Paths and indices of maximal tail dependence Ričardas Zitikis University of Western Ontario, London, Canada
[email protected]
12 December 2015
CMStatistics 2015, London, UK -- R. Zitikis
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Tails do matter!
Müller-Lyer illusion Franz Carl Müller-Lyer (1857–1916)
Just in case you thought otherwise, the shafts of the arrows are of the same length 12 December 2015
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Talk is based on Furman, Su, and Zitikis (2015) Paths and indices of maximal tail dependence ASTIN Bulletin, 45 (3), 661—678
Jianxi Su
Edward Furman 12 December 2015
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Bivariate distributions 𝐹𝑋,𝑌 𝑥, 𝑦 = 𝐏[𝑋 ≤ 𝑥, 𝑌 ≤ 𝑦] 𝑣 = 𝐹𝑌 (𝑦) marginal cdf
𝑢 = 𝐹𝑋 (𝑥) marginal cdf
𝐶(𝑢, 𝑣) copula captures dependence
Hence 𝐹𝑋,𝑌 𝑥, 𝑦 = 𝐶(𝐹𝑋 𝑥 , 𝐹𝑌 (𝑦)) 12 December 2015
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Copula tails • Financiers When 𝑢, 𝑣 ↓ 0, then 𝐶(𝑢, 𝑣) ~ ? • Actuaries
When 𝑢, 𝑣 ↑ 1, then 𝐶(𝑢, 𝑣) ~ ? • Classical indices 𝐶(𝑢,𝑢) 𝑢 ↓0 𝑢
Lower index λ𝐿 = lim Upper index λ𝑈 = 12 December 2015
𝐶(𝑢,𝑢) lim 𝑢 ↓0 𝑢
( 𝐶 survival copula )
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Lower-tail indices • They are based on 𝐶 𝑢, 𝑢 = 𝐏[ 𝑈 ≤ 𝑢, 𝑉 ≤ 𝑢] = 𝐏[ 𝑈, 𝑉 ∈ [0, 𝑢] × 0, 𝑢 ] • Why the cube 0, 𝑢 × 0, 𝑢 ?
• Why along the diagonal? 12 December 2015
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From cubes to rectangles • Find φ(𝑢) and ψ(𝑢) that maximize 𝐏[ 𝑈, 𝑉 ∈ [0, φ(𝑢)] × 0, 𝜓 𝑢 ] • On the same “footing” as the classical index Area([0, φ(𝑢)] × 0, 𝜓 𝑢 ) = φ(𝑢) ψ(𝑢) = Area([0, 𝑢] × 0, 𝑢 ) ) = 𝑢2 • Hence, ψ 𝑢 = 𝑢2 / φ(𝑢) and so pathφ 𝑢 ≝ φ 𝑢 , 𝑢2 / φ(𝑢) • Classics: φ 𝑢 = 𝑢 12 December 2015
⇒
pathφ 𝑢 = 𝑢, 𝑢 CMStatistics 2015, London, UK -- R. Zitikis
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New notion of tail dependence • Find φ 𝑢 , called function of max-dependence, such that 𝐶 pathφ 𝑢
is maximal
• Find minimal κ∗ ≥ 0, called index of max-dependence, such that 𝐶 pathφ 𝑢
≈sv
∗ κ 𝑢
when 𝑢 ↓ 0
≈sv means “up to a slowly varying function”
• Note: classical κ ≥ 0 is such that 𝐶 𝑢, 𝑢 ≈sv 𝑢κ when 𝑢 ↓ 0 12 December 2015
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Fatal shocks and the Marshall-Olkin copula • Motivation Company consists of two business lines, L1 & L2 Poisson process sends a fatal shock to only line L1 Another Poisson process sends a fatal shock to only line L2 Yet another Poisson sends a fatal shock to the entire company (L1 & L2)
• Notation T1 is the failure time of line L1 T2 is the failure time of line L2 Joint survival function of T1 and T2 𝑆T1,T2 𝑥, 𝑦 = exp − λ1 𝑥 − λ2 𝑦 − λ12 max 𝑥, 𝑦 where λ1 , λ2 , λ12 are the arrival rates of the three (independent) Poissons 12 December 2015
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Marshall-Olkin copula Joint survival function of T1 & T2 𝑆T1,T2 𝑥, 𝑦 = 𝐶𝑎,𝑏 (𝑆T1 𝑥 , 𝑆T2 𝑦 ) where 𝐶𝑎,𝑏 (𝑢, 𝑣) = min{ 𝑢1−𝑎 𝑣, 𝑢𝑣 1−𝑏 }
and
12 December 2015
λ12 𝑎= λ1 + λ12
and
λ12 𝑏= λ2 + λ12
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Marshall-Olkin copula • Function of max-dependence φ 𝑢 = 𝑢2𝑏/(𝑎+𝑏) and so 𝐶 pathφ 𝑢
=
• Classical φ 𝑢 = 𝑢 gives
12 December 2015
∗ κ 𝑢
with
κ∗
=2−
2𝑎𝑏 𝑎+𝑏
𝜅 = 2 − min 𝑎, 𝑏 𝜅 ≥ κ∗ always 𝜅 = κ∗ only when 𝑎 = 𝑏
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Marshall-Olkin copula
12 December 2015
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Mixture of M-O copulas • Symmetric copula with off-diagonal max-dependence functions 1 1 𝐶𝑎,𝑏 𝑢, 𝑣 + 𝐶𝑏,𝑎 𝑢, 𝑣 2 2 • Functions of max-dependence φ 𝑢 = 𝑢2𝑏/(𝑎+𝑏) φ 𝑢 = 𝑢2𝑎/(𝑎+𝑏) • Index of maximal dependence ∗
κ =2 12 December 2015
2𝑎𝑏 − 𝑎+𝑏
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Mixture of M-O copulas
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Concluding notes • Be careful when using classical indices of tail dependence because they are good reflectors of (maximal) tail dependence only for special dependence structures • Example: For the Gaussian copula, the path of maximal taildependence is diagonal The proof is (very) complex: Furman, Kuznetsov, Su, and Zitikis (2015). Tail Dependence of the Gaussian Copula Revisited. Available at SSRN: http://ssrn.com/abstract=2558675
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