Gumbel: The Gumbel Hougaard Copula

Description

Density function, distribution function, random generation, generator and inverse generator function for the Gumbel Copula with parameters alpha. The 4 classic estimation methods for copulas.

Usage

dgumbel(u, v=NULL, alpha, dim=2, warning = FALSE)
pgumbel(u, v=NULL, alpha, dim=2)
rgumbel(n, alpha, dim=2, method=1)
phigumbel(t, alpha=1)
invphigumbel(t, alpha=1)
gumbel.MBE(x, y, marg = "exp")
gumbel.EML(x, y, marg = "exp")
gumbel.IFM(x, y, marg = "exp")
gumbel.CML(x, y)

Value

dgumbel gives the density,

pgumbel gives the distribution function,

rgumbel generates random deviates,

phigumbel gives the generator,

invphigumbel gives the inverse generator.

gumbel.EML, gumbel.IFM, gumbel.MBE and gumbel.CML

returns the vector of estimates.

Invalid arguments will result in return value NaN.

Arguments

u: vector of quantiles if argument v is provided or matrix of quantiles if argument v is not provided
v: vector of quantiles, needed if u is not a matrix
n: number of observations. If length(n) > 1, the length is taken to be the number required.
alpha: parameter of the Copula. Must be greater than $1$.
dim: an integer specifying the dimension of the copula.
t: dummy variable of the generator $\phi$ or the inverse generator $\phi^-1$. could be a n-dimensional array.
method: an integer code for the method used in simulation. 1 is the common frailty approach, 2 uses the K function (only valid with dim=2).
x,y: vectors of observations, realizations of random variable $X$ and $Y$.
marg: a character string specifying the marginals of vector $(X,Y)$. It must be either "exp"(default value) or "gamma".
warning: a logical (default value FALSE) if you want warnings.

Author

A.-L. Caillat, C. Dutang, M. V. Larrieu and T. Nguyen

Details

The Gumbel Hougaard Copula with parameter alpha is defined by its generator $$\phi(t) = (-ln(t))^alpha.$$ The generator and inverse generator are implemented in phigumbel and invphigumbel respectively. As an Archimedean copula, its distribution function is $$ C(u_1, ...,u_n) = \phi^{-1}(\phi(u_1)+...+\phi(u_n)) = exp(-( (-ln(u_1))^alpha+...+(-ln(u_n))^alpha )^1/\alpha).$$ pgumbel and dgumbel computes the distribution function (expression above) and the density ($n$ times differentiation of expression above with respect to $u_i$). As there is no explicit formulas for the density of a Gumbel copula, dgumbel is not yet impemented for argument dim>3. This two functions works with a dim-dimensional array with the last dimension being equalled to dim or with a matrix with dim columns (see examples).

Random generation is carried out with 2 algorithms the common frailty algorithm (method=1) and the 'K' algorithm (method=2). The common frailty algorithm (cf. Marshall & Olkin(1988)) can be sum up in three lines

generate $y_1$, $y_2$ from exponential distribution of mean 1,
generate $\theta$ from a stable distribution with parameter$(1/alpha,1,1,0)$,
$u_1 <- phi(y_1/\theta)$ and $u_2 <- phi(y_2/\theta)$.

This algorithm works with any dimension. See Chambers et al(1976) for stable random generation. The 'K' algorithm use the fact the distribution function of random variable $C(U,V)$ is $K(t) = t-\phi(y)/\phi'(t)$. The algorithm is

generate $v_1$, $t$ from uniform distribution,
generate $v_2$ from the $K$ distribution i.e. $v_2<-K^{-1}(t)$,
$u_1<-\phi^{-1}(\phi(v_1)v_2)$ and $u_2<-\phi^{-1}(\phi(v_1)(1-v_2)) $.

Warning, the 'K' algorithm does NOT work with dim>2.

We implements the 4 usual method of estimation for copulas, namely the Exact Maximum Likelihood (gumbel.EML), the Inference for Margins (gumbel.IFM), the Moment-base Estimation (gumbel.MBE) and the Canonical Maximum Likelihood (gumbel.CML). For the moment, only two types of marginals are available : exponential distribution (marg="exp") and gamma distribution (marg="gamma").

References

Nelsen, R. (2006), An Introduction to Copula, Second Edition, Springer.

Marshall & Olkin(1988), Families of multivariate distributions, Journal of the American Statistical Association.

Chambers et al (1976), A method for simulating stable random variables, Journal of the American Statistical Association.

Nelsen, R. (2005), Dependence Modeling with Archimedean Copulas, booklet available at www.lclark.edu/~mathsci/brazil2.pdf.

Examples

Run this code

	#dim=2
	u<-seq(0,1, .1)
	v<-u
	#classic parametrization
	#independance case (alpha=1)
	dgumbel(u,v,1)
	pgumbel(u,v,1)
	#another parametrization
	dgumbel(cbind(u,v), alpha=1)
	pgumbel(cbind(u,v), alpha=1)

	#dim=3 - equivalent parametrization
	x <- cbind(u,u,u)
	y <- array(u, c(1,11,3))
	pgumbel(x, alpha=2, dim=3)
	pgumbel(y, alpha=2, dim=3)
	dgumbel(x, alpha=2, dim=3)
	dgumbel(y, alpha=2, dim=3)

	#dim=4
	x <- cbind(x,u)
	pgumbel(x, alpha=3, dim=4)
	y <- array(u, c(2,1,11,4))
	pgumbel(y, alpha=3, dim=4)
	
	
	#independence case
	rand <- t(rgumbel(200,1))
	plot(rand[1,], rand[2,], col="green", main="Gumbel copula")
	
	#positive dependence
	rand <- t(rgumbel(200,2))
	plot(rand[1,], rand[2,], col="red", main="Gumbel copula")
	
	#comparison of random generation algorithms
	nbsimu <- 10000
	#Marshall Olkin algorithm
	system.time(rgumbel(nbsimu, 2, dim=2, method=1))[3]
	#K algortihm
	system.time(rgumbel(nbsimu, 2, dim=2, method=2))[3]
	
	#pseudo animation
	if (FALSE) {
	anim <-function(n, max=50)
	{
	for(i in seq(1,max,length.out=n)) 
	{
		u <- t(rgumbel(10000, i, method=2))
		plot(u[1,], u[2,], col="green", main="Gumbel copula",
            xlim=c(0,1), ylim=c(0,1), pch=".")
		cat()
	}	
	}
	anim(20, 20)
	}
	
	#3D plots

	#plot the density
 	x <- seq(.05, .95, length = 30)
	y <- x
	z <- outer(x, y, dgumbel, alpha=2)
		
	persp(x, y, z, theta = 30, phi = 30, expand = 0.5, col = "lightblue",
      ltheta = 100, shade = 0.25, ticktype = "detailed",
      xlab = "x", ylab = "y", zlab = "density")
	
	#with wonderful colors
	#code of P. Soutiras
	zlim <- c(0, max(z))
	ncol <- 100
	nrz <- nrow(z)
	ncz <- ncol(z)
	jet.colors <- colorRampPalette(c("#00007F", "blue", "#007FFF", 
	"cyan", "#7FFF7F", "yellow", "#FF7F00", "red", "#7F0000")) 
	couleurs <- tail(jet.colors(1.2*ncol),ncol)
	fcol <- couleurs[trunc(z/zlim[2]*(ncol-1))+1]
	dim(fcol) <- c(nrz,ncz)
	fcol <- fcol[-nrz,-ncz]
	persp(x, y, z, col=fcol, zlim=zlim, theta=30, phi=30, ticktype = "detailed",
        box = TRUE, 	xlab = "x", ylab = "y", zlab = "density")
	

	#plot the distribution function
	z <- outer(x, y, pgumbel, alpha=2)
	persp(x, y, z, theta = 30, phi = 30, expand = 0.5, col = "lightblue",
        ltheta = 100, shade = 0.25, ticktype = "detailed",
        xlab = "u", ylab = "v", zlab = "cdf")
	
	

	#parameter estimation
	#true value : lambdaX=lambdaY=1, alpha=2
	simu <- qexp(rgumbel(200, 2))
	gumbel.MBE(simu[,1], simu[,2])
	gumbel.IFM(simu[,1], simu[,2])
	gumbel.EML(simu[,1], simu[,2])
	gumbel.CML(simu[,1], simu[,2])

	#true value : lambdaX=lambdaY=1, alphaX=alphaY=2, alpha=3
	simu <- qgamma(rgumbel(200, 3), 2, 1)
	gumbel.MBE(simu[,1], simu[,2], "gamma")
	gumbel.IFM(simu[,1], simu[,2], "gamma")
	gumbel.EML(simu[,1], simu[,2], "gamma")
	gumbel.CML(simu[,1], simu[,2])

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