# Estimating Tail Probabilities of Random Sums of Phase-Type Scale Mixture Random Variables +
## 6 Discussion and Outlook
Blue dots: ConAK2+IS+CV (conditioning and IS on Erlang using an exponential PDF). Cyan diamonds: ConAK2+IS+CV (conditioning and IS on Erlang using an Erlang PDF). (**a**) Estimates of \(\ell(u)\) as a function of \(u\) on a logarithmic scale. (**b**) Empirical logarithmic rates as a function of \(u\). (**c**) Empirical relative errors as a function of \(u\). 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 \(\widehat{\ell}_{\text{ConAK2+IS+CV}}\) with importance sampling density being Erlang. Despite positive indications from the empirical logarithmic efficiency estimates, the empirical relative errors for the estimator \(\widehat{\ell}_{\text{ConAK2+IS+CV}}\) (importance sampling using an exponential PDF) had an increasing trend and were fluctuating. Among the other two better-performing estimators, the estimator \(\widehat{\ell}_{\text{ConAK1+IS+CV}}\) had the lowest empirical relative errors, and empirical logarithmic rates which tended to 1 faster. In Example 6, where the scaling random variable was log-normal-distributed, we observed similar trends as in Example 5. In this example, we observed that the estimators \(\widehat{\ell}_{\text{ConAK1+IS+CV}}\), \(\widehat{\ell}_{\text{ConAK2+CV}}\), and \(\widehat{\ell}_{\text{ConAK2+IS+CV}}\) (importance sampling using an exponential PDF) gave sharp estimates. Empirical logarithmic rates of all the estimators appeared to tend to 1 as \(u\) became large. The estimators \(\widehat{\ell}_{\text{ConAK1+IS+CV}}\), \(\widehat{\ell}_{\text{ConAK2+CV}}\), and \(\widehat{\ell}_{\text{ConAK2+IS+CV}}\) (importance sampling using an exponential PDF) had lower empirical relative errors than estimator \(\widehat{\ell}_{\text{ConAK2+IS+CV}}\) with importance sampling density being Erlang. The estimators \(\widehat{\ell}_{\text{ConAK1+IS+CV}}\) and \(\widehat{\ell}_{\text{ConAK2+CV}}\) performed similarly, and showed better performance with empirical relative errors closer to 0, and higher empirical logarithmic rates in this example. In all the above examples, the estimator \(\widehat{\ell}_{\text{ConAK2+IS+CV}}\) (importance sampling using an exponential PDF) appeared to perform better than the estimator \(\widehat{\ell}_{\text{ConAK2+IS+CV}}\) (importance sampling using an Erlang PDF), with sharp estimates, empirical logarithmic rates closer to 1, and lower empirical relative errors. Finally, the examples given suggest that the estimator \(\widehat{\ell}_{\text{ConAK2+CV}}\) performs well when the scaling random variable has a heavier tail. The efficiency study suggests that \(\widehat{\ell}_{\text{ConAK1}}\) and \(\widehat{\ell}_{\text{ConAK2}}\) will attain the same efficiency as the Asmussen-Kroese estimator for the classes of distributions in Proposition 1. However, when conditioning on the scaling random variable, we do not always obtain reliable estimates due to the numerical limitations. In this case, the numerical experiments suggest that further exploiting importance sampling on the scaling random variable of the last term of the summands will provide more reliable estimates. For the log-normal case which does not fall in the cases in Proposition 1, we do not apply Proposition 1 to conclude theoretical efficiency for the scaling random variable being log-normal-distributed. However, the numerical experiment suggests that the estimator \(\widehat{\ell}_{\text{ConAK1+IS+CV}}\) performs the best. In conclusion, numerical studies suggest that, in general, the estimator \(\widehat{\ell}_{\text{ConAK1+IS+CV}}\) will provide more reliable estimates when estimating tail probabilities of random sums of PH scale mixture random variables. When the scaling random variables have much heavier tail (e.g., log-normal case), the estimator \(\widehat{\ell}_{\text{ConAK2+CV}}\) performs better than in other cases.