Estimators proposed by Krupskii and Joe under second order expansion for the coefficient of tail dependence \(\eta\) and the joint tail orthant probability
Arguments
- data
- a matrix of observations 
- q
- vector of quantile levels 
- ptail
- tail probability smaller than - q. Default to- NULL
- mqu
- marginal quantile levels for semiparametric estimation; data above this are modelled using a generalized Pareto distribution. If missing, empirical estimation is used throughout 
- type
- integer indicating the estimator type 
- ties.method
- method for ties 
Value
a list with elements
- pquantile level for estimation
- etaestimated coefficient of tail dependence \(\eta\)
- eta_sdestimated standard error of \(\eta\)
- k1parameter of the tail expansion
- patproportion of observations above the threshold
- lambdatail dependence coefficient (sic)
- tailprobtail probability, if- ptailis provided
Note
EXPERIMENTAL. The numerical optimization of the likelihood surface is difficult, as the function is ill-behaved. Visual inspection of estimates is necessary to check for non-convergence.
Examples
d <- 2
rho <- 0.9
Sigma <- matrix(rho, d, d) + diag(1 - rho, d)
eta_true <- 1/sum(Sigma)
data <- mev::mvrnorm(
   n = 1e4,
   mu = rep(0, d),
  Sigma = Sigma)
q <- seq(0.95, 0.995, by = 0.005)
taildep <- kjtail(data = data, q = q)
with(taildep,
 plot(x = 1-pat,
      y = eta,
      ylim = c(0,1),
      panel.first = {abline(h = (1+rho)/2)}))
