Fitting bivariate MGL copula models for mixed data

MGL.mle.mixed is used to fit bivariate mixed copula regression models via maximum likelihood (ML) method for continuous and semi-continuous variables.

MGL.mle.mixed(
  obs,
  U,
  U_,
  f,
  copula = c("MGL", "MGL180", "MGL-EV", "MGL-EV180", "Gumbel", "Normal", "MGB2", "t"),
  umin,
  hessian = TRUE,
  initpar,
  ...
)

Arguments

obs

two-dimensional matrix for loss observations (y1, y2).

U

two-dimenstional matrix for pseudo copula data with values in \([0,1]\) for (F(y1), F(y2)).

U_

two-dimensional matrix for pseudo copula data for the data (F(y1), F(y2-1)).

f

values of the density function for marginal distribution.

copula

copula 'MGL', 'MGL180', "MGL-EV", "MGL-EV180", "MGB2", "Normal" , "t".

umin

threshold value used in the semi-continuous data.

hessian

Logical. Should a numerically differentiated Hessian matrix be returned?

initpar

Initial values for the parameters to be optimized over.

...

additional arguments, see nlm for more details.

Value

A list containing the following components:

  • loglike: the value of the estimated maximum of the loglikelihood function.

  • copula: the name of the fitted copula. "MGL180" and "MGL-EV180" denote the survival MGL and MGL-EV copula respectively.

  • estimates: the point at which the maximum value of the loglikelihood is obtained.

  • se: the standard errors of the estimators.

  • AIC, BIC: the goodness fit of the regression models.

  • hessian: the hessian at the estimated maximum of the loglikelihood (if requested).

Details

The estimation method is performed via nlm function. Y1: continuous variable Y2: semi-continuous variable when Y2>umin, it is continuous and Y2<=umin is discrete.

For a portfolio of \(n\) observations \((y_{i1},y_{i2}; \; i=1,\ldots,n)\), the joint density function of \((Y_1,Y_2)\) can be written as $$ f_{Y_{1},Y_2}(y_{i1},y_{i2})=\begin{cases} f_{Y_1}(y_{i1})[ h_{2|1}(F_{Y_{1}}(y_{i1}),F_{Y_{2}}(y_{i2})) - h_{2|1}(F_{Y_{1}}(y_{i1}),F_{Y_{2}}(y_{i2}-1)) ], & y_{i2}\le umin,\\ f_{Y_1}(y_{i1})f_{Y_2}(y_{i2})c(F_{Y_{1}}(y_{i1}), F_{Y_{2}}(y_{i2})), & y_{i2} > umin, \end{cases} $$ where the density \(f_{Y_j}(\cdot)\) and cdf \(F_{Y_j}(\cdot)\) of the marginal distributions (\(i=1,2\)) are specified respectively. Here \(h_{2|1}(u_1, u_2)=\partial C(u_1,u_2)/\partial u_1\) is the \(h\)-function of bivariate copula.

copula:

  • "MGB2" is multivariate GB2.

  • "Normal" and "t" denote the Gaussian copula and Student-t copula respectively.

  • "MGL" and "MGL-EV" denote the MGL and MGL-EV copula respectively.

  • "MGL180" and "MGL-EV180" denote the survival MGL and survival MGL-EV copula respectively.

  • "Gumbel" is Gumbel copula.#'

Examples

library(rMGLReg)
# load the Chinese earthquake data set
u <- cbind(earthqCHI$u1, earthqCHI$u2) # cdf of marginal distribution
u_ <- cbind(earthqCHI$u1, earthqCHI$u2_) # cdf of marginal distribution for Y1 and Y2 - 1
y <- cbind(earthqCHI$y1, earthqCHI$y2)  # observations
f <- cbind(earthqCHI$f1, earthqCHI$f2) # pdf of marginal distribution
obs <- y
U <- u
U_ <- u_
umin <- 20
m.MGLMGA180 <- MGL.mle.mixed(obs = y, U = U, U_ = U_,
umin = umin, f = f,
copula = "MGL180",
method = "L-BFGS-B", initpar = c(2))
m.MGLMGA180
#> $loglike
#> [1] -3009.457
#> 
#> $copula
#> $copula$name
#> [1] "MGL180"
#> 
#> 
#> $estimates
#> [1] 2.762763
#> 
#> $se
#> [1] 0.2469525
#> 
#> $hessian
#>           [,1]
#> [1,] -16.39732
#> 
#> $AIC
#> [1] 6020.915
#> 
#> $BIC
#> [1] 6024.588
#>