# Scandinavian Actuarial Journal

### Volume 2004, Number 3 / May 2004

On Mixed and Compound Mixed Poisson Distributions
Pages 161-188
Demetrios L. Antzoulakos and Stathis Chadjiconstandtinidis

This paper computes recursive formulae for the tth order cumulative distribution function and tth order tail probability of a compound mixed Poisson distribution assuming that the derivative of the log of the mixing density is a ratio of polynomials. These results extend the famous recursions for aggregate distributions developed by Panjer, Sundt, Willmott and others. The higher order cumulative distribution functions are basically multiple integrals of the density, so the density is the 0th order cumulative; the distribution (integral of density) is the 1st order and so forth. Using integration by parts the 2nd order cumulative is related to a limited expected value (see Lee's PCAS paper on Graphical Excess of Loss Pricing). The tail probability functions are similarly derived from excess reinsurance costs.

The methods in the paper are applied to produce explicit formulae for shifted gamma, scaled beta, generalized Pareto and generalized inverse Gaussian (Sichel's distribution) mixing distributions.

The author's abstract is available on the journal's web site.

Extreme Value Theory and Archimedean Copulas
Pages 211-228
Mario V. Wüthrich

Wüthrich's paper is concerned with the distribution of the maximum of n identically distributed observations in the case where the observations are not independent. The celebrated Fisher-Tippett Extreme Value Theorem that says that if the (suitably normalized) distribution of maximums of independent, identically distributed variables converges to a non-degenerate limit distribution F then F must be one of the three extreme value distributions (EVDs): a Weibull, a Frechet or a Gumbel. The author extends this result to the case where the underlying distributions are continuous and have a dependence relationship specified by an Archimedian copula. His result shows that the normalized maximums of dependent variables must still converge to a Weibull, Frechet or Gumbel distribution.

An Archimedian copula with distribution C is generated by a function z : [0,1] → [0,∞] if C(x1, …, xn) = z-1(z(x1)+…+z(xn)). The case of independence corresponds to z = log. Other Archimedian copulas include the Clayton with z(t)=t-a - 1, the Gumbel with z(t)=(-log(t))a, and the Frank. Archimedian copulas can be extended to an arbitrary number of marginals, which makes them a natural sub-class of all copulas to consider for the extreme value theorem.

After proving the extended extreme value theorem, Wüthrich considers the domains of attraction for each individual extreme value distribution. This involves determining to which EVD the scaled maximums of a given distribution will converge. He also determines the appropriate scaling constants for each of the EVDs.

The surprising result of the paper is that a positive dependence structure leads to a slowdown of the growth of the maximum. Because the dependence structure leads to positive upper tail dependence it is, as the author observes, "harder for the maximum to escape". Once there are some small observations in the sample it is very likely the other values are also small. Thus, if one is only interested in the behavior of the maximum then positive upper tail dependence results in less risk than independence. However, if one is interested in the aggregate distribution then dependence is obviously much more dangerous.

The author's abstract is available on the journal's web site.

Stephen Mildenhall