# Estimating Tail Probabilities of Random Sums of Phase-Type Scale Mixture Random Variables +
## 6 Discussion and Outlook  
In this paper, we proposed straightforward simulation methods for estimating tail probabilities of random sums of scale mixture of PH-distributed summands. We combined Asmussen-Kroese estimation with the conditional Monte Carlo method and further exploited importance sampling applied to the last term of the summands. Since the method can perform poorly when the number of summands is large, we implemented an additional control variate for \(N\) to improve the accuracy of this estimator. When combining Asmussen-Kroese estimation with the conditional Monte Carlo method, we can either condition on the scaling random variable or PH random variable. The estimators are denoted as \(\widehat{\ell}_{\text{ConAK1}}\) and \(\widehat{\ell}_{\text{ConAK2}}\), respectively. In the given examples, we compared the estimators \(\widehat{\ell}_{\text{ConAK1}+\text{IS}+\text{CV}}\), \(\widehat{\ell}_{\text{ConAK2}+\text{CV}}\), \(\widehat{\ell}_{\text{ConAK2}+\text{IS}+\text{CV}}\) (importance sampling using an exponential PDF), and \(\widehat{\ell}_{\text{ConAK2}+\text{IS}+\text{CV}}\) (importance sampling using an Erlang PDF). We took the distribution of \(N\) to be geometrically distributed in most cases, with one extra example for \(N\) as deterministic. We observed that in Example 3, where the scaling random variable was exponentially distributed, the estimators \(\widehat{\ell}_{\text{ConAK1}+\text{IS}+\text{CV}}\) and \(\widehat{\ell}_{\text{ConAK2}+\text{IS}+\text{CV}}\) with importance sampling density being either exponential or Erlang gave sharp estimates and empirical logarithmic rates which tended to 1 as \(u\) became large, and also had lower empirical relative errors than estimator \(\widehat{\ell}_{\text{ConAK2}+\text{CV}}\). The estimator \(\widehat{\ell}_{\text{ConAK2}+\text{CV}}\) performed poorly due to the calculation capacity issue, as we observed that when \(u\) exceeded around 1500, the empirical relative errors were 0 in Figure 3c. The poor performance of estimator \(\widehat{\ell}_{\text{ConAK2}+\text{CV}}\) is not surprising as Proposition 1 does not provide theoretical efficiency guarantees when the scaling random variable has a Weibull tail and the number of summands is random. In Example 4, where the scaling random variable was Weibull-distributed, we took the number of summands \(N\) being geometrically distributed or being deterministic. The results are summarised in Figures 4 and 5. We observed that all the estimators in both examples performed similar. The estimators \(\widehat{\ell}_{\text{ConAK1}+\text{IS}+\text{CV}}\), \(\widehat{\ell}_{\text{ConAK2}+\text{CV}}\), and \(\widehat{\ell}_{\text{ConAK2}+\text{IS}+\text{CV}}\) (importance sampling using an exponential PDF) gave sharp estimates and empirical logarithmic rates which tended to 1 as \(u\) became large, and also had lower empirical relative errors than estimator \(\widehat{\ell}_{\text{ConAK2}+\text{IS}+\text{CV}}\) with importance sampling density being Erlang. Despite positive indications from the empirical logarithmic efficiency estimates, the empirical relative errors for the estimator \(\widehat{\ell}_{\text{ConAK2}+\text{CV}}\) had an increasing trend and were fluctuating. Among the other two better-performing estimators, the estimator \(\widehat{\ell}_{\text{ConAK1}+\text{IS}+\text{CV}}\) had the lowest empirical relative errors, and empirical logarithmic rates which tended to 1 faster. In Example 5, where the scaling random variable was Pareto-distributed, the estimators \(\widehat{\ell}_{\text{ConAK1}+\text{IS}+\text{CV}}\), \(\widehat{\ell}_{\text{ConAK2}+\text{CV}}\), and \(\widehat{\ell}_{\text{ConAK2}+\text{IS}+\text{CV}}\) (importance sampling using an  
Figure 7: The random variable \(N\) is geometrically distributed with success probability \(p=0.2\), \(W_{i}\overset{\text{i.i.d. }}{\sim}\text{LogN}(2,2.25)\), \(X_{i}\overset{\text{i.i.d. }}{\sim}\text{Erlang}(2,3)\), where \(i=1,\ldots,N\). Black squares: ConAK1+IS+CV (conditioning and IS on log-normal). Red stars: ConAK2+CV (conditioning on Erlang).