# Estimating Tail Probabilities of Random Sums of Phase-Type Scale Mixture Random Variables +
## 5 Numerical Experiments  
In this section, we provide a variety of numerical experiments to illustrate the proposed simulation methods. In all illustrative cases, we consider each scaling random variable \(W\) to have support on all of \([0,\infty)\) and each \(X\) to be Erlang-distributed with shape parameter being \(k\) and rate parameter being \(\beta\). Erlang distributions are selected as an example PH distribution because they play an important role in the class of PH distributions ([18]). The class of generalised Erlang distributions (a series of \(k\) exponentials with their own rates \(\beta_{i}\) where \(i\in\{1,2,\ldots,k\}\)) is dense in the set of all probability distributions on the non-negative half-line. We remind the reader that in all cases, the resulting product \(Z=W\cdot X\) is heavy-tailed since the scaling random variable has unbounded support. We explore the estimators \(\widehat{\ell}_{\text{ConAK2+CV}}\) which are the conditional Asmussen-Kroese estimator with control variates for the number of summands, and the estimators \(\widehat{\ell}_{\text{ConAK1+IS+CV}}\) and \(\widehat{\ell}_{\text{ConAK2+IS+CV}}\), which are the conditional Asmussen-Kroese estimator with importance sampling on the last term of the summands and control variate for the number of summands. The figures we include in this paper are the estimates of \(\ell(u)\) for \(u\) in the range \([0,10^{4})\), empirical logarithmic rates obtained by estimating \(\log(\mathbb{V}\text{ar}(\widehat{\ell}(u)))/2\log(\ell(u))\), and empirical relative errors obtained by estimating \(\mathbb{V}\text{ar}(\widehat{\ell}(u))/\ell(u)^{2}\). We explore the estimators combined with control variates for the number of summands to the exclusion of the estimators not combined with control variates in later examples, because combining with control variates will slightly improve the efficiency of the proposed estimators. We provide a preliminary comparison for the estimators \(\widehat{\ell}_{\text{ConAK1}}\), \(\widehat{\ell}_{\text{ConAK1+CV}}\), \(\widehat{\ell}_{\text{ConAK1+IS}}\) and \(\widehat{\ell}_{\text{ConAK1+IS+CV}}\) for the scaling random variable being exponential, and the estimators \(\widehat{\ell}_{\text{ConAK2}}\), \(\widehat{\ell}_{\text{ConAK2+CV}}\), \(\widehat{\ell}_{\text{ConAK2+IS}}\) and \(\widehat{\ell}_{\text{ConAK2+IS+CV}}\) for the scaling random variable being Weibull, aiming to rule out estimators that do not perform well in later examples. The details of density selection regarding importance sampling for the preliminary examples are discussed in Examples 3 and 4, and the results are summarised in Figures 1 and 2, respectively. We observe that in Figure 1, estimators \(\widehat{\ell}_{\text{ConAK1}}\) and \(\widehat{\ell}_{\text{ConAK1+CV}}\) perform poorly due to numerical instability. When \(u\) exceeds around \(2600\), it is beyond the capability of data calculation. Thus, we do not include the estimators \(\widehat{\ell}_{\text{ConAK1}}\) and \(\widehat{\ell}_{\text{ConAK1+CV}}\) in later examples. In Figure 2, all estimators appear to provide sharp estimates. We also observe that the empirical logarithmic rate tends to \(1\) when \(u\) gets large, which suggests that the estimators are logarithmically efficient. Nevertheless, the empirical relative errors for the estimators without importance sampling is increasing and not as stable when \(u\) becomes large. Since we could observe a slight improvement when combining with control variates from both Figures 1 and 2, we include the estimators combined with control variates exclusively. **Example 3** (Light-tailed Scaling: Exponential).