tails of copulas
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Tails of Copulas. Gary G Venter. Correlation Issues. Correlation is stronger for large events Can model by copula methods Quantifying correlation Degree of correlation Part of spectrum correlated. Modeling via Copulas. Correlate on probabilities  PowerPoint PPT PresentationTRANSCRIPT
Tails of CopulasGary G Venter
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Correlation IssuesCorrelation is stronger for large events Can model by copula methodsQuantifying correlationDegree of correlationPart of spectrum correlated
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Modeling via CopulasCorrelate on probabilitiesInverse map probabilities to correlate lossesCan specify where correlation takes place in the probability rangeConditional distribution easily expressedSimulation readily available
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What is a copula?A way of specifying joint distributionsA way to specify what parts of the marginal distributions are correlatedWorks by correlating the probabilities, then applying inverse distributions to get the correlated marginal distributionsFormally they are joint distributions of unit uniform variates, as probabilities are uniform on [0,1]
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Formal RulesF(x,y) = C(FX(x),FY(y))Joint distribution is copula evaluated at the marginal distributionsExpresses joint distribution as interdependency applied to the individual distributionsC(u,v) = F(FX1(u),FY1(v))u and v are unit uniforms, F maps R2 to [0,1]FYX(y) = C1(FX(x),FY(y)) Derivative of the copula is the conditional distributionE.g., C(u,v) = uv, C1(u,v) = v = Pr(V [0,1], e.g., h(u,v) = (uv)3/5Draw u,v,w from [0,1]If h(u,v)>w, drop v and set v=uSimulate from u and v, which might be u
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Simulated Pareto (1,4) h(u)=u0.3 (Partial Power Copula)
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Partial Cutoff Copula h(u)=(u>k)
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Partial Perfect Copula FormulasFor case h(u,v)=h(u)h(v)H(u)=h(u)C(u,v) = uv H(u)H(v) + H(1)H(min(u,v))C1(u,v) = v h(u)H(v) + H(1)h(u)(v>u)
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Taush(u)=ua, t(a)= (a+1)4/3 +8/[(a+1)(a+2)2(a+3)]h(u)=(u>k), t(k) = (1 k)4 h(u)=h0.5, t(h) = (h2+2h)/3 h(u)= h0.5ua(u>k), t(h,a,k) = h2(1ka+1)4(a+1)4/3+8h[(a+2)2(1ka+3)(1ka+1)(a+1)(a+3)(1ka+2)2]/dwhere d = (a+1)(a+2)2(a+3)
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Quantifying Tail ConcentrationL(z) = Pr(Uz)L(z) = C(z,z)/z R(z) = [1 2z +C(z,z)]/(1 z)L(1) = 1 = R(0)Action is in R(z) near 1 and L(z) near 0lim R(z), z>1 is R, and lim L(z), z>0 is L
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LR Function(L below , R above)
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R usually above tau
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Example: ISO Loss and LAEFreez and Valdez find Gumbel fits best, but only assume ParetosKlugman and Parsa assume Frank, but find better fitting distributions than ParetoAll moments less than tail parameter converge
Loss Median
Loss Tail
Expense Median
Expense Tail
Frees & Valdez
12,000
1.12
5500
2.12
Klugman & Parsa
12,275
1.05
5875
1.58
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Can Try Joint Burr, from HRTF(x,y) = 1(1+(x/b)p)a (1+(y/d)q)a +[1+(x/b)p +(y/d)q]a E.g. F(x,y)=1[1+x/14150]1.11[1+(y/6450)1.5]1.11 +[1+x/14150 +(y/6450)1.5]1.11Given loss x, conditional distribution is Burr:FYX(yx) = 1[1+(y/dx)1.5]2.11 with dx = 6450 +11x 2/3
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Example: 2 States Hurricanes
MD & DE Joint Empirical Probabilities
DE vs. MD copula

0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
1.000

0.200
0.400
0.600
0.800
1.000
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L and R Functions, Tau = .45R looks about .25, which is >0,