We wish to thank the two discussants for their interesting and thought provoking discussions of the paper. Our responses to their comments are now given.
Both Jose Garrido and David Stanford have discussed the phase-type (PH) class in some detail, and we wish to make some further remarks both about PH distributions and their relationship to the mixed Erlang class. We also wish to thank David Stanford for his additional contributions on PH distributions.
Although the PH class is a proper subclass of the mixed Erlang class and is hence strictly smaller, our view is that this fact is somewhat irrelevant from a modelling standpoint for a few reasons. First, both classes are dense in the class of positive continuous distributions and hence are capable of approximating a wide variety of distributional shapes, etc. As pointed out by Jose Garrido however, both classes are not appropriate for use in modelling situations where the extreme right tail of the distribution is of particular interest and is known to be heavy (i.e. subexponential). Second it is our view that each of the PH and the mixed Erlang class have their own advantages in various situations. The PH class is extremely useful in modelling in connection with quite complex stochastic modelling, allowing for tractable and mathematically elegant conclusions, often under fairly weak assumptions. Their common usage both in queueing and ruin theory is a testament to this fact. As pointed out by David Stanford, they are very convenient for use in conjunction with probabilistic reasoning, and the use of the matrix calculus often allows for concise and mathematically elegant formulae. Drawbacks to their usage are the requirement that the matrix calculus must be learned and the lack of a simple and efficient numerical algorithm when they are fitted to positve data.
Conversely, the mixed Erlang class, while less useful in more complex stochastic modelling, is extremely convenient for use in standard risk and ruin theoretic models by virtue of the fact that the ensuing results (without exception) involve elementary mathematics only. As these results involve infinite series, no root finding is needed as with partial fraction expansions of Laplace transforms as is often the case with the class of combinations of exponentials, or as is needed for calculation of eigenvalues for evaluation of matrix exponentials under the PH methodology. As is hopefully clear from the paper, computationally convenient formulae exist within the standard risk theoretic context for a wide variety of quantities of interest, including (rather remarkably) evaluation of finite time ruin probabilities (or equivalently, transient waiting time probabilities in a queueing context).
Jose Garrido mentioned that the mixing weights for the mixed Erlang class are often complex. We do not disagree with this statement, but wish to give our view that it is normally the case that these mixing weights may be evaluated numerically in a straightforward manner. A good example is given by Example 4 in the paper, where a Panjer-type recursion may be used to calculate the mixing weights recursively. In connection with aggregate claims analysis, we wish to clarify our view that evaluation of the mixing weights has been reduced to evaluation of the discrete compound distribution with probability generating function (3.10) in the paper (as noted by David Stanford). Consequently, this problem has been reduced to a simpler, well-known problem in risk theory. As such, all techniques that are available for the solution of this standard problem are at our disposal in this situation. These include, among others, Panjer-type recursions and the fast Fourier transform. Thus, we disagree that the mixed Erlang class 'does not present clear advantages over the PH or other classes closed under convolutions', because the mixed Erlang class allows for evaluation in the continuous case using the computationally simpler discrete methodology.
Turning now to ruin theory, we completely agree with the observations made by Jose Garrido. We also wish to point out that the mixed Erlang representation (4.22) for the distribution of the deficit where the mixing weights are given by (4.27) was done previously for the classical compound Poisson risk model (Willmot and Lin, 2001), but has been extended here to the more general Sparre Andersen risk model. Furthermore, for explicit numerical evaluation of finite time ruin probabilities