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Carl Schwarz

I think the major issue for numerical stability is that you want your software to be robust. Most users of software are not statisticians and so don’t know how to evaluate a fit (e.g. non-sensical standard errors, non-convergence) and assume that if the software runs, the output must be correct. So good algorithms are needed to ensure that the software converges even under extreme cases (e.g. sample size of 4!) that we cannot predict in advance. The software should rely on the user checking the reported vcv matrix and or the curvature of the likelihood surface at the final convergence point (e.g. after 100 iterations when it stops due to a very flat likelihood).

David Fox

The reviewer’s statement “the brief criticism given is not helpful” is bewildering. It’s a statement of fact based on observation. The full sentence in our MS reads:

“While offering a high degree of flexibility, experience with these distributions during that time has repeatedly highlighted numerical stability and convergence issues when estimating parameters using maximum likelihood”.

As the person responsible for introducing the Burr family into the SSD lexicon together with colleagues GB and RvD, we have more than 60 years of accumulated experience with this distribution and the Burrlioz software. On this basis, I think we are well-placed to make such commentary (criticisms?)!!

Further, and to the issue at hand: I (we?) agree with the statement “Every distribution used for SSDs has limitations”. That is completely understood. But for me these limitations are of varying complexity and consequence. A disrtibution like the log-logistic is, for the most part, well-behaved. Any limitations with this distribution in the context of SSDs is more to do with the fit provided and not the estimation of parameters. I regard the former as a second-order issue and the latter as a first-order issue. So, failure to converge or multiple optima for me are more serious (in terms of consequences) than a failure to fit.