nvmix (version 0.0-8)

dependencemeasures: Dependence Measures for grouped normal variance mixture copulas

Description

Computation of rank correlation coefficients Spearman's rho and Kendall's tau for grouped normal variance mixture copulas as well as computation of the (lower and upper) tail dependence coefficient of a grouped t copula.

Usage

corgnvmix(scale, qmix, method = c("kendall", "spearman"), groupings = 1:2, 
          ellip.kendall = FALSE, control = list(), verbose = TRUE, ...)
lambda_gStudent(df, scale, control = list(), verbose = TRUE)

Arguments

scale

\(n\)-vector giving the \(\rho\) parameters of the copula.

qmix

specification of the mixing variables; see pgnvmix().

method

character indicating if Spearman's rho or Kendall's tau is to be computed.

groupings

vector specifying the grouping structure; either rep(1, 2) (ungrouped) or 1:2 (grouped case).

ellip.kendall

logical if the formula for Kendalll's tau for elliptical copulas shall be used; see details below.

df

either scalar or \(2\)-vector giving the degrees-of- freedoms for the t copula; if provided as scalar, the copula is an (ungrouped) t copula and lambda_gStudent() uses a closed formula.

control

list specifying algorithm specific parameters; see get_set_param().

verbose

logical indicating whether a warning is given if the required precision has not been reached.

…

additional arguments (for example, parameters) passed to the underlying mixing distribution when qmix is a character string or function.

Value

lambda_gStudent() and corgnvmix() return a numeric \(n\)-vector with the computed dependence measure with corresponding attributes "abs. error" and "rel. error"(error estimates of the RQMC estimator) and "numiter" (number of iterations).

Details

For grouped normal variance mixture copulas, including the grouped t, there is no closed formula for Kendall's tau and Spearman's rho. The function corgnvmix() approximates these dependence measures by numerically approximating an integral representation for these measures.

If no grouping is present (i.e., when groupings = rep(1, 2)), the copula is an elliptical copula for which the formula \(\tau = 2asin(\rho)/pi\) holds. This formula holds only approximately in the grouped case; the quality of the approximation depends on how different the mixing variables for the two components are. When the mixing distributions are not too far apart and when the copula parameter is not close to 1, this approximation is ``very accurate``, as demonstrated in Daul et al (2003).

In the ungrouped case, lambda_gStudent() computes the tail dependence coefficient \(lambda\) based on the known formula 2 * pt( -sqrt( (df + 1)*(1 - rho) / (1 + rho)), df = df + 1) for the tail dependence coefficient of a t copula.

In the grouped case, RQMC methods are used to efficiently approximate the integral given in Eq. (26) of Luo and Shevchenko (2010).

References

Hintz, E., Hofert, M. and Lemieux, C. (2020), Grouped Normal Variance Mixtures. Risks 8(4), 103.

Hintz, E., Hofert, M. and Lemieux, C. (2021), Normal variance mixtures: Distribution, density and parameter estimation. Computational Statistics and Data Analysis 157C, 107175.

Luo, X. and Shevchenko, P. (2010). The t copula with multiple parameters of degrees of freedom: bivariate characteristics and application to risk management. Quantitative Finance 10(9), 1039-1054.

Daul, S., De Giorgi, E. G., Lindskog, F. and McNeil, A (2003). The grouped t copula with an application to credit risk. Available at SSRN 1358956.

See Also

dgStudentcopula(), pgStudentcopula(), rgStudentcopula()

Examples

# NOT RUN {
### Examples for corgnvmix() ###################################################

## Create a plot displaying Spearman's rho for a grouped t copula as a function 
## of the copula parameter for various choices of the degrees-of-freedom 
qmix <- "inverse.gamma"
df <- matrix( c(1, 2, 1, 5, 1, Inf), ncol = 2, byrow = TRUE)
l.df <- nrow(df)
scale <- seq(from = 0, to = 1, length.out = 99)
set.seed(1) # for reproducibility 
kendalls <- sapply(seq_len(l.df), function(i) 
   corgnvmix(scale, qmix = qmix, method = "kendall", df = df[i, ]))
## Include the elliptical approximation (exact when df1 = df2) 
kendall_ell <- corgnvmix(scale, method = "kendall", ellip.kendall = TRUE)
## Plot
lgnd <- character(l.df + 1)
lgnd[1] <- "elliptical (equal df)"
plot(NA, xlim = c(0, 1), ylim = c(0, 1), xlab = expression(rho),
     ylab = "Kendall's tau")
lines(scale, kendall_ell, lty = 1)
for(i in 1:l.df){
   lines(scale, kendalls[, i], col = i + 1, lty = i + 1)
   lgnd[i+1] <- paste0("df1 = ", df[i, 1], ", df2 = ", df[i, 2])
} 
legend("topleft", lgnd, col = 1:(l.df + 1), lty = 1:(l.df + 1), bty = 'n')


### Examples for lambda_gStudent() #############################################

## Create a plot displaying 'lambda' as a function of the copula parameter
## for various choices of the degrees-of-freedom 
df <- c(3, 6, 9)
df_ <- list( rep(df[1], 2), rep(df[2], 2), rep(df[3], 2), # ungrouped 
             c(df[1], df[2]), c(df[1], df[3]), c(df[2], df[3])) # grouped  
l.df_ <- length(df_)
scale <- seq(from = -0.99, to = 0.99, length.out = 112) # scale parameters
set.seed(1) # for reproducibilty
lambdas <-
   sapply(seq_len(l.df_), function(i) lambda_gStudent(df_[[i]], scale = scale))
lgnd <- character(length(df_))
plot(NA, xlim = range(scale), ylim = range(lambdas), xlab = expression(rho), 
     ylab = expression(lambda))
for(i in seq_len(l.df_)){
   lines(scale, lambdas[, i], col = i, lty = i)
   lgnd[i] <- if(df_[[i]][1] == df_[[i]][2]) paste0("df = ", df_[[i]][1]) else
      paste0("df1 = ", df_[[i]][1], ", df2 = ", df_[[i]][2])
}
legend("topleft", lgnd, col = seq_len(l.df_), lty = seq_len(l.df_),
       bty = 'n')
## If called with 'df' a 1-vector, closed formula for lambda is used => check
lambda.true <- sapply(1:3, function(i) lambda_gStudent(df_[[i]][1], scale = scale))
stopifnot(max(abs( lambda.true - lambdas[, 1:3])) < 4e-4)
# }