# Estimating Tail Probabilities of Random Sums of Phase-Type Scale Mixture Random Variables +
## 4 Efficiency of the Modified Amussen-Kroese Estimator
random variables with continuous distribution function \(B\in\mathcal{S}\), independent of \(N\)._  
**Condition 2**. : _There exists a function \(a(u)\) that satisfies one of the following:_  
1. \(\lim_{u\to\infty}\frac{\widehat{\ell}(u)}{a(u)}=N\) _almost surely;_
2. \(\lim_{u\to\infty}\frac{\widehat{\ell}(u)}{a(u)}=\mathbb{E}[N]\) _._  
**Condition 3**. : _There exists a random variable \(U\) with \(\mathbb{E}[U^{2}]<\infty\), such that for all \(u>0\),_  
\[\limsup_{u\to\infty}\frac{\mathbb{V}\text{ar}(\widehat{\ell}(u))}{\ell(u)^{2} }\leq K<\infty.\]3  
_almost surely._  
**Lemma 2**. : _If Condition 2 (1) and Condition 3 are fulfilled, we obtain_  
\[\limsup_{u\to\infty}\frac{\mathbb{V}\text{ar}(\widehat{\ell}(u))}{\ell(u)^{2} }\leq K<\infty.\]4  
_If Condition 2 (2) and Condition 3 are fulfilled, we obtain_  
\[\limsup_{u\to\infty}\frac{\mathbb{V}\text{ar}(\widehat{\ell}(u))}{\ell(u)^{2} }\leq K<\infty.\]5  
Proof. : \[\limsup_{u\to\infty}\frac{\mathbb{V}\text{ar}(\widehat{\ell}(u))}{\ell(u)^{2} }\leq K<\infty.\]6 (16)  
If Condition 3 is fulfilled, by dominated convergence, we then interchange limit and expectation in (16) to obtain the result. Note that if Condition 1 is fulfilled, then (14) is bounded. We list the conditions and lemma above which are mirrored from [13] for the reader's convenience. **Proposition 1**. : _The estimators \(\widehat{\ell}_{\text{ConAKI}}(u)\) and \(\widehat{\ell}_{\text{ConAK2}}(u)\)_  
* _Have bounded relative error when_ \(N\) _is light-tailed, and_ \(W\in\mathcal{D}\cap\mathcal{L}\)_;_
* _Are logarithmically efficient when_ \(N\) _is deterministic, and_ \(W\) _has a Weibullian tail with shape parameter_ \(\in(0,\log(3/2)/\log(4/3))\)_._  
Proof. : Based on the law of total variance \(\mathbb{V}\text{ar}(Y)=\mathbb{E}[\mathbb{V}\text{ar}(Y\mid X)]+\mathbb{V} \text{ar}[\mathbb{E}[Y\mid X]]\), we can conclude that the conditional Monte Carlo estimator has lower variance than the estimator \(Y\). Thus, we can obtain \(\mathbb{V}\text{ar}\left[\widehat{\ell}_{\text{ConAKI}}(u)\right]\leq\mathbb{ V}\text{ar}\left[\widehat{\ell}_{\text{AK}}(u)\right]\) and \(\mathbb{V}\text{ar}\left[\widehat{\ell}_{\text{ConAK2}}(u)\right]\leq\mathbb{ V}\text{ar}\left[\widehat{\ell}_{\text{AK}}(u)\right]\) for conditional Asmussen-Kroese estimators \(\widehat{\ell}_{\text{ConAK1}}(u)\) and \(\widehat{\ell}_{\text{ConAK2}}(u)\). The Asmussen-Kroese estimator (2) was proven to be logarithmically efficient when \(Z\) has a Weibullian tail in [7], and to have bounded relative error in [13] for the following three cases:  
* \(N\) is light-tailed, and \(Z\in\mathcal{D}\cap\mathcal{L}\). * \(N\) is light-tailed, and \(Z\sim\log\text{normal}(\mu,\sigma^{2})\). * \(N\) is bounded, and \(Z\) is Weibull-distributed with shape parameter \(\in(0,\log(3/2)/\log(3))\). Here, we extended the result to scaling random variables in similar cases for the estimators \(\widehat{\ell}_{\text{ConAK1}}(u)\) and \(\widehat{\ell}_{\text{ConAK2}}(u)\), which are as follows:  
* \(W\in\mathcal{D}\cap\mathcal{L}\) As a consequence of Corollary 3.3 in [17], if \(F\) is a non-degenerate phase-type distribution, and the scaling distribution \(H\in\mathcal{D}\cap\mathcal{L}\), then the phase-type scale mixture distribution \(B\in\mathcal{D}\cap\mathcal{L}\). Therefore, the estimators \(\widehat{\ell}_{\text{ConAK1}}(u)\) and \(\widehat{\ell}_{\text{ConAK2}}(u)\) attain the same efficiency as the Asmussen-Kroese estimator, as they have lower variance. * \(W\in\mathcal{W}(\alpha,\beta,\gamma,C)\) By Lemma 1, we observe that a phase-type distribution has a Weibullian tail with shape parameter being \(1\), and if the scaling random variable \(W\in\mathcal{W}(\alpha,\beta,\gamma,C)\), then \(Z\in\mathcal{W}(\alpha_{Z},\beta_{Z},\gamma_{Z},C_{Z})\). Furthermore, the shape parameter of \(Z\) is \(\alpha_{Z}=\frac{\alpha}{\alpha+1}\in(0,1)\); thus, \(Z\) is heavy-tailed.