a distribution. For each of the three statistics, comparing the values obtained from the observed PDS with the corresponding distribution of simulated values immediately yields the significance of possible deviations such as that of a QPO, or the goodness of the fit, as indicated by the AD and KS statistics. As in G12, in addition to the KS, we chose the AD statistic because it is sensitive to a few outliers from the expected distribution. For each GRB the choice between the two competing models was determined by the likelihood ratio test (LRT) in the Bayesian implementation described by V10. As for the aforementioned statistics, from the posterior predictive distribution we sampled the pdf of the statistic defined as where denotes the model obtained with the parameters at the mode of the posterior distribution of a generic model . The statistics in Eq. ( A.9 ) is then sampled using the simulated PDS () and compared with the observed value. For the LRT test, we performed simulations and accepted the bpl model when the probability of chance improvement was lower than 1% (see Sect. 2.3 ). As remarked in Sect. 2.3 , while this type of LRT accounts for the whole parameter space of the simpler model, the same does not hold for the alternative one, as is the case when we use the Bayes factor. Consequently, the rejection of pl does not necessarily imply that bpl is a better option. However, for the limited scope of this investigation, on independent grounds bpl represents a sensible alternative (Sect. A.1.1 ). Finally, after we determined the best–fitting model, we sampled through the MCMC simulation the joint posterior distribution of the model parameters and provided mean and standard deviation for each of them. As an example, Fig. A.1 shows the marginal posterior distributions of all the pairs of parameters of the bpl model for 050713A. Fig. A.1 Example of marginal posterior distributions for the pairs of parameters of the bpl model obtained from simulated posterior simulations for 050713A (only 2000 points are shown for clarity). Solid lines show the contour levels. The goodness of fit is established by the -values associated to the AD and KS statistics. In addition, we also compare the distribution of of the observed PDS against the expected distribution, as shown in Fig. A.2 for 050713A. Fig. A.2 Distribution of for 050713A (Fig. 1 ). The dashed line shows the
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