Copulas Univariate Functions

Gary G Venter

Conditioning with Copulas Let C1(u,v) denote the first partial derivative of

C(u,v). F(x,y) = C(FX(x),FY(y)), distribution of Y|X=x is given by:

FY|X(y) = C1(FX(x),FY(y)) C(u,v) = uv, the conditional distribution of V given

that U=u is C1(u,v) = v = Pr(V<v|U=u).

If C1 is simple enough to invert algebraically, then the simulation of joint probabilities can be done using the derived conditional distribution. That is, first simulate a value of U, then simulate a value of V from C1.

Copulas Univariate Functions

Gary G Venter

Tails of Copulas

ASTIN 2001

Copulas Univariate Functions

Gary G Venter

Kendall correlation

is a constant of the copula = 4E[C(u,v)] – 1 = 2dE[C(u1, . . .,ud)] – 1

2d – 1 – 1

Copulas Univariate Functions

Gary G Venter

Frank’s Copula

Define gz = e-az – 1 Frank’s copula with parameter a 0 can be

expressed as: C(u,v) = -a-1ln[1 + gugv/g1]

C1(u,v) = [gugv+gv]/[gugv+g1]

c(u,v) = -ag1(1+gu+v)/(gugv+g1)2

(a) = 1 – 4/a + 4/a2 0a t/(et-1) dt

For a<0 this will give negative values of . v = C1

-1(p|u) = -a-1ln{1+pg1/[1+gu(1–p)]}

Copulas Univariate Functions

Gary G Venter

0.01

0.1

1

10Frank's Copula Density on Log Scale =.5

Copulas Univariate Functions

Gary G Venter

Gumbel Copula

C(u,v) = exp{- [(- ln u)a + (- ln v)a]1/a}, a 1. C1(u,v) = C(u,v)[(- ln u)a + (- ln v)a]-1+1/a(-ln u)a-1/u c(u,v) = C(u,v)u-1v-1[(-ln u)a +(-ln v)a]-2+2/a[(ln u)(ln

v)]a-1 {1+(a-1)[(-ln u)a +(-ln v)a]-

1/a} (a) = 1 – 1/a Simulate two independent uniform deviates u and v Solve numerically for s>0 with ues = 1 + as The pair [exp(-sva), exp(-s(1-v)a)] will have the

Gumbel copula distribution

Copulas Univariate Functions

Gary G Venter

0.001

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Copulas Univariate Functions

Gary G Venter

Heavy Right Tail Copula

C(u,v) = u + v – 1 + [(1 – u)-1/a + (1 – v)-1/a – 1]-a a>0

C1(u,v) = 1 – [(1 – u)-1/a + (1 – v)-1/a – 1] -a-1(1 – u)-1-1/a

c(u,v) = (1+1/a)[(1–u)-1/a +(1– v)-1/a –1] -a-2[(1–u)(1–v)]-1-1/a

(a) = 1/(2a + 1) Can solve conditional distribution for v

Copulas Univariate Functions

Gary G Venter

0.1

1

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100

Heavy Right Tail Copula Log Scale = .5

Copulas Univariate Functions

Gary G Venter

Joint Burr

F(x) = 1 – (1 + (x/b)p)-a and G(y) = 1 – (1 + (y/d)q)-a

F(x,y) = 1 – (1 + (x/b)p)-a – (1 + (y/d)q)-a + [1 + (x/b)p + (y/d)q]-a

The conditional distribution of y|X=x is also Burr:

FY|X(y|x) = 1 – [1 + (y/dx)q]-(a+1), where dx = d[1 +

(x/b)p/q]

Copulas Univariate Functions

Gary G Venter

Partial Perfect Correlation Copula Generator

Assume logical values 0 and 1 are arithmetic also

h : unit square unit interval H(x) = 0

xh(t)dt 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) c(u,v) = 1 – h(u)h(v) + H(1)h(u)(u=v)

Copulas Univariate Functions

Gary G Venter

h(u) = (u>a)

H(u) = (u – a)(u>a) (a) = (1 – a)4

PP Max Data Pairs t = .5

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Copulas Univariate Functions

Gary G Venter

h(u) = ua

H(u) = ua+1/(a+1) (a) = 1/[3(a+1)4] +

8/[(a+1)(a+2)2(a+3)]

PP (uv)̂ a Data Pairs =.5

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Copulas Univariate Functions

Gary G Venter

The Normal Copula N(x) = N(x;0,1) B(x,y;a) = bivariate normal distribution function, = a Let p(u) be the percentile function for the standard normal: N(p(u)) = u, dN(p(u))/du = N’(p(u))p’(u) = 1 C(u,v) = B(p(u),p(v);a) C1(u,v) = N(p(v);ap(u),1-a2) c(u,v) = 1/{(1-a2)0.5exp([a2p(u)2-2ap(u)p(v)+a2p(v)2]/[2(1-a2)])} (a) = 2arcsin(a)/ a: 0.15643 0.38268 0.70711 0.92388 0.98769

0.10000 0.25000 0.50000 0.75000 0.90000

Copulas Univariate Functions

Gary G Venter

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Normal Copula Log Scale =.5

Copulas Univariate Functions

Gary G Venter

Tail Concentration Functions

L(z) = Pr(U<z,V<z)/z2 R(z) = Pr(U>z,V>z)/(1 – z)2

L(z) = C(z,z)/z2 1 - Pr(U>z,V>z) = Pr(U<z) + Pr(V<z) -

Pr(U<z,V<z) = z + z – C(z,z). Then R(z) = [1 – 2z +C(z,z)]/(1 – z)2

Generalizes to multi-variate case

Copulas Univariate Functions

Gary G Venter

HRT L and R Functions for = .1, .5, and .9

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10

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1000

Gumbel L and R functions for = .1, .5, and .9

1

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1000Frank L and R Functions for = .1, .5, .9

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1000

Normal L and R Functions for = .1, .5, .9

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1000

PP Power L and R Functions for = .1, .5, .9

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PP Max L and R Functions for = .1, .5, and .9

1

10

100

1000

Copulas Univariate Functions

Gary G Venter

Cumulative Tau

–1+4010

1 C(u,v)c(u,v)dvdu

J(z) = –1+40z0

z C(u,v)c(u,v)dvdu/C(z,z)2

Generalizes to multi-variate case

Copulas Univariate Functions

Gary G Venter

Frank Cumulative Tau = .1, .5, .9

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Gumbel Cumulative Tau = .1, .5, .9

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PP Power Cumulative Tau = .1, .5, .9

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HRT Cumulative Tau = .1, .5, .9

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PP Max Cumulative Tau = .1, .5, .9

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Normal Cumulative Tau, t = .1, .5, .9

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Copulas Univariate Functions

Gary G Venter

Cumulative Conditional Mean

M(z) = E(V|U<z) = z-10z0

1 vc(u,v)dvdu M(1) = ½ A pairwise concept

Copula Distribution Function

K(z) = Pr(C(u,v)<z) Generalizes to multi-variate case

Copulas Univariate Functions

Gary G Venter

Frank M(z) for = .1, .5, .9

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HRT M(z), = .1, .5, .9

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Normal M(z) for = .1, .5, .9

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PP Max M(z), = .1, .5, .9

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PP Power M(z), = .1, .5, .9

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Gumbel M(z) for = .1, .5, .9

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Copulas Univariate Functions

Gary G Venter

MD & DE Joint Empirical Probabilities

DE vs. MD copula

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HRT Gumbel Frank NormalParameter 0.968 1.67 4.92 0.624Ln Likelihood 124 157 183 176Tau 0.34 0.40 0.45 0.43

Copulas Univariate Functions

Gary G Venter

LR Function for DE/MD and Fits

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100

Data

Frank

Normal

PP Pow er

J(z) Data and Fits

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Normal

Frank

M(z) Data and Fits

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Frank

Normal

Empirical K as Function of Frank K

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Frank