MGL-fitting

Case I - modelling Danish fire insurance data without covariates

  • We fit the bivariate copula and regression models to the Danish fire insurance data set which was collected from the Copenhagen Reinsurance Company and comprises 2167 fire losses over the period 1980-1990.

  • The claims have been adjusted for inflation to reflect 1985 values and are expressed in millions of Danish Krone and can be found in the R package: fitdistrplus.

  • The total claims in the multivariate data set is divided into building loss, contents loss and profit loss.

  • The pseudo-copula data \((u_{i1}, u_{i2})\) (\(i=1,...n\)) is based on the kernel smoothing method.

Load the data 

library(rMGLReg)
library(fitdistrplus)
library(splines)
library(snpar)
  data("danishmulti")
  dt <- data.table::data.table(danishmulti)
  dtnew <- dt[Building>0&Contents>0]
  y1 <- dtnew$Building
  y2 <- dtnew$Contents
  y <- cbind(y1, y2)
  
  # empirical cdf
  u1 <- snpar::kde(y[,1], kernel = "gaus", 
             xgrid = y[,1],
             h = 0.2)$Fhat
  u2 <- snpar::kde(y[,2], kernel = "gaus", 
             xgrid = y[,2],
             h = 0.2)$Fhat
  U <- cbind(u1, u2) # two-dim 
  plot(U); plot(log(U))

  • We report the estimation results, AIC and BIC values of the survival MGL and the survival MGL-EV copula, along with four other families of copulas with positive upper tail indices, the MGB2 copula, the Gumbel copula, the Student \(t\) copula, and the Gaussian copula.

  • The Gumbel copula is an extreme-value copula and also belongs to the Archimedean family, whereas the Student \(t\) copula and Gaussian copula belongs to the elliptical copulas.

fitting the copula models 

m.norm <- MGL.mle(U = U,
                        copula  = "Normal", hessian = TRUE,
                        initpar = 0.5)

m.t <- MGL.mle(U = U,
                     copula  = "t", hessian = TRUE,
                     initpar = c(0.5, 4))
m.gumbel <- MGL.mle(U = U,
                     copula  = "Gumbel", hessian = TRUE,
                          initpar = c(2))
m.MGLMGA180 <- MGL.mle(U = U,
                    copula  = "MGL180", hessian = TRUE,
                             initpar = c(1))
m.MGB2 <- MGL.mle(U = U,
                      copula  = "MGB2", hessian = TRUE,
                        initpar = c(0.1, 2, 0.4))
m.MGLEV180 <- MGL.mle(U = U,
                         copula  = "MGL-EV180", hessian = TRUE,
                         initpar = c(2))
recap <- function(x){
  res <- c(alpha = x$estimates,
           se = x$se,
           loglike = x$loglike,
           AIC = x$AIC, BIC = x$BIC)
  if(length(res) < 6)
    res <- c(res[1], NA, NA,res[2], NA, NA, res[3:5])
  if (length(res) > 6 & length(res) < 9)
    res <- c(res[1:2], NA, res[3:4], NA, res[5:7])
  res <- as.matrix(res)
  colnames(res) <- x$copula$name
  res}
res.all <- round(cbind(recap(m.norm),
                       recap(m.t),
                       recap(m.gumbel),
                       recap(m.MGLMGA180),
                       recap(m.MGB2),
                       recap(m.MGLEV180)
), 4)
out.com <- t(res.all)
out.com <- out.com[order(out.com[,9], decreasing = T),]
knitr::kable(out.com, digits = 2)