For reinsurance contract simulated annual losses, an inequality relating their standard deviation and mean is found, σ f ≥ m f ffiffiffiffiffiffiffiffiffiffi ffi μ A C ðÞ μ A ðÞ r , where the coefficient in the inequality is the square root of the ratio of numbers of zero losses years to numbers of non-zero losses years. The largest such coefficient is also proved to be the universal upper bound. As direct application of this inequality, bounds for other risk measures of reinsurance contract, the TVaR (average of the annual losses that are larger than a given loss), the probability of attaching (greater than a given attachment loss), and the probability of exceeding (the annual loss limit) are obtained, which in turn reveal the capability upper limit of the simulation approach.

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Probability distributions with identical shape factor asymptotic limit formulas are defined as asymptotic equivalent distributions. The GB1, GB2, and Generalized Gamma distributions are examples of asymptotic equivalent distributions, which have similar fitting capabilities to data distribution with comparable parameters values. These example families are also asymptotic equivalent to Kumaraswamy, Weibull, Beta, ExpGamma, Normal, and LogNormal distributions at various parameters boundaries. The asymptotic analysis that motivated the asymptotic equivalent distributions definition is further generalized to contour analysis, with contours not necessarily parallel to the axis. Detailed contour analysis is conducted for GB1 and GB2 distributions for various contours of interest. Methods combing induction and symbolic deduction are crafted to resolve the dilemma over conflicting symbolic asymptotic limit results. From contour analysis build on graphical and analytical reasoning, we find that the upper bound of the GB2 distribution family, having the maximum shape factor for given skewness, is attained by the Double Pareto distribution.

Propose use kurtosis divided by skewness squared as shape factor, and use the global or conditional minimum/maximum of this shape factor for selecting and differentiating distribution families. Semi-empirical formulas for that lower/upper bound are calculated for various distribution families, with the aid of Computer Algebra System, for fitting hard to match distributions. Previous studies show high CV distribution is hard to fit and simulate, this study extends that conclusion to cases with low CV but still hard to match EP curves, characterized by having shape factors close to 1. The maximal likelihood approach of distribution fit can tell us which distribution family is better suited for an empirical distribution, but the shape factor range information can tell us why a distribution cannot fit well, or is not suitable at all. So the shape factor, in a sense, determines the EP curve shape.

The twisted Wang transform distribution family, defined as the composition of parameter shifted inverse CDF function with an original CDF function, is found to be most suitable for matching low shape factor distributions, characterizing hard to fit or to simulate reinsurance portfolio losses for some perils from our previous study. Among them, the best form for matching a hard-to-fit empirical loss distribution for a specific peril, is the Exponential Fractional Extra Power 0 Distribution in (0,1) with CDF:.The simplest yet still a good form of this family is the Transformed Hyperbolic Tangent Distribution with CDF:,which has analytical formulas for the moments. The twisted Wang transform distribution family is compared and confirmed to be superior to all other well-known distribution families through extensive numerical optimization practice, distribution forms guesses, and computer-aided exploration experiments.