Tail Approximations for Sums of Dependent Regularly Varying Random Variables Under Archimedean Copula Models

Abstract

In this paper, we compare two numerical methods for approximating the probability that the sum of dependent regularly varying random variables exceeds a high threshold under Archimedean copula models. The first method is based on conditional Monte Carlo. We present four estimators and show that most of them have bounded relative errors. The second method is based on analytical expressions of the multivariate survival or cumulative distribution functions of the regularly varying random variables and provides sharp and deterministic bounds of the probability of exceedance. We discuss implementation issues and illustrate the accuracy of both procedures through numerical studies.

Appendices

Appendix A: Proofs for the Conditional Distributions of Propositions 3 and 15

1.1 A.1 Proof of Proposition 3

From the conditional cumulative distribution function $F_{U_{j}|Z,U_{j-1}, {\ldots } ,U_{1}}(u_{j}|z,u_{j-1},{\ldots } ,u_{1})$ with j = 1, we derive the conditional cumulative distribution function $F_{U_{1}|Z}$

$$F_{U_{1}|Z}(u_{1}|z)=\left( 1-\frac{{\Phi}^{\leftarrow} (u_{1})}{{\Phi}^{\leftarrow} (z)}\right)^{n-1}, $$

for z < u₁ < 1. By using the expression of the probability density function of Z in Proposition 2, and because the marginal probability density function of U₁ is 1 on (0,1), the conditional probability density function of Z|U₁ is given by

$$f_{Z|U_{1}}(z|u_{1})=\frac{({\Phi}^{\leftarrow} )^{(1)}(u_{1})}{(n-2)!}\left( {\Phi}^{\leftarrow} (u_{1})-{\Phi}^{\leftarrow} (z)\right)^{n-2}({\Phi}^{\leftarrow} )^{(1)}(z){\Phi}^{(n)}\left( {\Phi}^{\leftarrow} (z)\right) , $$

for 0 < z < u₁. The conditional cumulative distribution function of Z is then obtained as follows:

$$\begin{array}{@{}rcl@{}} F_{Z|U_{1}}(z|u_{1}) &=&\left( {\Phi}^{\leftarrow} \right)^{(1)}(u_{1})\frac{ (-1)^{n-2}}{(n-2)!}{{\int}_{0}^{z}}\left( {\Phi}^{\leftarrow} (v)-{\Phi}^{\leftarrow} (u_{1})\right)^{n-2}\left( {\Phi}^{\leftarrow} \right)^{(1)}(v){\Phi}^{(n)}\\ &&\times \left( {\Phi}^{\leftarrow} (v)\right) \mathrm{d}v \\ &=&-\left( {\Phi}^{\leftarrow} \right)^{(1)}(u_{1})\frac{(-1)^{n-2}}{(n-2)!} {\int}_{{\Phi}^{\leftarrow} (z)}^{\infty} \left( v-{\Phi}^{\leftarrow} (u_{1})\right)^{n-2}{\Phi}^{(n)}(v)\mathrm{d}v \\ &=&-\left( {\Phi}^{\leftarrow} \right)^{(1)}(u_{1})\frac{(-1)^{n-2}}{(n-2)!} {\int}_{{\Phi}^{\leftarrow} (z)}^{\infty} \left( v-{\Phi}^{\leftarrow} (u_{1})\right)^{n-2}\mathrm{d}\left( {\Phi}^{(n-1)}(v)\right) \\ &=&-\left( {\Phi}^{\leftarrow} \right)^{(1)}(u_{1})\frac{(-1)^{n-2}}{(n-2)!} \left( v-{\Phi}^{\leftarrow} (u_{1})\right)^{n-2}{\Phi}^{(n-1)}(v)|_{{\Phi}^{\leftarrow} (z)}^{\infty} \\ &&+\left( {\Phi}^{\leftarrow} \right)^{(1)}(u_{1})\frac{(-1)^{n-2}}{(n-2)!} {\int}_{{\Phi}^{\leftarrow} (z)}^{\infty} (n-2)\left( v-{\Phi}^{-1}(u_{1})\right)^{n-3}{\Phi}^{(n-1)}(v)\mathrm{d}v. \end{array} $$

Note that $\lim \limits _{v\rightarrow \infty } \left (v-{\Phi }^{\leftarrow } (u_{1})\right )^{j}{\Phi }^{(j)}(v)= 0$ for all j = 1,…, n − 2. The conditional cumulative distribution function of Z given U₁ is derived iteratively:

$$\begin{array}{@{}rcl@{}} F_{Z|U_{1}}(z|u_{1}) &=&\left( {\Phi}^{\leftarrow} \right)^{(1)}(u_{1})\big[ \frac{(-1)^{n-2}}{(n-2)!}\left( {\Phi}^{\leftarrow} (z)-{\Phi}^{\leftarrow} (u_{1})\right)^{n-2}{\Phi}^{(n-1)}({\Phi}^{\leftarrow} (z)) \\ &&-\frac{(-1)^{n-3}}{(n-3)!}{\int}_{{\Phi}^{\leftarrow} (z)}^{\infty} \left( v-{\Phi}^{\leftarrow} (u_{1})\right)^{n-3}{\Phi}^{(n-1)}(v)\mathrm{d}v\big] \\ &=&\left( {\Phi}^{\leftarrow} \right)^{(1)}(u_{1})\sum\limits_{j = 0}^{n-2} \frac{(-1)^{j}}{j!}\left( {\Phi}^{\leftarrow} (z)-{\Phi}^{\leftarrow} (u_{1})\right)^{j}{\Phi}^{(j + 1)}({\Phi}^{\leftarrow} (z)) \end{array} $$

for z ∈ (0, u₁).

1.2 A.2 Proof of Proposition 15

The multivariate survival distribution function of Y = (Y₁,..., Y_n) is given by

$$\Pr (Y_{1}>y_{1},...,Y_{n}>Y_{n})=C\left( \overline{F}_{1}(y_{1}),..., \overline{F}_{n}(y_{n})\right) , $$

and therefore we deduce that

$$F_{Y}\left( y_{1},...,y_{n}\right) =\sum\limits_{1\leq i_{1},\ldots ,i_{j}\leq n}(-1)^{j}\ C(\overline{F}_{i_{1}}(y_{i_{1}}),...,\overline{F} _{i_{j}}(y_{i_{j}})). $$

We now calculate the derivative of F_Y(y₁,…, y_n) with respect to each and every component of y_−i. Among the 2ⁿ elements of the previous sum, there will only be two elements that will be different from 0 after taking the derivatives (n − 1) times. Hence we get

$$\begin{array}{@{}rcl@{}} \frac{\partial^{(n-1)}F_{Y}\left( y_{1},...,y_{n}\right)} {\partial y_{1}...\partial y_{i-1}\partial y_{i + 1}...\partial y_{n}} &=&{\Phi}^{(n-1)}\left( \sum\limits_{j\neq i}{\Phi}^{\leftarrow} (\overline{F} _{j}(y_{j}))\right) \prod\limits_{j\neq i}\left[ \left( {\Phi}^{\leftarrow} \right)^{(1)}(\overline{F}_{j}(y_{j}))f_{j}(y_{j})\right] \\ &&-{\Phi}^{(n-1)}\left( \sum\limits_{j = 1}^{n}{\Phi}^{\leftarrow} (\overline{F} _{j}(y_{j}))\right) \prod\limits_{j\neq i}\left[ \left( {\Phi}^{\leftarrow} \right)^{(1)}(\overline{F}_{j}(y_{j}))f_{j}(y_{j})\right] . \end{array} $$

Let us now note that the joint probability density function of Y_−i is given by

$$f(\mathbf{y}_{-i})={\Phi}^{(n-1)}\left( \sum\limits_{j = 1,j\neq i}^{n}{\Phi}^{\leftarrow} (\overline{F}_{j}(y_{j}))\right) \prod\limits_{j = 1,j\neq i}^{n} \left[ \left( {\Phi}^{\leftarrow} \right)^{(1)}(\overline{F} _{j}(y_{j}))f_{j}(y_{j})\right] . $$

We derive that the conditional cumulative distribution function of $Y_{i}^{\ast } =\left (Y_{i}|\mathbf {Y}_{-i}=\mathbf {y}_{-i}\right )$ is finally characterized by

$$\begin{array}{@{}rcl@{}} \Pr (Y_{i}\leq y_{i}|\mathbf{Y}_{-i}=\mathbf{y}_{-i}) &=&1-\frac{ (-1)^{n-1}{\Phi}^{(n-1)}(\sum\limits_{j = 1}^{n}{\Phi}^{\leftarrow} (\overline{F} _{j}(y_{j})))\prod\limits_{j\neq i}\left[ \left( {\Phi}^{\leftarrow} \right)^{(1)}(\overline{F}_{j}(y_{j}))f_{j}(y_{j})\right]} {(-1)^{n-1}{\Phi}^{(n-1)}(\sum\limits_{j\neq i}{\Phi}^{\leftarrow} (\overline{F} _{j}(y_{j})))\prod\limits_{j\neq i}\left[ \left( {\Phi}^{\leftarrow} \right)^{(1)}(\overline{F}_{j}(y_{j}))f_{j}(y_{j})\right]} \\ &=&1-\frac{{\Phi}^{(n-1)}\left( \sum\limits_{j = 1}^{n}{\Phi}^{\leftarrow} (\overline{F}_{j}(y_{j}))\right)} {{\Phi}^{(n-1)}\left( \sum\limits_{j\neq i}{\Phi}^{\leftarrow} (\overline{F}_{j}(y_{j}))\right)} . \end{array} $$

Appendix B: Proofs on the Asymptotic Properties of the Estimators

1.1 B.1 Proof of Proposition 11

Let us first recall that the relative error of an unbiased estimator Z(s) is defined by $e(Z(s))=\sqrt {\mathbb {E[}Z^{2}(s)]}/z(s)$. Therefore $e^{2}(Z(s))= 1+\mathbb {V}ar(Z(s))/z^{2}(s)$. Let us now remark that

$$\begin{array}{@{}rcl@{}} \mathbb{V}ar(Z_{NR1}^{X}(s)) &=&\mathbb{V}ar\left( \sum\limits_{i = 1}^{n}\left( \overline{F}_{i}(s/n)-\overline{F}_{i}(s)\right) I_{\{S_{n}^{X^{i}}>s,{X_{i}^{i}}=M_{n}^{X^{i}}\}}\right) \\ &=&\sum\limits_{i = 1}^{n}\left( \overline{F}_{i}(s/n)-\overline{F}_{i}(s)\right)^{2} \mathbb{V}ar\left( I_{\{S_{n}^{X^{i}}>s,{X_{i}^{i}}=M_{n}^{X^{i}}\}}\right) . \end{array} $$

It follows that

$$\begin{array}{@{}rcl@{}} \mathbb{V}ar(Z_{NR1}^{X}(s)) &\leq &\sum\limits_{i = 1}^{n}\left( \overline{F} _{i}(s/n)\right)^{2} \\ &\sim &\sum\limits_{i = 1}^{n}n^{2\alpha_{i}}\left( \overline{F}_{i}(s)\right)^{2} \\ &\leq &\left( \sum\limits_{i = 1}^{n}n^{2\alpha_{i}}\right) \left( z(s)\right)^{2}, \end{array} $$

which is enough to conclude that the relative error is bounded as s tends to infinity. The variance of $Z_{NR1}^{Y}(s)$ can be bounded similarly and the same conclusion holds.

1.2 B.2 Proof of Proposition 14

Because Φ^(n− 2) is differentiable, the survival distribution function of the radius R is

$$\overline{F}_{R}(x)=\sum\limits_{j = 0}^{n-1}(-1)^{j}\frac{x^{j}}{j!}{\Phi}^{(j)}(x). $$

Using the the fact that the function Φ(x) = x^−βl_Φ(x) is a regularly varying function at infinity, we have, for j = 1,…, (n − 1),

$$\lim\limits_{x\rightarrow \infty} \frac{(-1)^{j}\ x^{j}\ {\Phi}^{(j)}(x)}{ {\Phi} (x)}=\beta (\beta + 1){\ldots} (\beta +j-1), $$

and we can deduce that

$$\lim\limits_{x\rightarrow \infty} \frac{\overline{F}_{R}(x)}{\Phi (x)} =\lim\limits_{x\rightarrow \infty} \frac{\sum\limits_{j = 1}^{n-1}(-1)^{j}\ \frac{x^{j}}{j!}\ {\Phi}^{(j)}(x)}{\Phi (x)}=\sum\limits_{j = 1}^{n-1}\frac{ \beta (\beta + 1){\ldots} (\beta +j-1)}{j!}. $$

We now define $g(r)=\sum \limits _{i = 1}^{n}\overline {F}_{i}^{\leftharpoonup } ({\Phi } (r))$ and ${L_{0}^{Y}}(s)=\inf \{r\in \mathcal {R}^{+}:g(r)\geq s\}$. Because $\overline {F}^{\leftarrow }$ and Φ are both non-increasing functions, we have

$$g(r)=\sum\limits_{i = 1}^{n}\overline{F}_{i}^{\leftarrow} ({\Phi} (r\times 1))\geq \sum\limits_{i = 1}^{n}\overline{F}_{i}^{\leftarrow} ({\Phi} (rW_{i})) $$

for all $\mathbf {W}\in \mathfrak {s}_{n}$, and then L0Y(s) ≤ L^Y(W, s) for all $\mathbf {W}\in \mathfrak {s}_{n}$. Moreover, from the definiton of ${L_{0}^{Y}}(s)$, we have

$$\max\limits_{i = 1,2,{\ldots} ,n}\overline{F}_{i}^{\leftarrow} ({\Phi} ({L_{0}^{Y}}(s)))\geq s/n, $$

and

$${\Phi} ({L_{0}^{Y}}(s))\leq \max\limits_{i = 1,2,{\ldots} ,n}\overline{F} _{i}(s/n)\leq n^{\alpha_{n}}\max\limits_{i = 1,2,{\ldots} ,n}\overline{F} _{i}(s)\leq n^{\alpha_{n}}z(s). $$

Thus, with $\lim \limits _{s\rightarrow \infty } {L_{0}^{Y}}(s)=\infty$, the second moment of $Z_{NR2}^{Y}(s)$ is bounded in the following way

$$\begin{array}{@{}rcl@{}} \mathbb{E}\left[ \left( Z_{NR2}^{Y}(s)\right)^{2}\right] &\leq &2\left( \left( \Pr ({M_{n}^{Y}}>s)\right)^{2}+E\left[ \left( \overline{F}_{R}(L^{Y}(\mathbf{W},s))^{2}\right) \right] \right) \\ &\leq &2\left( \left( \Pr ({M_{n}^{Y}}>s)\right)^{2}+\left( \overline{F} _{R}({L_{0}^{Y}}(s))\right)^{2}\right) \\ &&\sim 2\left( \left( \Pr ({M_{n}^{Y}}>s)\right)^{2}+\left( \sum\limits_{j = 1}^{n-1}\frac{\beta (\beta + 1){\ldots} (\beta +j-1)}{j!}\right)^{2}\left( {\Phi} ({L_{0}^{Y}}(s))\right)^{2}\right) \\ &\leq &2\left( \left( z(s)\right)^{2}+\left[ \sum\limits_{j = 1}^{n-1}\frac{ \beta (\beta + 1){\ldots} (\beta +j-1)}{j!}\right]^{2}\times n^{2\alpha_{n}}\left( z(s)\right)^{2}\right) \\ &\leq &2\left( 1+n^{2\alpha_{n}}\left( \sum\limits_{j = 1}^{n-1}\frac{\beta (\beta + 1){\ldots} (\beta +j-1)}{j!}\right)^{2}\right) \left( z(s)\right)^{2} \end{array} $$

and the result follows.

1.3 B.3 Proof of Proposition 18

We start with an inequality between Φ and F_R. Since Φ is an n-monotone function, (n − 1)-times differentiable and since the random variable R has the cumulative distribution function given in Theorem 7, we have

$$\frac{\Phi (ax)}{(1-a)^{(n-1)}}\geq \bar{F}_{R}(x),\ \forall x\in \mathbb{R} ^{+} \ \text{and} \ a\in (0,1). $$

Indeed, because Φ is non-increasing function then there exists μ ∈ (ax, x) such that

$${\Phi} (ax)=\sum\limits_{k = 0}^{n-2}(1-a)^{k}\ \frac{x^{k}}{k!}\ (-1)^{k}{\Phi}^{(k)}(x)+(1-a)^{(n-1)}\ \frac{x^{(n-1)}}{(n-1)!}\ (-1)^{(n-1)}\ {\Phi}^{{ (n-1)}}(\mu ). $$

Since Φ is an n-monotone function, (− 1)^(n− 2)Φ^(n− 2)(x) is a convex function, and (− 1)^(n− 1)Φ^(n− 1)(x) is a non-increasing function. This implies that (− 1)^(n− 1)Φ^(n− 1)(μ) ≥ (− 1)^(n− 1)Φ^(n− 1)(x) because μ ≤ x. Thus we have

$${\Phi} (ax)\geq \sum\limits_{k = 0}^{n-1}(1-a)^{k}(-1)^{k}\ \frac{x^{k}}{k!}\ {\Phi}^{(k)}(x), $$

and

$$ \frac{\Phi (ax)}{(1-a)^{(n-1)}}\geq \sum\limits_{k = 0}^{n-1}(1-a)^{(k-n + 1)}(-1)^{k}\frac{x^{k}}{k!}{\Phi}^{(k)}(x)\geq \bar{F}_{R}(x). $$

(15)

To prove Proposition 18, first note that $Z_{NR3,2}^{Y}(s)$ is bounded by $\bar {F}_{R}\left (L_{\lambda }^{Y}(\mathbf {W},s)\right )$ since

$$Z_{NR3,2}^{Y}(s)=F_{R}\left( U^{Y}(\mathbf{W},s)\right) -F_{R}\left( L_{\lambda}^{Y}(\mathbf{W},s)\right) \leq \bar{F}_{R}\left( L_{\lambda} ^{Y}(\mathbf{W},s)\right) . $$

Moreover, from the definition of $\overline {F}_{R}\left (L_{\lambda }^{Y}(\mathbf {W},s)\right )$ and $M_{2}\{\left ({\Phi }^{\leftarrow } (\bar {F} _{i}(\lambda s))/W_{i}\right )_{i = 1,{\ldots } ,n}\}$, there exists two indexes i₁, i₂ ∈ 1, 2,…, n such that

$$L_{\lambda}^{Y}(\mathbf{W},s)\geq M_{2}\left\{ \left( \frac{{\Phi}^{\leftarrow} (\overline{F}_{i}(\lambda s))}{W_{i}}\right)_{i = 1,\ldots ,n}\right\} =\frac{{\Phi}^{\leftarrow} (\overline{F}_{i_{1}}(\lambda s))}{ W_{i_{1}}}\vee \frac{{\Phi}^{\leftarrow} (\overline{F}_{i_{2}}(\lambda s))}{ W_{i_{2}}}. $$

Therefore,

$$\left\{ \begin{array}{c@{ \geq} c} W_{i_{1}}L_{\lambda}^{Y}(\mathbf{W},s) & {\Phi}^{\leftarrow} (\overline{F} _{i_{1}}(\lambda s)) \\ W_{i_{2}}L_{\lambda}^{Y}(\mathbf{W},s) & {\Phi}^{\leftarrow} (\overline{F} _{i_{2}}(\lambda s)) \end{array} \right. $$

implies

$$\left\{ \begin{array}{c@{ \leq} c} {\Phi} \left( W_{i_{1}}L_{\lambda}^{Y}(\mathbf{W},s)\right) & \overline{F} _{i_{1}}(\lambda s) \\ {\Phi} \left( W_{i_{2}}L_{\lambda}^{Y}(\mathbf{W},s)\right) & \overline{F} _{i_{2}}(\lambda s) \end{array} \right. . $$

Appling (15) with $a=W_{i_{j}}$, j = 1, 2 and $x=L_{\lambda }^{Y}(\mathbf {W},s)$, we have

$$\left\{ \begin{array}{c@{ \leq} c} (1-W_{i_{1}})^{n-1}\bar{F}_{R}\left( L_{\lambda}^{Y}(\mathbf{W},s)\right) &{\Phi} \left( W_{i_{1}}L_{\lambda}^{Y}(\mathbf{W},s)\right) \\ (1-W_{i_{2}})^{n-1}\bar{F}_{R}\left( L_{\lambda}^{Y}(\mathbf{W},s)\right) &{\Phi} \left( W_{i_{2}}L_{\lambda}^{Y}(\mathbf{W},s)\right) \end{array} \right. . $$

For j = 1, 2 we have $\overline {F}_{i_{j}}(\lambda s)\!\sim \! \lambda ^{-\alpha _{i_{j}}}\overline {F}_{i_{j}}(s)$ as s tends to ∞, and $\overline {F} _{i_{j}}(s)\!\leq \! z(s)$, which yields

$$\begin{array}{@{}rcl@{}} \mathbb{E}\left[ \left( Z_{NR3,2}^{Y}(s)\right)^{2}\right] &\leq& \mathbb{E} \left[ \overline{F}_{R}\left( L_{\lambda}^{Y}(\mathbf{W},s)\right)^{2} \right]\\ &\leq& \mathbb{E}\left[ \left( (1-W_{i_{1}})^{-(n-1)}\wedge (1-W_{i_{2}})^{-(n-1)}\right)^{2}\right] \lambda^{-2\alpha_{n}}\left( z(s)\right)^{2} \\ &\leq &2^{2n-2}\lambda^{-2\alpha_{n}}\left( z(s)\right)^{2}. \end{array} $$

1.4 B.4 Proof of Proposition 21

We have

$$\Pr ({S_{n}^{Y}}>s,{M_{n}^{Y}}\leq \kappa s)=\Pr ({S_{n}^{Y}}>s,{M_{n}^{Y}}\leq \kappa s,M_{n-1}^{Y}>\frac{1-\kappa} {n-1}\ s). $$

If we estimate this probability conditionally on $\mathbf {W}\in \mathfrak {s} _{n}$ by the same method of estimating $Z_{NR3,2}^{Y}(s)$, the value of λ in this case is $\frac {1-\kappa } {n-1}\in (0,1/n)$, the second moment of this estimator is upper bounded by $2^{2n-2}\left (\frac {1-\kappa } {n-1}\right )^{-2\alpha _{n}}\times \lbrack z^{Y}(s)]^{2}$. Thus, the variance of $Z_{NR3,2}^{Y}(s)$ is bounded by

$$\begin{array}{@{}rcl@{}} \mathbb{V}ar(Z_{NR4}^{Y}(s)) &\leq &2\sum\limits_{i = 1}^{n}[\big(\bar{F} _{i}(\kappa s)-\bar{F}_{i}(s)\big)]^{2}\mathbb{V}ar\left( \mathbb{I} _{\{S_{n}^{Y\kappa i}>s,Y_{i}^{\kappa i}=M_{n}^{Y\kappa i}\}}\right)\\ &&+ 2^{2n-1}\left( \frac{1-\kappa} {n-1}\right)^{-2\alpha_{n}}[z^{Y}(s)]^{2} \\ &\leq &2\sum\limits_{i = 1}^{n}\left( \overline{F}_{i}(\kappa s)\right)^{2}+ 2^{2n-1}\left( \frac{1-\kappa} {n-1}\right)^{-2\alpha_{n}}\left( z^{Y}(s)\right)^{2} \\ &\leq &\left( 2\kappa^{-2\alpha_{n}}+ 2^{2n-1}\left( \frac{1-\kappa} {n-1} \right)^{-2\alpha_{n}}\right) \left( z^{Y}(s)\right)^{2}. \end{array} $$