rvinecopulib (version 0.6.3.1.1)

bicop_distributions: Bivariate copula distributions

Description

Density, distribution function, random generation and h-functions (with their inverses) for the bivariate copula distribution.

Usage

dbicop(u, family, rotation, parameters, var_types = c("c", "c"))
pbicop(u, family, rotation, parameters, var_types = c("c", "c"))
rbicop(n, family, rotation, parameters, qrng = FALSE)
hbicop(
  u,
  cond_var,
  family,
  rotation,
  parameters,
  inverse = FALSE,
  var_types = c("c", "c")
)

Value

dbicop() gives the density, pbicop() gives the distribution function, rbicop() generates random deviates, and hbicop() gives the h-functions (and their inverses).

The length of the result is determined by n for rbicop(), and the number of rows in u for the other functions.

The numerical arguments other than n are recycled to the length of the result.

Arguments

u

evaluation points, a matrix with at least two columns, see Details.

family

the copula family, a string containing the family name (see bicop for all possible families).

rotation

the rotation of the copula, one of 0, 90, 180, 270.

parameters

a vector or matrix of copula parameters.

var_types

variable types, a length 2 vector; e.g., c("c", "c") for both continuous (default), or c("c", "d") for first variable continuous and second discrete.

n

number of observations. If `length(n) > 1``, the length is taken to be the number required.

qrng

if TRUE, generates quasi-random numbers using the bivariate Generalized Halton sequence (default qrng = FALSE).

cond_var

either 1 or 2; cond_var = 1 conditions on the first variable, cond_var = 2 on the second.

inverse

whether to compute the h-function or its inverse.

Details

See bicop for the various implemented copula families.

The copula density is defined as joint density divided by marginal densities, irrespective of variable types.

H-functions (hbicop()) are conditional distributions derived from a copula. If \(C(u, v) = P(U \le u, V \le v)\) is a copula, then $$h_1(u, v) = P(V \le v | U = u) = \partial C(u, v) / \partial u,$$ $$h_2(u, v) = P(U \le u | V = v) = \partial C(u, v) / \partial v.$$ In other words, the H-function number refers to the conditioning variable. When inverting H-functions, the inverse is then taken with respect to the other variable, that is v when cond_var = 1 and u when cond_var = 2.

Discrete variables

When at least one variable is discrete, more than two columns are required for u: the first \(n \times 2\) block contains realizations of \(F_{X_1}(x_1), F_{X_2}(x_2)\). The second \(n \times 2\) block contains realizations of \(F_{X_1}(x_1^-), F_{X_1}(x_1^-)\). The minus indicates a left-sided limit of the cdf. For, e.g., an integer-valued variable, it holds \(F_{X_1}(x_1^-) = F_{X_1}(x_1 - 1)\). For continuous variables the left limit and the cdf itself coincide. Respective columns can be omitted in the second block.

See Also

bicop_dist(), bicop()

Examples

## evaluate the copula density
dbicop(c(0.1, 0.2), "clay", 90, 3)
dbicop(c(0.1, 0.2), bicop_dist("clay", 90, 3))

## evaluate the copula cdf
pbicop(c(0.1, 0.2), "clay", 90, 3)

## simulate data
plot(rbicop(500, "clay", 90, 3))

## h-functions
joe_cop <- bicop_dist("joe", 0, 3)
# h_1(0.1, 0.2)
hbicop(c(0.1, 0.2), 1, "bb8", 0, c(2, 0.5))
# h_2^{-1}(0.1, 0.2)
hbicop(c(0.1, 0.2), 2, joe_cop, inverse = TRUE)

## mixed discrete and continuous data
x <- cbind(rpois(10, 1), rnorm(10, 1))
u <- cbind(ppois(x[, 1], 1), pnorm(x[, 2]), ppois(x[, 1] - 1, 1))
pbicop(u, "clay", 90, 3, var_types = c("d", "c"))