# Estimating Tail Probabilities of Random Sums of Phase-Type Scale Mixture Random Variables +
## 4 Efficiency of the Modified Amussen-Kroese Estimator  
In this section, we investigate the asymptotic relative error of the estimators \(\widehat{\ell}(u)\). Since \(\ell(u)=\mathbb{E}[\widehat{\ell}(u)]\), the estimator \(\widehat{\ell}(u)\) is consistent and we have the following second-order efficiency measures (see [15]),  
1. The estimator is said to be logarithmically efficient if the following condition on the logarithmic rates holds: \[\liminf_{u\to\infty}\frac{\log(\mathbb{V}\text{ar}(\widehat{\ell}(u)))}{2 \log(\ell(u))}\geq 1.\]
2. The estimator is said to have bounded relative error if \[\limsup_{u\to\infty}\frac{\mathbb{V}\text{ar}(\widehat{\ell}(u))}{\ell(u)^{2} }\leq K<\infty.\] where \(K\) is a constant that does not depend on \(u\). 3. The estimator is said to have (asymptotically) vanishing relative error if \[\limsup_{u\to\infty}\frac{\mathbb{V}\text{ar}(\widehat{\ell}(u))}{\ell(u)^{2 }}=0.\]  
We focus on subexponential distributions and related distributions. The definitions of these relevant distributions are listed as follows. Definition 2: A distribution function \(F\) with support on \((0,\infty)\) is subexponential, if \(\forall\ n\geq 2\), it satisfies  
\[\lim_{x\to\infty}\frac{\overline{F}^{\,n}(x)}{\overline{F}(x)}=n,\]_where \(\overline{F}^{*n}(x)\) denotes the \(n\)-fold convolution of \(\overline{F}\). Denote the class of subexponential distribution functions by \(\mathcal{S}\)._  
**Definition 3**. : _A distribution function \(F\) with support on \((0,\infty)\) is dominatedly varying, if \(\forall\ y\in(0,1)\), it satisfies_  
\[\limsup_{x\to\infty}\frac{\overline{F}(yx)}{\overline{F}(x)}<\infty.\]  
_Denote the class of dominatedly varying distribution functions by \(\mathcal{D}\)._  
**Definition 4**. : _A distribution function \(F\) with support on \((0,\infty)\) is long-tailed, if \(\forall\ y\geq 0\), it satisfies_  
\[\lim_{x\to\infty}\frac{\overline{F}(x-y)}{\overline{F}(x)}=1.\]  
_Denote the class of long-tailed distribution functions by \(\mathcal{L}\)._  
We also exploit the definitions of Weibullian tail from [16]. **Definition 5** ([16]). : _A distribution \(F\) on \([0,\infty)\) is said to have a Weibullian tail_  
\[\overline{F}(z)\sim Cz^{\gamma}exp[-\beta z^{\alpha}],\text{where }\alpha, \beta,C>0,\gamma\in(-\infty,\infty).\]  
We write \(Z\in\mathcal{W}(\alpha,\beta,\gamma,C)\) if \(Z\) satisfies Definition 5. The following examples are two examples of widely used distributions that have Weibullian tails. **Example 1**. : _The CDF of a Weibull distribution with shape parameter \(b\) and scale parameter \(a\) is_  
\[\overline{H}(w)=e^{-(w/a)^{b}},\ x\geq 0,\]  
_denoted as Weibull(\(b\),\(a\)), and \(W\in\mathcal{W}(b,a^{-b},0,1)\)._  
**Example 2**. : _The asymptotic form of the tail of a general PH distribution has the form_  
\[\overline{F}(x)\sim Cx^{k}e^{-\eta x},\]  
_where \(C,\eta>0\) and \(k\in\{0,1,2\dots\}\) (cf. [1]), so that we can write \(X\in\mathcal{W}(1,\eta,k,C)\)._  
Lemma 1 in [16] provides the way to calculate the parameters of product of two random variables having Weibullian tails. **Lemma 1** ([16]). : _Let \(W\in\mathcal{W}(\alpha_{1},\beta_{1},\gamma_{1},C_{1})\), \(X\in\mathcal{W}(\alpha_{2},\beta_{2},\gamma_{2},C_{2})\) be independent non-negative random variables. Then, \(W\cdot X\in\mathcal{W}(\alpha,\beta,\gamma,C)\) with_  
\[\alpha=\frac{\alpha_{1}\alpha_{2}}{\alpha_{1}+\alpha_{2}},\] \[\limsup_{u\to\infty}\frac{\mathbb{V}\text{ar}(\widehat{\ell}(u))}{\ell(u)^{2} }\leq K<\infty.\]0 \[\limsup_{u\to\infty}\frac{\mathbb{V}\text{ar}(\widehat{\ell}(u))}{\ell(u)^{2} }\leq K<\infty.\]1 \[\limsup_{u\to\infty}\frac{\mathbb{V}\text{ar}(\widehat{\ell}(u))}{\ell(u)^{2} }\leq K<\infty.\]2  
Suppose that we have the following conditions for the random variables involved. **Condition 1**. : _Assume that \(N\) is an integer-valued random variable with \(\mathbb{E}[N^{2}]<\infty\) and \(\mathbf{X}=(X_{1},X_{2},X_{3},\dots)\) is a sequence of i.i.d.