# Estimating Tail Probabilities of Random Sums of Phase-Type Scale Mixture Random Variables +
## 5 Numerical Experiments
: _Consider now \(X_{i}\overset{i.i.d}{\sim}\text{Erlang}(k,\beta)\), and the scaling random variable \(W_{i}\overset{i.i.d}{\sim}\text{LogN}(\mu,\sigma^{2})\) with PDF_  
Figure 1:4  
_with corresponding CDF_  
Figure 1:5  
_where \(i=1,\ldots,N\), and \(\Phi(\cdot)\) is the CDF of a standard normal random variate._  
1. _When conditioning on the scaling random variable, the resulting hazard-rate twisted PDF of the scaling random variable is_ Figure 1:6  
Figure 1:7 The random variable \(N\) is geometrically distributed with success probability \(p=0.2\), \(W_{i}\overset{i.i.d}{\sim}\text{Pareto}(8)\), \(X_{i}\overset{i.i.d}{\sim}\text{Erlang}(2,3)\), where \(i=1,\ldots,N\). Black squares: ConAK1+IS+CV (conditioning and IS on Pareto). 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 \(u\). _The likelihood ratio is given by_  
Figure 1:8  
_where \(\theta(N)=1-1/\Lambda((Z_{N-1}^{\star}\vee(u-S_{N-1}))/\mathbb{E}[X_{N}])\). The likelihood ratio for (7), conditioning on \(N\), is given by_  
Figure 1:9  
_where \(\theta(N)=1-1/\Lambda((Z_{N-1}^{\star}\vee(u-S_{N-1}))/\mathbb{E}[X_{N}])\). We note that from (11), we have \(\overline{H}_{\theta}(W_{N})=\big{[}\overline{H}(W_{N})\big{]}^{1-\theta}\), and so it is straightforward to generate the twisted scaling random variables by using the inverse transform method._
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_ Figure 2:0 _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_ \(\theta(N)=\beta-\frac{k\mathbb{E}[W_{N}]}{Z_{N-1}^{\star}\vee(u-S_{N-1})}\)_, 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_ Figure 2:1 _An alternative option is to perform importance sampling using the exponential PDF. Then, the likelihood ratio is given by_ Figure 2:2 _We solve_ \(\zeta^{\prime}(\theta(N))=\frac{Z_{N-1}^{\star}\vee(u-S_{N-1})}{\mathbb{E}[W_ {N}]}\) _using_ \(\zeta(\theta(N))=\ln\frac{\beta}{\beta-\theta(N)}\)_, conditioning on_ \(N\)_, resulting in_ \(\theta(N)=\beta-\frac{k\mathbb{E}[W_{N}]}{Z_{N-1}^{\star}\vee(u-S_{N-1})}\)_, and_ \(\beta_{\theta}=\frac{\mathbb{E}[W_{N}]}{Z_{N-1}^{\star}\vee(u-S_{N-1})}\)_. The likelihood ratio for (_10_), conditioning on_ \(N\)_, is given by_ Figure 2:3  
_We take a sample size of_ \(10^{5}\)_,_ \(W_{i}\overset{i.i.d}{\sim}LogN(\mu,\sigma^{2})\)_,_ \(X_{i}\overset{i.i.d}{\sim}Erlang(k,\beta)\)_, where_ \(i=1,\ldots,N\)_, and_ \(N\sim\text{Geo}(p)\) _with_ \(\mu=2\)_,_ \(\sigma=1.5\)_,_ \(k=2\)_,_ \(\beta=3\)_, and_ \(p=0.2\)_. The results are shown in Figure_ 7_. Proposition_ 1 _does not provide a theoretical efficiency guarantee for this example as the log-normal distribution is neither_ \(\mathcal{D}\cap\mathcal{L}\) _nor has a Weibullian tail. From numerical example results, we can observe that estimators_ \(\widehat{\ell}_{\text{ConAK}1+\text{IS}+\text{CV}}\)_,_ \(\widehat{\ell}_{\text{ConAK}2+\text{CV}}\)_, and_ \(\widehat{\ell}_{\text{ConAK}2+\text{IS}+\text{CV}}\) _(importance sampling density using an exponential PDF) perform well._ \(\widehat{\ell}_{\text{ConAK}2+\text{IS}+\text{CV}}\) _(importance sampling density using an exponential PDF) turns out to have slightly lower empirical logarithmic rates and higher empirical relative errors than estimators_ \(\widehat{\ell}_{\text{ConAK}1+\text{IS}+\text{CV}}\) _and_ \(\widehat{\ell}_{\text{ConAK}2+\text{CV}}\)_.