Professors Willmot and Lin are to be commended for a very clear, elegant and comprehensive review of risk analytic methods with mixed Erlang distributions.
The paper draws from recent work of the authors on risk models with countable mixtures of Erlang distributions, in particular, Willmot and Woo (2007, NAAJ) and Lee and Lin (2009; to appear). It is worth noting that the authors propose the mixed Erlang as a claim severity model, and that a statistical parameter estimation technique is also presented.
Like for mixtures/combinations of exponentials (see Keatinge, 1999 or Dufresne, 2007) and phase-type (PH) distributions (Neuts, 1981;O'Cinneide, 1990;Latouche and Ramaswami, 1999), the mixed Erlang class in (1.2) of the paper is closed under convolutions. This makes it a natural choice for risk models, which are based typically on random sums of independent risks. The paper argues some advantages of mixed Erlangs over combinations of exponentials or PH distributions. We revisit their arguments here in a more detailed and critical analysis:
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The authors show that mixed Erlang distributions form a wider class than previously thought, as it includes generalized Erlang distributions, or distributions that do not admit a representation as a PH distribution or a combination of exponentials. As in the PH case, often these mixed Erlang representations can be far from simple; see the expressions for the resulting mixing discrete probabilities q j in Examples 3 and 4 of the paper. In this respect, Example 5 is an interesting simpler case, as it shows that a logarithmic mixture of Erlangs representation is possible for the effective claim size distribution (Willmot, 1989a), after discounting.
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Section 3 then shows how the mixed Erlang structure is preserved for functionals of interest for the claim severity distribution; moments, the excess-of-loss or the equilibrium distribution, or the resulting mean residual lifetime. Depending on the numerical difficulties in evaluating the series in (3.4)-(3.8), these results can be used for the light tail members of the class, but become less advantageous for heavy-tail claim severities.
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The case of aggregate claims is discussed next. Again the mixed Erlang structure is preserved, this time under convolutions. The paper does not (and cannot) elaborate much on the calculation of the mixing probabilities {c 1 , c 2 , . . .} of the aggregate claims distribution, for general q j mixing probabilities. The reason is that, as for convolutions of general claim severity distributions, the calculations here are tractable only when the Laplace transform inversion is possible or the claim frequency distribution { p 0 , p 1 , . . .} allows for Panjer-type recursions. For this purpose, the mixed Erlang class does not present clear advantages over the PH or other classes closed under convolutions. That said, the application to stop-loss premiums and higher order stop-loss is promising, as seen by the nice formulas in (3.15)-(3.16) that lead to possible numerical evaluations. An interesting possibility for future research would be to use the nice structure for the effective claim size distribution, in Example 5, to study the distribution of the discounted aggregate claims.
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The ruin results in Section 4 are for general Sparre Andersen risk model (claim arrivals follow an ordinary renewal process). It is known that the Laplace transform Ḡ (x) (in ) of the time to ruin T can be written in (4.3), as the tail of a compound geometric distribution, where the geometric parameter is the integrated discounted density h (x) of the surplus immediately prior to ruin, and the secondary distribution is the integrated tail of the ladder height distribution b (x) (Willmot, 2007). The paper shows in (4.6) that here b (x) is also a mixed Erlang. A bit of the wizardry at which the authors are proven masters produces 'explicit', (4.15), and recursive formulas, (4.16), for the alternative form of Ḡ (x) in (4.12). Clearly the coefficients C j ( ) in (4.15) are explicit only to the extent to which in (4.1), the q j ( ) in (4.7) and c n ( ) in (4.11) exist in closed form. The paper does give illustrations in the special cases of Coxian inter-arrival times and of the Poisson classical model. These results are known already, although derived here using similar, yet different techniques due to the mixed Erlang structure.
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Section 5 complements nicely the paper with a proposed iterative estimation scheme for the mixed Erlang parameters {q 1 , q 2 , . . .} and . First, the countable mixture needs to be truncated at J terms, which becomes an additional parameter to be estimated. An EM algorithm is used for that purpose, which differs from the one developed in Asmussen et al. (1996) for PH distributions. The paper proposes a clever adaptation of Tijms' formula to provide good starting values for J and