# Estimating Tail Probabilities of Random Sums of Phase-Type Scale Mixture Random Variables +
## 5 Numerical Experiments
: _Let the scaling random variable \(W_{i}\) be Pareto-distributed, with PDF_  
\[h_{\theta}(w)=\mu_{\theta}\operatorname{e}^{-\mu_{\theta}w},\ \ w\geq 0\,,\]4  
_and consider \(X_{i}\stackrel{{ i.i.d}}{{\sim}}Erlang(k,\beta)\), where \(i=1,\ldots,N\). Note that Pareto distributions \(\in\mathcal{D}\cap\mathcal{L}\); see [5]._  
1. _When conditioning on the scaling random variable, the resulting hazard-rate twisted PDF of the scaling random variable is_ \[h_{\theta}(w)=\mu_{\theta}\operatorname{e}^{-\mu_{\theta}w},\ \ w\geq 0\,,\]5 _Hence, the likelihood ratio is given by_ \[h_{\theta}(w)=\mu_{\theta}\operatorname{e}^{-\mu_{\theta}w},\ \ w\geq 0\,,\]6 _Thus, conditioning on_ \(N\)_, we set_ \(\theta(N)=1-1/\Lambda((Z_{N-1}^{\star}\vee(u-S_{N-1}))/\mathbb{E}[X_{N}])\)_. Since_ \(\Lambda(W_{N})=\delta\ln(W_{N})\) _in this case, the likelihood ratio of single element for (_7_), conditioning on_ \(N\)_, is_ \[h_{\theta}(w)=\mu_{\theta}\operatorname{e}^{-\mu_{\theta}w},\ \ w\geq 0\,,\]7
2. _When conditioning on the PH random variable, the resulting PDF of the Erlang random variable is Erlang_\((k,\beta_{\theta})\)_; then, the likelihood ratio is given by_ \[h_{\theta}(w)=\mu_{\theta}\operatorname{e}^{-\mu_{\theta}w},\ \ w\geq 0\,,\]8 _We solve_ \(\zeta^{\prime}(\theta(N))=\frac{Z_{N-1}^{\star}\vee(u-S_{N-1})}{\mathbb{E}[W_ {N}]}\) _using_ \(\zeta(\theta(N))=k\ln\frac{\beta}{\beta-\theta(N)}\)_, conditioning on_ \(N\)_, resulting in_ \(\theta(N)=\beta-\frac{\mathbb{E}[W_{N}]}{Z_{N-1}^{\star}\vee(u-S_{N-1})}\)_, and_ \(\beta_{\theta}=\frac{k\mathbb{E}[W_{N}]}{Z_{N-1}^{\star}\vee(u-S_{N-1})}\)_. The likelihood ratio for (_10_), conditioning on_ \(N\)_, is given by_ \[h_{\theta}(w)=\mu_{\theta}\operatorname{e}^{-\mu_{\theta}w},\ \ w\geq 0\,,\]9  
Figure 1:0 The number of summands \(N=30\), \(W_{i}\stackrel{{\mathrm{i.i.d}}}{{\sim}}\mathrm{Weibull}(0.35,2)\), \(X_{i}\stackrel{{\mathrm{i.i.d}}}{{\sim}}\mathrm{Erlang}(2,3)\), where \(i=1,\ldots,N\). Black squares: ConAK1+IS+CV (conditioning and IS on Weibull). Red stars: ConAK2+CV (conditioning on Erlang). 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 \(u\). An alternative option is to perform importance sampling using the exponential PDF. Then, the likelihood ratio for is given by_  
Figure 1:1  
_We solve \(\zeta^{\prime}(\theta(N))=\frac{Z_{N-1}^{*}\vee(u-S_{N-1})}{\mathbb{E}[W_{N}]}\) using \(\zeta(\theta(N))=\ln\frac{\beta}{\beta-\theta(N)}\), conditioning on \(N\), resulting in \(\theta(N)=\beta-\frac{\mathbb{E}[W_{N}]}{Z_{N-1}^{*}\vee(u-S_{N-1})}\), and \(\beta_{\theta}=\frac{\mathbb{E}[W_{N}]}{Z_{N-1}^{*}\vee(u-S_{N-1})}\). The likelihood ratio for (10), conditioning on \(N\), is given by_  
Figure 1:2  
_We take a sample size of \(10^{5}\), \(W_{i}\overset{i.i.d}{\sim}\text{Pareto}(\delta)\), \(X_{i}\overset{i.i.d}{\sim}\text{Erlang}(k,\beta)\), where \(i=1,\ldots,N\), and \(N\sim\text{Geo}(p)\) with \(\delta=8\), \(k=2\), \(\beta=3\), and \(p=0.2\). The results are shown in Figure 1:3 Proposition 1 suggests that the estimator \(\widehat{\ell}_{\text{ConAK2+CV}}\) will have bounded relative error, since \(\text{Pareto}\in\mathcal{D}\cap\mathcal{L}\). From numerical example results, we can observe that the estimator \(\widehat{\ell}_{\text{ConAK1+IS+CV}}\) performs well. The estimator \(\widehat{\ell}_{\text{ConAK2+CV}}\) has a few larger empirical relative error estimates as \(u\) is varied. Estimators \(\widehat{\ell}_{\text{ConAK2+IS+CV}}\) (importance sampling density using an Erlang PDF and using an exponential PDF) appear to fluctuate more when \(u\) becomes larger, and they have increasing empirical relative errors._  
**Example 6** (Heavy-tailed Scaling: Log-normal).