R: Cardioid Distribution

Card {VGAM} | R Documentation

## Cardioid Distribution

### Description

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

### Usage

    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.

### Note

Convergence problems might occur with `rcard`.

### Author(s)

Thomas W. Yee and Kai Huang

### See Also

`cardioid`.

### Examples

    ## Not run:
    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), col = "blue",
         ylab = paste("[dp]card(mu=", mu, ", rho=", rho, ")"),
         main = "Blue is density, orange is cumulative distribution function",
         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

    ## End(Not run)

[Package VGAM version 1.0-3 Index]