beta.Blomqvist {copula}

Sample and Population Version of Blomqvist's Beta for Archimedean Copulas

Description

Compute the population (beta.()) and sample (betan()) version of Blomqvist's beta for an Archimedean copula.

See the reference below for definitions and formulas.

Usage

beta.(cop, theta, d, scaling=FALSE)
betan(u, scaling=FALSE)

Arguments

cop

an Archimedean copula (of dimension d) to be estimated.

theta

copula parameter.

d

dimension.

scaling

logical, if true, the factors 2^(d-1)/(2^(d-1)-1) and 2^(1-d) in Blomqvist's beta are omitted.

u

For betan: (n\times d)-matrix of d-dimensional observations distributed according to the copula.

Value

beta.:

a number, being the population version of Blomqvist's beta for the corresponding Archimedean copula;

betan:

a number, being the sample version of Blomqvist's beta for the given data.

References

Schmid and Schmidt (2007), Nonparametric inference on multivariate versions of Blomqvist's beta and related measures of tail dependence, Metrika 66, 323–354.

See Also

acopula

Examples

beta.(copGumbel, 2.5, d = 5)

d.set <- c(2:6, 8, 10, 15, 20, 30)
cols <- adjustcolor(colorRampPalette(c("red", "orange", "blue"),
                                     space = "Lab")(length(d.set)), 0.8)
## AMH:
for(i in seq_along(d.set))
   curve(Vectorize(beta.,"theta")(copAMH, x, d = d.set[i]), 0, .999999,
         main = "Blomqvist's beta(.) for  AMH",
         xlab = quote(theta), ylab = quote(beta(theta, AMH)),
         add = (i > 1), lwd=2, col=cols[i])
mtext("NB:  d=2 and d=3 are the same")
legend("topleft", paste("d =",d.set), bty="n", lwd=2, col=cols)

## Gumbel:
for(i in seq_along(d.set))
   curve(Vectorize(beta.,"theta")(copGumbel, x, d = d.set[i]), 1, 10,
         main = "Blomqvist's beta(.) for  Gumbel",
         xlab = quote(theta), ylab = quote(beta(theta, Gumbel)),
         add=(i > 1), lwd=2, col=cols[i])
legend("bottomright", paste("d =",d.set), bty="n", lwd=2, col=cols)

## Clayton:
for(i in seq_along(d.set))
   curve(Vectorize(beta.,"theta")(copClayton, x, d = d.set[i]), 1e-5, 10,
         main = "Blomqvist's beta(.) for  Clayton",
         xlab = quote(theta), ylab = quote(beta(theta, Gumbel)),
         add=(i > 1), lwd=2, col=cols[i])
legend("bottomright", paste("d =",d.set), bty="n", lwd=2, col=cols)

## Joe:
for(i in seq_along(d.set))
   curve(Vectorize(beta.,"theta")(copJoe, x, d = d.set[i]), 1, 10,
         main = "Blomqvist's beta(.) for  Joe",
         xlab = quote(theta), ylab = quote(beta(theta, Gumbel)),
         add=(i > 1), lwd=2, col=cols[i])
legend("bottomright", paste("d =",d.set), bty="n", lwd=2, col=cols)

## Frank:
for(i in seq_along(d.set))
   curve(Vectorize(beta.,"theta")(copFrank, x, d = d.set[i]), 1e-5, 50,
         main = "Blomqvist's beta(.) for  Frank",
         xlab = quote(theta), ylab = quote(beta(theta, Gumbel)),
         add=(i > 1), lwd=2, col=cols[i])
legend("bottomright", paste("d =",d.set), bty="n", lwd=2, col=cols)

## Shows the numeric problems:
curve(Vectorize(beta.,"theta")(copFrank, x, d = 29), 35, 42, col="violet")