% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/distributions.R
\name{gumbel_biv}
\alias{gumbel_biv}
\alias{dgumbel_biv}
\alias{sgumbel_biv}
\alias{pgumbel_biv}
\alias{rgumbel_biv}
\title{Gumbel Bivariate Exponential Distribution}
\usage{
dgumbel_biv(x1, x2, k1 = 1, k2 = 1, theta = 0, log.p = FALSE)

sgumbel_biv(x1, x2, k1 = 1, k2 = 1, theta = 0, log.p = FALSE)

pgumbel_biv(x1, x2, k1 = 1, k2 = 1, theta = 0)

rgumbel_biv(n, k1 = 1, k2 = 1, theta = 0)
}
\arguments{
\item{x1, x2}{Non-negative numeric values or vectors.}

\item{k1, k2}{Positive rate parameters.}

\item{theta}{Non-negative association parameter; must satisfy
\eqn{0 \le \theta \le k_1 k_2}.}

\item{log.p}{Logical; if \code{TRUE} probabilities are given as
\eqn{\log(p)}.}

\item{n}{Number of random observations.}
}
\value{
\describe{
  \item{\code{dgumbel_biv}}{Numeric vector of density values.}
  \item{\code{pgumbel_biv}}{Numeric vector of joint CDF values.}
  \item{\code{sgumbel_biv}}{Numeric vector of joint survival function values.}
  \item{\code{rgumbel_biv}}{A two-column matrix with columns \code{X1} and
    \code{X2} containing the simulated observations.}
}
}
\description{
Density, distribution (survival) function, and random generation for the
Gumbel bivariate exponential distribution with parameters \eqn{k_1},
\eqn{k_2}, and association parameter \eqn{\theta}.

The joint survival function is
\deqn{\bar{F}(x_1,x_2) = \exp(-k_1 x_1 - k_2 x_2 - \theta x_1 x_2),
  \quad x_1,x_2 > 0,\; k_1,k_2 > 0,\; 0 \le \theta \le k_1 k_2.}
}
\examples{
# Survival function
sgumbel_biv(1, 2, k1 = 1, k2 = 1, theta = 0.5)

# Density
dgumbel_biv(0.5, 0.5, k1 = 1, k2 = 1.5, theta = 0.3)

# Random sample
set.seed(42)
dat <- rgumbel_biv(100, k1 = 1, k2 = 1, theta = 0.5)
head(dat)

}
\references{
Gumbel E.J. (1960). Bivariate exponential distributions. *Journal of the
American Statistical Association*, 55(292), 698--707.

Jayalekshmi S., Rajesh G., Nair N.U. (2022).
\doi{10.1080/03610926.2022.2085874}
}
