Cardioid Distribution

Density, distribution function, quantile function and random generation for the cardioid distribution.

dcard(x, mu, rho, log = FALSE)
pcard(q, mu, rho, lower.tail = TRUE, log.p = FALSE)
qcard(p, mu, rho, tolerance = 1e-07, maxits = 500,
      lower.tail = TRUE, log.p = FALSE)
rcard(n, mu, rho, ...)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. Same as in runif.

mu, rho

See cardioid for more information.

tolerance, maxits, ...

The first two are control parameters for the algorithm used to solve for the roots of a nonlinear system of equations; tolerance controls for the accuracy and maxits is the maximum number of iterations. rcard calls qcard so the ... can be used to vary the two arguments.

log

Logical. If log = TRUE then the logarithm of the density is returned.

lower.tail, log.p

Same meaning as in pnorm or qnorm.

Details

See cardioid, the VGAM family function for estimating the two parameters by maximum likelihood estimation, for the formula of the probability density function and other details.

Value

dcard gives the density, pcard gives the distribution function, qcard gives the quantile function, and rcard generates random deviates.

Author

Thomas W. Yee and Kai Huang

Note

Convergence problems might occur with rcard.

See also

cardioid.

Examples

if (FALSE) { # \dontrun{
mu <- 4; rho <- 0.4; x <- seq(0, 2*pi, len = 501)
plot(x, dcard(x, mu, rho), type = "l", las = 1, ylim = c(0, 1),
     ylab = paste("[dp]card(mu=", mu, ", rho=", rho, ")"),
     main = "Blue is density, orange is the CDF", col = "blue",
     sub = "Purple lines are the 10,20,...,90 percentiles")
lines(x, pcard(x, mu, rho), col = "orange")

probs <- seq(0.1, 0.9, by = 0.1)
Q <- qcard(probs, mu, rho)
lines(Q, dcard(Q, mu, rho), col = "purple", lty = 3, type = "h")
lines(Q, pcard(Q, mu, rho), col = "purple", lty = 3, type = "h")
abline(h = c(0,probs, 1), v = c(0, 2*pi), col = "purple", lty = 3)
max(abs(pcard(Q, mu, rho) - probs))  # Should be 0
} # }