Shape factor asymptotic analysis II

Probability distribution is the core for stochastic modeling. What type of distribution is more suitable for an application is one of the fundamental questions to ask. Kurtosis, as a high order characteristic of distribution, bring about some confusions: people are indecisive about whether it measures more of the peakedness of the distribution near its center or more of the tail concentration away from distribution center. Some researchers even conclude that the kurtosis does not provide any useful information other than the mean and standard deviation and is practically worthless. However, from the perspective of application, usefulness is as important as or is more important than meaningfulness.

The shape factor, defined as the kurtosis divided by squared skewness, which we proposed and studied in a series of papers, can be think of as a modified or normalized kurtosis. Its range or extreme values are found to be useful for differentiate distribution families, and helpful for discovering relationship among distributions.

As a simple example for the usefulness of the shape factor, the Beta distribution, which is prevalent in reinsurance loss modeling, can be roughly described as having shape factor values in the interval [1, 1.5]. Then any empirical distribution with shape factor outside or near the endpoint of this interval is not appropriate for modeling as Beta distribution.

Our methodology here, the “extrema analysis”-find the range or extremum-can also be used to resolve the controversial about kurtosis. From numerical experiments with distributions having symmetric finite range piecewise linear probability distribution functions (PDF), we know that with central peakedness only, the upper bound of the kurtosis is 2.4. On the other hand, with only tail concentration, the upper bound of kurtosis is 1.8. However, with central peakedness and tail concentration together, the upper bound of kurtosis is infinite. So both of them and their combination contributed to the values of kurtosis, which measures the PDF overall or aggregated curvatures or concavities.

With this background on our shape factor concept, this paper posted on https://mpra.ub.uni-muenchen.de/110827/, specifically contributed the induction and deduction combined technique to overcome symbolic limit errors for the shape factor study of GB1 and GB2. Another concept of our research output need to mention is the asymptotic equivalent distributions, which change the school of thought for distribution fitting from distribution family selection to focus on asymptotic equivalent distribution class, diminishing the importance of individual distribution forms. Its figure 26 and 27 may be of general interest for one who want to find or confirm that whether some distribution families are most suitable for a specific application empirical distribution.