Fitting bivariate/multivariate MGL and MGL-EV copula regression models.

MGL.reg is used to fit bivariate MGL and MGL-EV copula regression models for two continuous variables.

MGL.reg(
  U,
  X,
  copula = c("MGL", "MGL180", "MGL-EV", "MGL-EV180", "Gumbel", "MGB2"),
  hessian = TRUE,
  initpar,
  ...
)

Arguments

U

two-dimensional matrix with values in \([0,1]\).

X

design matrix

copula

'MGL', 'MGL180', "MGL-EV", "MGL-EV180", "Gumbel", "MGB2".

hessian

Logical. Should a numerically differentiated Hessian matrix be returned?

initpar

Initial values for the parameters to be optimized over.

...

additional arguments, see optim 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 optim function. Y1 and Y2 are both continuous variables.

copula: "MGL180" and "MGLEV180" denote the survival MGL and survival MGL-EV copula respectively.

• For "Gumbel" regression model, the copula parameter $$\delta_i = \exp(X_i^T\beta) + 1,$$ where \({X}_{i}=(1,x_{i1}...,x_{ik})\) denotes the vector of covariates and \(\beta\) is the vector of coefficients to be estimated in the copula regression.

• For "MGL", "MGL180", "MGL-EV", "MGL-EV180" regression model, the copula parameter $$\delta_i = \exp(X_i^T\beta).$$

• For "MGB2", the copula parameter $$\q_i = \exp(X_i^T\beta)$$ and \((p_1,p_2)\) remain to be constant.

• Note that the regression modelling can be extended to the high-dimensional case when copula is "MGL180" and "MGL".

Examples

# 10-dimensional regression models
set.seed(111)
d <- 10
n <- 1000 # sample size
beta.true <- c(-0.6, 0.5, 0.2) # true regression coefficients
x1 <- rnorm(n, 0, 1)
x2 <- rnorm(n, 0, 1)
X <- model.matrix(~ x1 + x2) # design matrix
delta.sim <- as.vector(exp(X%*%beta.true)) # true copula parameters
Usim <- matrix(0, nrow = n, ncol = d)
for (i in 1:n){
  Usim[i, ] <- rcMGL.multi(n = 1, d = d, pars = delta.sim[i])
}
m.MGLMGA <- MGL.reg(U = Usim, copula = "MGL",
                       X = X, method = "Nelder-Mead",
                       initpar = c(-0.32, 0.001, 0.001)
)
m.MGLMGA
#> $loglike
#> [1] 729.0535
#> 
#> $copula
#> $copula$name
#> [1] "MGL"
#> 
#> 
#> $estimates
#> [1] -0.6091929  0.4555460  0.1676800
#> 
#> $se
#> [1] 0.03939535 0.03466884 0.03400156
#> 
#> $hessian
#>           [,1]       [,2]      [,3]
#> [1,] -913.5135  -544.5927 -119.1106
#> [2,] -544.5927 -1159.3980  -22.2301
#> [3,] -119.1106   -22.2301 -883.3534
#> 
#> $AIC
#> [1] -1452.107
#> 
#> $BIC
#> [1] -1437.384
#>