Authors’ Reply to ‘Letter to the Editor regarding folded models and the paper by Brazauskas and Kleefeld (2011)’

Abstract

We thank Dr. Scollnik for reading our paper carefully and for pointing out an important issue in the numerical example of the paper. Yes, we agree that our comparison of the newly proposed model with its closest competitors was “quick” and a bit unfair to the other distributions. Indeed, using the logtransformed data instead of original data for comparing the fits of various distributions gives a homecourt advantage to the folded-t7 (FT7) model. As one can see from Tables 1 and 2 in Scollnik (2012), the logarithmic transformation changes the values of statistical performance measures for the truncated generalized Pareto distribution (GPD), as it should, and makes the GPD a much more competitive model for the data under consideration. Moreover, we see that the fit of the truncated lognormal model is borderline and that of the truncated composite lognormal-Pareto (LNPa) is excellent. Using the fminsearch function in MATLAB for finding maximum likelihood estimators, we were able to replicate (within a small margin of rounding error) all numbers in Table 2 of the discussion paper. The direct fit of the GPD to the Norwegian fire claims now clearly passes the χ2 test and the values of its (appropriately transformed) negative log-likelihood, NLL, and the Akaike information criterion (AIC) are substantially smaller. However, while the GPD looks more competitive now, it is still uniformly outperformed by the FT7 model, according to the NLL, AIC and the χ 2 criteria. Consequently, since the truncated lognormal model yields inferior fit when compared to that of the GPD, it is also uniformly outperformed by the FT7 model. Further, since the LNPa model has three parameters (all other distributions under consideration have at most two parameters), it was not viewed in our paper as one of the “closest competitors”. Nonetheless, it certainly fits the Norwegian data very well and thus merits further investigation. To this end, we first note that, under reasonable circumstances, one would expect the model with more parameters to fit the given data set better and to have a smaller NLL than a more parsimonious model. Therefore, in such situations information-based decision rules, such as the AIC and Schwarz Bayesian criterion (SBC), come in handy. According to the AIC measure, the penalty to the LNPa model for having an additional parameter is relatively small and thus the LNPa outperforms the FT7 model (AICLNPa = 1688.521 < 1690.834 = AICFT7). On the other hand, according to the SBC measure, the conclusion is opposite: the FT7 outperforms the LNPa model (SBCFT7 = 1700.270 < 1702.674 = SBCLNPa). Note that in all these comparisons the FT7 was treated as a two-parameter model, although its degrees of freedom were fixed (ν = 7). When both parameters are estimated using