The slope of the gamma-ray burst variability/peak luminosity correlation

First using a sample of 32 gamma-ray bursts (GRBs) with known redshift and then a sample of 551 BATSE GRBs with derived pseudo-redshift, the time variability/peak luminosity correlation (V versus L), originally found by Reichart et al. using a sample of 18 GRBs, was tested. For both samples the correlation is still found is but less relevant because of a much higher spread of the data. Assuming a straight line in the logL-logV plane (logL = m logV + b), both Guidorzi et al. and Guidorzi found that the line slope for both samples is much lower than that derived by Reichart et al.: m = 1.3+0.8-0.4 and 0.85 +/- 0.02, to be compared with m = 3.3+1.1-0.9 by Reichart et al. Reichart & Nysewander discuss our results and attribute the different slope to the fact that we do not take into account the variance of the sample in the fit (also called the slop), and demonstrate that, using the method presented by Reichart, the expanded data set in the logL-logV plane is still well described by a line with slope m = 3.4+0.9-0.6. Here we compare the results of two methods accounting for the slope of the sample: the method implemented by Reichart and that by D'Agostini. We demonstrate that the first method provides an inconsistent estimate of the parameter when the sample variance is comparable to the interval of values covered by the GRB variability. We also demonstrate that, using the second method, the slope of the logL-logV correlation is still consistent with that derived by us earlier and inconsistent with that derived by Reichart & Nysewander. Finally we discuss the implications on the interpretations proposed for the V-L correlation and show that our results are in agreement with the peak energy/variability correlation found by Lloyd-Ronning & Ramirez-Ruiz and the peak energy/peak luminosity correlation found by Yonetoku et al. and Ghirlanda et al