Extremal dependence of random scale constructions

Abstract

A bivariate random vector can exhibit either asymptotic independence or dependence between the largest values of its components. When used as a statistical model for risk assessment in fields such as finance, insurance or meteorology, it is crucial to understand which of the two asymptotic regimes occurs. Motivated by their ubiquity and flexibility, we consider the extremal dependence properties of vectors with a random scale construction (X1,X2) = R(W1,W2), with non-degenerate R > 0 independent of (W1,W2). Focusing on the presence and strength of asymptotic tail dependence, as expressed through commonly-used summary parameters, broad factors that affect the results are: the heaviness of the tails of R and (W1,W2), the shape of the support of (W1,W2), and dependence between (W1,W2). When R is distinctly lighter tailed than (W1,W2), the extremal dependence of (X1,X2) is typically the same as that of (W1,W2), whereas similar or heavier tails for R compared to (W1,W2) typically result in increased extremal dependence. Similar tail heavinesses represent the most interesting and technical cases, and we find both asymptotic independence and dependence of (X1,X2) possible in such cases when (W1,W2) exhibit asymptotic independence. The bivariate case often directly extends to higher-dimensional vectors and spatial processes, where the dependence is mainly analyzed in terms of summaries of bivariate sub-vectors. The results unify and extend many existing examples, and we use them to propose new models that encompass both dependence classes.

References

  1. Abdous, B., Fougères, A. -L., Ghoudi, K.: Extreme behaviour for bivariate elliptical distributions. Can J Stat 33(3), 317–334 (2005)

    MathSciNet  Article  Google Scholar 

  2. Asmussen, S., Hashorva, E., Laub, P.J., Taimre, T.: Tail asymptotics of light-tailed Weibull-like sums. Probab. Math. Stat. 37(2), 235–256 (2017). https://doi.org/10.19195/0208-4147.37.2.3

    MathSciNet  MATH  Google Scholar 

  3. Balkema, G., Embrechts, P.: High risk scenarios and extremes: A geometric approach. Zürich Lectures in Advanced Mathematics. European Mathematical Society, Zürich (2007)

  4. Balkema, G., Nolde, N.: Asymptotic independence for unimodal densities. Adv. Appl. Probab. 42(2), 411–432 (2010)

    MathSciNet  Article  Google Scholar 

  5. Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.L: Statistics of extremes. Wiley, New York (2004)

    Google Scholar 

  6. Belzile, L., Nešlehová, J.: Extremal attractors of Liouville copulas. J. Multivar. Anal. 160, 68–92 (2018)

    MathSciNet  Article  Google Scholar 

  7. Breiman, L.: On some limit theorems similar to the arc-sin law. Theory of Probability & Its Applications 10(2), 323–331 (1965)

    MathSciNet  Article  Google Scholar 

  8. Capéraà, P., Fougères, A. -L., Genest, C.: Bivariate distributions with given extreme value attractor. J. Multivar. Anal. 72(1), 30–49 (2000)

    MathSciNet  Article  Google Scholar 

  9. Charpentier, A., Fougères, A. -L., Genest, C., Nešlehová, J.: Multivariate archimax copulas. J. Multivar. Anal. 126, 118–136 (2014)

    MathSciNet  Article  Google Scholar 

  10. Cline, D.B.: Convolution tails, product tails and domains of attraction. Probab. Theory Relat. Fields 72(4), 529–557 (1986)

    MathSciNet  Article  Google Scholar 

  11. Cline, D.B., Samorodnitsky, G.: Subexponentiality of the product of independent random variables. Stoch. Process. Appl. 49(1), 75–98 (1994)

    MathSciNet  Article  Google Scholar 

  12. Coles, S.G., Tawn, J.A.: Modelling extreme multivariate events. Journal of the Royal Statistical Society: Series B (Statistical Methodology), pp. 377–392 (1991)

  13. Davis, R.A., Mikosch, T.: Extreme value theory for space–time processes with heavy-tailed distributions. Stoch. Process. Appl. 118(4), 560–584 (2008)

    MathSciNet  Article  Google Scholar 

  14. Dȩbicki, K., Farkas, J., Hashorva, E.: Extremes of randomly scaled Gumbel risks. J. Math. Anal. Appl. 458(1), 30–42 (2018)

    MathSciNet  Article  Google Scholar 

  15. de Haan, L., Zhou, C.: Extreme residual dependence for random vectors and processes. Adv. Appl. Probab. 43, 217–242 (2011)

    MathSciNet  Article  Google Scholar 

  16. Dombry, C., Ribatet, M.: Functional regular variations, Pareto processes and peaks over threshold. Statistics and its Interface 8, 9–17 (2015)

    MathSciNet  Article  Google Scholar 

  17. Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling extremal events: for insurance and finance, volume 33. Springer Science & Business Media (2013)

  18. Ferreira, A., de Haan, L.: The generalized Pareto process; with a view towards application and simulation. Bernoulli 20(4), 1717–1737 (2014)

    MathSciNet  Article  Google Scholar 

  19. Foss, S., Korshunov, D., Zachary, S.: Convolutions of long-tailed and subexponential distributions. J. Appl. Probab. 46(3), 756–767 (2009)

    MathSciNet  Article  Google Scholar 

  20. Fougères, A. -L., Soulier, P.: Limit conditional distributions for bivariate vectors with polar representation. Stoch Model 26(1), 54–77 (2010)

    MathSciNet  Article  Google Scholar 

  21. Hashorva, E.: On the residual dependence index of elliptical distributions. Statist. Probab. Lett. 80(13), 1070–1078 (2010)

    MathSciNet  Article  Google Scholar 

  22. Hashorva, E.: Exact tail asymptotics in bivariate scale mixture models. Extremes 15(1), 109–128 (2012)

    MathSciNet  Article  Google Scholar 

  23. Hashorva, E., Hüsler, J.: On multivariate Gaussian tails. Ann. Inst. Stat. Math. 55(3), 507–522 (2003)

    MathSciNet  Article  Google Scholar 

  24. Hashorva, E., Pakes, A.G., Tang, Q.: Asymptotics of random contractions. Insurance Math. Econom. 47(3), 405–414 (2010)

    MathSciNet  Article  Google Scholar 

  25. Heffernan, J.E., Resnick, S.I.: Limit laws for random vectors with an extreme component. Ann. Appl. Probab. 17(2), 537–571 (2007)

    MathSciNet  Article  Google Scholar 

  26. Heffernan, J.E., Tawn, J.A.: A conditional approach for multivariate extreme values (with discussion). J. R. Stat. Soc. Ser. B (Stat Methodol.) 66(3), 497–546 (2004)

    MathSciNet  Article  Google Scholar 

  27. Hult, H., Lindskog, F.: Multivariate extremes, aggregation and dependence in elliptical distributions. Adv. Appl. Probab. 34(03), 587–608 (2002)

    MathSciNet  Article  Google Scholar 

  28. Huser, R.G., Opitz, T., Thibaud, E.: Bridging asymptotic independence and dependence in spatial extremes using Gaussian scale mixtures. Spatial Statistics 21(A), 166–186 (2017)

    MathSciNet  Article  Google Scholar 

  29. Huser, R.G., Wadsworth, J.L: Modeling spatial processes with unknown extremal dependence class. J. Am. Statist. Assoc 114(525), 434–444 (2019). https://doi.org/10.1080/01621459.2017.1411813

    MathSciNet  Article  Google Scholar 

  30. Johnson, N.L., Kotz, S., Balakrishnan, N.: Continuous univariate distributions, 2nd edn., vol. 1. Wiley, Chichester (1994)

  31. Johnson, N.L., Kotz, S., Balakrishnan, N.: Continuous univariate distributions, 2nd edn., vol. 2. Wiley, New York (1995)

  32. Krupskii, P., Huser, R., Genton, M.: Factor copula models for replicated spatial data. J. Am. Statist. Assoc. 113(521), 467–479 (2018). https://doi.org/10.1080/01621459.2016.1261712

    MathSciNet  Article  Google Scholar 

  33. Ledford, A.W., Tawn, J.A.: Modelling dependence within joint tail regions. J. R. Stat. Soc. Ser. B (Stat Methodol.) 59(2), 475–499 (1997)

    MathSciNet  Article  Google Scholar 

  34. McNeil, A.J., Nešlehová, J.: Multivariate Archimedean copulas, d-monotone functions and 1 −norm symmetric distributions. Ann. Stat. 37(5B), 3059–3097 (2009)

    MathSciNet  Article  Google Scholar 

  35. Nolde, N.: Geometric interpretation of the residual dependence coefficient. J. Multivar. Anal. 123, 85–95 (2014)

    MathSciNet  Article  Google Scholar 

  36. Opitz, T.: Extremal t processes: Elliptical domain of attraction and a spectral representation. J. Multivar. Anal. 122, 409–413 (2013)

    MathSciNet  Article  Google Scholar 

  37. Pakes, A.G.: Convolution equivalence and infinite divisibility. J. Appl. Probab. 41(02), 407–424 (2004)

    MathSciNet  Article  Google Scholar 

  38. Resnick, S.I.: Extreme values, regular variation and point processes. Springer, Berlin (1987)

    Google Scholar 

  39. Resnick, S.I.: Heavy-tail phenomena: probabilistic and statistical modeling. Springer Science & Business Media, Berlin (2007)

    Google Scholar 

  40. Rootzén, H., Segers, J., Wadsworth, J.L.: Multivariate peaks over thresholds models. Extremes 21, 115–145 (2018)

    MathSciNet  Article  Google Scholar 

  41. Rootzén, H., Tajvidi, N.: Multivariate generalized Pareto distributions. Bernoulli, pp. 917–930 (2006)

  42. Seifert, M.I.: On conditional extreme values of random vectors with polar representation. Extremes 17(2), 193 (2014)

    MathSciNet  Article  Google Scholar 

  43. Wadsworth, J.L., Tawn, J.A., Davison, A.C., Elton, D.M.: Modelling across extremal dependence classes. J. R. Stat. Soc. Ser. B (Stat Methodol.) 79(1), 149–175 (2017)

    MathSciNet  Article  Google Scholar 

  44. Watanabe, T.: Convolution equivalence and distributions of random sums. Probab. Theory Relat. Fields 142(3), 367–397 (2008)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgments

Financial support from the Swiss National Science Foundation grant 200021-166274 (Sebastian Engelke) and UK Engineering and Physical Sciences Research Council grant EP/P002838/1 (Jennifer Wadsworth) is gratefully acknowledged. Thomas Opitz was partially funded by the French national programme LEFE/INSU.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jennifer Wadsworth.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A: Lemmas and proofs

A.1 Additional lemmas

The following Lemma 8 is widely known as Breiman’s lemma and is useful in several contexts throughout Section 7.

Lemma 8 (Breiman’s lemma, see Breiman (1965), Cline and Samorodnitsky (1994), and Pakes (2004), Lemma 2.1)

SupposeXF,YGare independentrandom variables. If\(\overline {F}\in \text {RV}_{-\alpha }^{\infty }\)withα ≥ 0 andY ≥ 0 with E(Yα+ε) < forsomeε > 0,then

$$\overline{F}_{XY}(x)\sim \mathrm{E}\left( Y^{\alpha}\right){\kern1.7pt} \overline{F}(x), \qquad x\to\infty.$$

Equivalently, if F ∈ETα and E(e(α+𝜖)Y) < , then \(\bar {F}_{X+Y}(x)=\overline {F\star G}(x)\sim \mathrm {E}\left (e^{\alpha Y}\right ){\kern 1.7pt} \overline {F}(x)\).

The following Lemma 9 provides some additional detail on the function τ, as defined in Section 2.

Lemma 9

If\(1-\tau _{1}(b_{1}-\cdot ) \in \text {RV}_{1/\gamma _{1}}^{0}\),forγ1 > 0,then\(b_{1} - \tau _{1}^{-1}(1-\cdot ) \in \text {RV}_{1/\gamma _{1}}^{0}\),andγ1 ≤ 1.Similarly, if\(1-\tau _{2}(b_{2}+\cdot ) \in \text {RV}_{1/\gamma _{2}}^{0}\),forγ2 > 0,then\(\tau _{2}^{-1}(1-\cdot )-b_{2} \in \text {RV}_{\gamma _{2}}^{0}\)andγ2 ≤ 1.

Proof

The argument is similar for both cases, so we focus on the first one. Let g(t) = 1 − τ1(b1t), which is decreasing as t → 0 and invertible, with \(g^{-1}(s)=b_{1} - \tau _{1}^{-1}(1-s)\). Since \(g \in \text {RV}^{0}_{1/\gamma _{1}}\), then \(g^{-1} \in \text {RV}^{0}_{\gamma _{1}}\) (Resnick (2007), Proposition 2.6(v), adapting to regular variation at zero). Now make the left-sided Taylor expansion

$$ \tau_{1}(b_{1}-t) = \tau_{1}(b_{1}) -t \tau_{1}^{\prime}(b_{1-}) + O(t^{2}), $$

and note that τ1(b1) = 1. Consequently, \(1-\tau _{1}(b_{1}-t) = t \tau _{1}^{\prime }(b_{1-}) + O(t^{2})\), and so finiteness of \(\tau _{1}^{\prime }(b_{1-})\) will imply that the index of regular variation of g is at least 1. Define the convex function μ : (0,) → (0,) by μ(x) = ν(1,x), so that τ(x) = 1/μ(1/x − 1). Since τ1 is increasing on (0,b1), μ is increasing on ((1 − b1)/b1,). We have

$$ \tau^{\prime}(x_{-}) = x^{-2} \mu^{\prime}((1/x-1)_{+}) / \mu(1/x-1)^{2}, $$

and so \(\tau _{1}^{\prime }(b_{1-}) = b_{1}^{-2} \mu ^{\prime }([(1-b_{1})/b_{1}]_{+}) / \mu ((1-b_{1})/b_{1})^{2}\). For h ∈ (0, 1), convexity entails

$$ \mu((1-b_{1})/b_{1} + h) \leq h \mu((1-b_{1})/b_{1}+1) +(1-h)\mu((1-b_{1})/b_{1}), $$

and so

$$ 0\leq \frac{\mu((1{\kern1.7pt}-{\kern1.7pt}b_{1})/b_{1} {\kern1.7pt}+{\kern1.7pt} h) {\kern1.7pt}-{\kern1.7pt} \mu((1{\kern1.7pt}-{\kern1.7pt}b_{1})/b_{1})}{h} \leq \mu((1-b_{1})/b_{1}+1)-\mu((1-b_{1})/b_{1}) \!<\!\infty. $$

Hence μ([(1 − b1)/b1]+) < , giving \(\tau _{1}^{\prime }(b_{1-})<\infty \). Thus the index of regular variation of g is at least 1. □

The following Lemma 10 clarifies the influence of negative values in convolutions of exponential-tailed distributions. It allows us to extend certain results from the literature formulated for nonnegative random variables to the real line.

Lemma 3 (Convolutions of exponential-tailed distributions with negative values)

Fori = 1, 2 and probabilitydistributionsFi ∈ETαdefined over\(\mathbb {R}\)withα > 0,denote\(p_{i}=\bar {F}_{i}(0)\in [0,1]\)theprobability of nonnegative values. Using the convention0/0 = 0,let\(F_{i}^{+}\)with\(1-F_{i}^{+}(x)=\bar {F}_{i}(x)/p_{i}\),x ≥ 0, denote the conditionaldistribution ofFiovernonnegative values, and let\(F_{i}^{-}(x)=F_{i}(x)/(1-p_{i})\),x < 0, denote the conditionaldistribution ofFiover negativevalues. We use the notation\(M_{H}(\alpha )={\int }_{-\infty }^{\infty }\exp (\alpha y)H(\mathrm {d}y)\)fora given distribution H. Then:

  1. 1.

    If\(\bar {F_{i}^{+}}(x)/\bar {F_{1}^{+}\star F_{2}^{+}}(x)\rightarrow 0\)whenxfori = 1, 2,then

    $$ \bar{F_{1}\star F_{2}} \sim p_{1}p_{2}\bar{F_{1}^{+}\star F_{2}^{+}}. $$
    (1)
  2. 2.

    If \(\bar {F_{1}^{+}\star F_{2}^{+}} \sim c_{1}\bar {F_{1}^{+}}+c_{2}\bar {F_{2}^{+}}\) with constants 0 ≤ c1,c2 < , then

    $$ \bar{F_{1}\star F_{2}}(x) \sim \bar{F}_{1}(x) \left( p_{2}c_{1}+(1-p_{2})M_{F_{2}^{-}}(\alpha)\right)+\bar{F}_{2}(x) \left( p_{1}c_{2}+(1-p_{1})M_{F_{1}^{-}}(\alpha)\right). $$
    (2)

    Specifically, if \(c_{1}=M_{F_{2}^{+}}(\alpha )\) and \(c_{2}=M_{F_{1}^{+}}(\alpha )\), then

    $$ \bar{F_{1}\star F_{2}}(x)\sim M_{F_{2}}(\alpha) \bar{F}_{1}(x)+M_{F_{1}}(\alpha) \bar{F}_{2}(x), $$

    and if \(c_{1}=M_{F_{2}^{+}}(\alpha )\) and c2 = 0, then

    $$ \bar{F_{1}\star F_{2}}(x)\sim M_{F_{2}}(\alpha) \bar{F}_{1}(x)+\bar{F}_{2}(x)(1-p_{1})m_{F_{1}^{-}}(\alpha). $$

Proof

We start with the mixture representation

$$ \bar{F_{1}\star F_{2}}(x)=p_{1}p_{2}\bar{F_{1}^{+}\star F_{2}^{+}}(x)+p_{1}(1-p_{2})\bar{F_{1}^{+}\star F_{2}^{-}}(x)+(1-p_{1})p_{2}\bar{F_{1}^{-}\star F_{2}^{+}}(x), \quad x \geq 0. $$

We can then use the equation \(p_{i}\bar {F_{i}^{+}}(x)=\bar {F_{i}}(x)\), x ≥ 0, and the following inequalities for {i1,i2} = {1, 2},

$$ 0=\bar{F_{i_{1}}^{-}\star F_{i_{2}}^{-}}(x) = \bar{F_{i_{1}}^{-}}(x)\leq \bar{F_{i_{1}}^{+}\star F_{i_{2}}^{-}}(x) \leq \bar{F_{i_{1}}^{+}}(x) \leq \bar{F_{i_{1}}^{+}\star F_{i_{2}}^{+}}(x) , \quad x\geq 0. $$

Moreover, Lemma 8 can be applied for mixed terms, yielding \(\overline {F_{i_{1}}^{+}\star F_{i_{2}}^{-}}\sim M_{F_{i_{2}}^{-}}(\alpha )\overline {F_{i_{1}}^{+}}\), and then Eqs. 38 and 39 follow from straightforward calculations. To determine the behavior for special cases of c1 and c2, observe that \((1-p_{i})M_{F_{i}^{-}}(\alpha )+p_{i}M_{F_{i}^{+}}(\alpha )=M_{F_{i}}(\alpha )\). □

A.2 Proof of lemmas in Section 7

Proof of Lemma 2

  1. 1.

    If b1 = b2 = 1 then P(W = 1) = 0 and \(\bar {F}_{W}(1-s)=\bar {F}_{Z}(\tau ^{-1}(1-s))\). By assumption, \(\bar {F}_{Z}(\tau ^{-1}(1-s)) = \ell _{Z}(1-\tau ^{-1}(1-s))(1-\tau ^{-1}(1-s))^{\alpha _{Z}}\), where \(\ell _{Z} \in {\text {RV}_{0}^{0}}\). Since \(1-\tau ^{-1}(1-s) \in \text {RV}_{\gamma }^{0}\) (Lemma 9) with limit zero, results on composition of regularly varying functions (Resnick (2007), Proposition 2.6 (iv)) implies the result, with αW = αZγ.

  2. 2.a)

    If b1 < 1 then

    $$ \bar{F}_{W}(1-s) - \P(W=1) = \P(Z\in[\tau_{1}^{-1}(1-s),b_{1}))+\P(Z\in (b_{2},\tau_{2}^{-1}(1-s)]), $$
    (3)

    and since Z has a Lebesgue density,

    $$ \begin{array}{@{}rcl@{}} &&\P(Z\in[\tau_{1}^{-1}(1-s),b_{1}))+\P(Z\in (b_{2},\tau_{2}^{-1}(1-s)])\\ &\sim& f_{Z}(b_{1})(b_{1} - \tau_{1}^{-1}(1-s)) + f_{Z}(b_{2})(\tau_{2}^{-1}(1-s) - b_{2}) \in \text{RV}_{\gamma}^{0}, \end{array} $$
    (4)

    again using Lemma 9.

  3. 2.b)

    A left-sided Taylor expansion of \(\tau _{1}^{-1}\) about 1 gives

    $$ \tau_{1}^{-1}(1-s) = b_{1} - (\tau_{1}^{-1})'(1_{-}) s + O(s^{2}), $$

    where O(s2)/s uniformly tends to 0 as s → 0, and similarly we can make a right-sided expansion for \(\tau _{2}^{-1}(1-s)\). Hence, using Eqs. 40 and 41, \( \bar {F}_{W}(1-s) - \P (W=1) = s \ell (s)\) with \(\lim _{s\to 0}\ell (s) = f_{Z}(b_{1}) (\tau _{1}^{-1})'(1_{-}) - f_{Z}(b_{2})(\tau _{2}^{-1})'(1_{-})\). Noting the link \(1/\tau _{1}^{\prime }(b_{1-}) = (\tau _{1}^{-1})'(1_{-})\), similarly for \(1/\tau _{2}^{\prime }(b_{2+})\), gives Eq. 24.

  4. 2.

    For the final part, we have

    $$ \begin{array}{@{}rcl@{}}\bar{F}_{W_{\wedge}}(\zeta(1-s)) &=& \bar{F}_{Z}(\tau_{1}^{-1}(\zeta(1-s))) - \bar{F}_{Z}(1-\tau_{1}^{-1}(\zeta(1-s)))\\ &\sim& f_{Z}(1/2)\left\{ 1 - 2\tau_{1}^{-1}(\zeta(1-s))\right\}. \end{array} $$

    Again by left-sided Taylor expansion of \(\tau _{1}^{-1}\) about ζ = τ(1/2), we have

    $$\tau_{1}^{-1}(\zeta(1-s)) = 1/2 - (\tau_{1}^{-1})'(\zeta_{-}) \zeta s + O(s^{2}),$$

    and so we obtain \(\bar {F}_{Z}(\tau _{1}^{-1}(\zeta (1-s))) - \bar {F}_{Z}(1-\tau _{1}^{-1}(\zeta (1-s))) = s \ell _{\wedge }(s)\) with \(\lim _{s\to 0}\ell _{\wedge }(s) = 2 f_{Z}(1/2)\zeta (\tau _{1}^{-1})'(\zeta _{-})\). Noting again that \(1/\tau _{1}^{\prime }(1/2_{-}) = (\tau _{1}^{-1})'(\zeta _{-})\), we arrive at Eq. 25.

Proof of Lemma 5

The result for nonnegative S and V is found in Theorem 4(v) of Cline (1986). The extension to negative values then follows from Lemma 10(1). □

Proof

of Lemma 6 The result is given in Theorem 6(ii,iii) of Cline (1986) for \(\bar {F}(0)=1\) and \(\bar {G}_{1}(0)=1\). For point 1, the extension to negative values in F and G1 follows from observing that Theorem 6(iii) of Cline (1986) implies

$$ \bar{F\star G_{1}}\sim \bar{F^{+}\star G_{1}^{+}}, \qquad x\rightarrow\infty, $$
(5)

where F+ is obtained from F by setting F+(0) = F(0) and \(\bar {F^{+}}(0)=1\), and the same construction is taken for \(G_{1}^{+}\); i.e., F+ and \(G_{1}^{+}\) arise from projecting negative values to 0. For point 2, the extension to negative values can be shown using Lemma 10(2). Indeed, the same limit \(M_{G_{1}}(\alpha )\) arises for \(\bar {F\star G_{1}}(x)/\bar {F}(x)\) if we project negative values in F and G1 to 0 or not. □

Appendix B: Tail classes and examples

Definitions of tail classes are given in Section 1.1. The following lemma summarizes important relationships between such tail classes. In this section, we refer to the class of heavy-tailed distributions by HT, and to superheavy-tailed distributions by SHT.

Lemma 4 (Relationships between tail classes)

The following relationships between distribution classes hold:

  1. 1.

    \(\text {RV}_{\alpha }^{\infty }\subset \text {CE}_{0}\)forα > 0,

  2. 2.

    \(\text {ET}_{0}\subsetneq \text {HT}\) .

  3. 3.

    For ETαwithα > 0,we have:

    • \(F(\exp (\cdot ))\in \text {ET}_{\alpha } \Leftrightarrow F \in \text {RV}_{\alpha }^{\infty }\),

    • CEα ⊂ETα,

    • ETα,β>− 1 ∩CE = .

  4. 4.

    For WTβ,we have:

    • \(\text {WT}_{1}\subset \bigcup _{\alpha >0}\text {ET}_{\alpha }\),

    • WTβ ⊂CE0forβ < 1,

    • LWTβ ⊂SHT forβ < 1.

  5. 5.

    By denotingF1F2if\(\overline {F}_{1}(x)/\overline {F}_{2}(x)\rightarrow 0\)forx,we have:

    • If\(\tilde {\alpha }<\alpha \),then\(\text {WT}_{\beta >1}\prec \text {ET}_{\alpha } \prec \text {ET}_{\tilde {\alpha }} \prec \text {WT}_{\beta <1}\prec \text {LWT}_{\beta >1}\prec \text {RV}_{\alpha >0}^{\infty }\prec \text {RV}_{\tilde {\alpha }}^{\infty }\prec \text {SHT}\).

    • \(\text {CE}_{\alpha >0}\prec \text {ET}_{\tilde {\alpha },\beta }\)for\(\tilde {\alpha }\leq \alpha \)andanyβ > 0.

We recall the membership in tail classes for well-known parametric distribution families in Table 1, see Johnson et al. (1994, 1995) for reference about parameters. Here we abstract away from the usual parameter symbols of these distributions to avoid conflicting notations with general tail parameters. We refer parameters as scl and loc if scl × X + loc has scale scl and location loc, where X has scale 1 and location 0. Another parameter shp may be related to shape for some distributions.

Table 4 Membership in tail classes (columns) for distribution families (rows)

Appendix C: Additional illustrations

Figure 4 illustrates further examples of norms ν and related functions τ(z) and τ(1 − z), as defined in Section 2.

Fig. 4
figure4

Further illustration of different norms ν and their related functions τ(z) (solid line) and τ(1 − z) (dashed line)

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Engelke, S., Opitz, T. & Wadsworth, J. Extremal dependence of random scale constructions. Extremes 22, 623–666 (2019). https://doi.org/10.1007/s10687-019-00353-3

Download citation

Keywords

  • Copula
  • Extreme value theory
  • Residual tail dependence
  • Tail dependence

AMS 2000 Subject Classifications

  • 60G70
  • 60E05
  • 62H20