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Joint density under a bivariate copula

dcopula.Rd

Computes the joint density (or log-density) of a bivariate distribution constructed from two arbitrary margins combined with a specified copula.

Usage

dcopula(d1, d2, p1, p2, copula = cgaussian(0), log = FALSE)

Arguments

d1, d2

Marginal density values. If log = TRUE, supply the log-density. If log = FALSE, supply the raw density.

p1, p2

Marginal CDF values. Need not be supplied on log scale.

copula

A function of two arguments (u, v) returning the log copula density \(\log c(u,v)\). You can either construct this yourself or use the copula constructors available (see details)

log

Logical; if TRUE, return the log joint density. In this case, d1 and d2 must be on the log scale.

Value

Joint density (or log-density) under the bivariate copula.

Details

The joint density is $$f(x,y) = c(F_1(x), F_2(y)) \, f_1(x) f_2(y),$$ where \(F_i\) are the marginal CDFs, \(f_i\) are the marginal densities, and \(c\) is the copula density.

The marginal densities d1, d2 and CDFs p1, p2 must be differentiable for automatic differentiation (AD) to work.

Available copula constructors are:

Examples

# Normal + Exponential margins with Gaussian copula
x <- c(0.5, 1); y <- c(1, 2)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dexp(y, 2, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pexp(y, 2)
dcopula(d1, d2, p1, p2, copula = cgaussian(0.5), log = TRUE)
#> [1] -2.818014 -4.809862

# Normal + Beta margins with Clayton copula
x <- c(0.5, 1); y <- c(0.2, 0.8)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dbeta(y, 2, 1, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pbeta(y, 2, 1)
dcopula(d1, d2, p1, p2, copula = cclayton(2), log = TRUE)
#> [1] -3.8093660 -0.1671136

# Normal + Beta margins with Gumbel copula
x <- c(0.5, 1); y <- c(0.2, 0.4)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dbeta(y, 2, 1, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pbeta(y, 2, 1)
dcopula(d1, d2, p1, p2, copula = cgumbel(1.5), log = TRUE)
#> [1] -1.807264 -1.274899

# Normal + Exponential margins with Frank copula
x <- c(0.5, 1); y <- c(1, 2)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dexp(y, 2, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pexp(y, 2)
dcopula(d1, d2, p1, p2, copula = cfrank(2), log = TRUE)
#> [1] -2.705857 -4.370568