tEVcop: The t-EV (Extreme Value) Copula

In wasquith/copBasic: General Bivariate Copula Theory and Many Utility Functions

The t-EV (Extreme Value) Copula

Description

The t-EV copula (Joe, 2014, p. 189) is a limiting form of the t-copula (multivariate t-distribution):

\mathbf{C}_{\rho,\nu}(u,v) = \mathbf{tEV}(u,v; \rho, \nu) = \mathrm{exp}\bigl(-(x+y) \times B(x/(x+y); \rho, \nu)\bigr)\mbox{,}

where x = -\log(u), y = -\log(v), and letting \eta = \sqrt{(\nu+1)/(1-\rho^2)} define

B(w; \rho, \nu) = w \times T_{\nu+1}\bigl(\eta[(w/[1-w])^{1/\nu}-\rho]\bigr) + (1-w) \times T_{\nu+1}\bigl(\eta[([1-w]/w)^{1/\nu}-\rho]\bigr)\mbox{,}

where T_{\nu+1} is the cumulative distribution function of the univariate t-distribution with \nu-1 degrees of freedom. As \nu \rightarrow \infty, the copula weakly converges to the Hüsler–Reiss copula (HRcop) because the t-distribution converges to the normal (see Examples for a study of this copula).

The \mathbf{tEV}(u,v; \rho, \nu) copula is a two-parameter option when working with extreme-value copula. There is a caveat though. Demarta and McNeil (2004) conclude that “the parameter of the Gumbel [GHcop] or Galambos [GLcop] A-functions [the Pickend dependence function and B-function by association] can always be chosen so that the curve is extremely close to that of the t-EV A-function for any values of \nu and \rho. The implication is that in all situations where the t-EV copula might be deemed an appropriate model then the practitioner can work instead with the simpler Gumbel or Galambos copulas.”

Usage

tEVcop(u, v, para=NULL, ...)

Arguments

u	Nonexceedance probability u in the X direction;
v	Nonexceedance probability v in the Y direction;
para	A vector (two element) of parameters in \rho and \nu order; and
...	Additional arguments to pass.

Value

Value(s) for the copula are returned.

Note

Note, Joe (2014) shows x = \log(u) (note absence of the minus sign)—this is not correct.

Author(s)

W.H. Asquith

References

Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.

Demarta, S., and McNeil, A.J., 2004, The t copula and related copulas: International Statistical Review, v. 33, no. 1, pp. 111–129, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/j.1751-5823.2005.tb00254.x")}

See Also

GHcop, GLcop, HRcop

Examples

## Not run: 
  tau <- 1/3 # Example from copula::evCopula.Rd
  tev.cop <- copula::tevCopula(copula::iTau(copula::tevCopula(), tau))
  copula::pCopula(c(0.1,.5), copula=tev.cop)         # 0.07811367
  tEVcop(0.1, 0.5, para=slot(tev.cop, "parameters")) # 0.07811367
## End(Not run)

## Not run: 
  nsim <- 2000; pargh <- c(5, 0.5, 0.5)
  UV <- simCOP(nsim, cop=GHcop, para=pargh)
  U <- lmomco::pp(UV[,1], sort=FALSE)
  V <- lmomco::pp(UV[,2], sort=FALSE)
  RT <- mleCOP(u=U, v=V, cop=tEVcop, init.para=c(0.5, log(4)),
               parafn=function(k) return( c(k[1], exp(k[2])) ) )
  partev <- RT$para

  FT <- simCOP(nsim, cop=tEVcop, para=RT$para)

  tauCOP(cop=GHcop,  para=pargh )
  tauCOP(cop=tEVcop, para=partev)

  tauCOP(cop=GHcop,  para=pargh ) # [1] 0.3003678
  tauCOP(cop=tEVcop, para=partev) # [1] 0.3178904

  densityCOPplot(cop=GHcop,  para=pargh)
  densityCOPplot(cop=tEVcop, para=partev, ploton=FALSE, contour.col="red") #
## End(Not run)

## Not run: 
  # A demonstration Joe (2014, p. 190) for which tEvcop() has
  # upper tail dependence parameter as
  para <- c(0.8, 10)
  lamU <- 2 * pt( -sqrt( (para[2]+1) * (1-para[1]) / (1+para[1]) ), para[2]+1)
  "tEVcop.copula" <- function(u,v, para=NULL, ...) {
        if(length(u) == 1) u <- rep(u,length(v)); if(length(v)==1) v <- rep(v,length(u))
        return(copula::pCopula(matrix(c(u,v), ncol=2),
               copula::tevCopula(param=para[1], df=para[2])))
  }
  lamU.copBasic <- taildepCOP(cop=tEVcop,        para)$lambdaU
  lamU.copula   <- taildepCOP(cop=tEVcop.copula, para)$lambdaU
  print(c(lamU, lamU.copBasic, lamU.copula))
  #[1] 0.2925185 0.2925200 0.2925200 # So, we see that they all match.
## End(Not run)

## Not run: 
  # Convergence of tEVcop to HRcop as nu goes to infinity.
  nu <- 10^(seq(-4, 2, by=0.1)) # nu right end point rho dependent
  rho <- 0.7 # otherwise, expect to see 'zeros' errors on the plot()
  # Compute Blomqvist Beta (fast computation is reason for choice)
  btEV <- sapply(nu, function(n) blomCOP(tEVcop, para=c(rho, n)))
  limit.thetas <- sqrt(2 / (nu*(1-rho))) # for nu --> infinity HRcop
  thetas <- sapply(btEV, function(b) {
       uniroot(function(l, blom=NA) { blom - blomCOP(HRcop, para=l) },
       interval=c(0,10), blom=b)$root })
  plot(limit.thetas, thetas, log="xy", type="b",
       xlab="Theta of HRcop via limit nu --> infinity",
       ylab="Theta from Blomqvist Beta equivalent HRcop to tEVcop")
  abline(0,1)
  mtext(paste0("Notice the 'weak' convergence to lower left, and \n",
               "convergence increasing with rho"))
  # Another reference of note
  # https://mediatum.ub.tum.de/doc/1145695/1145695.pdf (p.39) #
## End(Not run)

wasquith/copBasic documentation built on Feb. 17, 2025, 11:39 a.m.