GeneralizedPareto: The Generalized Pareto Distribution

In ArcadeAntics/essentials: Essential Functions not Included in Base R

The Generalized Pareto Distribution

Description

Density, distribution function, quantile function and random generation for the generalized pareto distribution with location equal to location, scale equal to scale and shape equal to shape.

Usage

dgpd(x, location = 0, scale = 1, shape = 0, log = FALSE)
pgpd(q, location = 0, scale = 1, shape = 0, lower.tail = TRUE, log.p = FALSE)
qgpd(p, location = 0, scale = 1, shape = 0, lower.tail = TRUE, log.p = FALSE)
rgpd(n, location = 0, scale = 1, shape = 0)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required.
location	vector of locations.
scale	vector of scale.
shape	vector of shapes.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are P[X \le x], otherwise P[X > x].

Details

If location, scale or shape are not specified they assume the default values of 0, 1 and 0, respectively.

The generalized pareto distribution with location \mu, scale \sigma > 0 and shape \xi has density

f(s) = \frac{1}{\sigma} (1 + \xi s)^{-(1 + 1/\xi)}

for s \ge 0 when \xi \ge 0, and for 0 \le s \le -1/\xi when \xi < 0, where s = \frac{(x - \mu)}{\sigma}. In the limit \xi → 0, the density simplifies to

f(s) = \frac{1}{\sigma} \exp(-s)

Value

dgpd gives the density, pgpd gives the distribution function, qgpd gives the quantile function, and rgpd generates random deviates.

The length of the result is determined by n for rgpd, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

For scale = 0 this gives the limit as \sigma decreases to 0, a point mass at \mu. scale < 0 is an error and returns NaN.

Examples

scales <- c( 1,  1,  1,  2,  2,  2)
shapes <- c( 1,  5, 20,  1,  5, 20)
legend.text <- as.expression(essentials::plapply(list(scales, shapes),
    function(scale, shape) {
        call("list",
             call("==", as.symbol("sigma"), scale),
             call("==", as.symbol("xi"), shape))
    }))
cols <- c("red", "green3", "blue", "red", "green3", "blue")
ltys <- c("solid", "solid", "solid", "dashed", "dashed", "dashed")
x <- seq.int(0, 5, length.out = 1001)


# we use plapply here instead of lapply because
# plapply allows us to name the looping arguments
ys <- essentials::plapply(
    list(scale = scales, shape = shapes),
    essentials::dgpd,
    x = x
)
graphics::par(mar = c(4.9, 4.5, 2.1, 0.4))
graphics::plot(
    xlim = range(x), ylim = range(ys),
    panel.first = graphics::grid(col = "gray69"),
    x = NA_real_, y = NA_real_,
    xlab = "x", ylab = ~f(list(x, mu, sigma, xi)),
    main = "Probability density function",
    bty = "L"
)
essentials::mfor(y, col, lty, list(ys, cols, ltys), {
    graphics::lines(x, y, col = col, lty = lty, lwd = 2)
})
graphics::legend(
    x = "topright",
    legend = legend.text,
    col = cols,
    lwd = 2,
    lty = ltys,
    bty = "n"
)
graphics::title(sub = ~"All with" ~ mu == 0, adj = 1)





ys <- essentials::plapply(
    list(scale = scales, shape = shapes),
    essentials::pgpd,
    q = x
)
graphics::plot(
    xlim = range(x), ylim = c(0, 1),
    panel.first = graphics::grid(col = "gray69"),
    x = NA_real_, y = NA_real_,
    xlab = "x", ylab = ~F(list(x, mu, sigma, xi)),
    main = "Cumulative probability function",
    bty = "L"
)
essentials::mfor(y, col, lty, list(ys, cols, ltys), {
    graphics::lines(x, y, col = col, lty = lty, lwd = 2)
})
graphics::legend(
    x = "topleft",
    legend = legend.text,
    col = cols,
    lwd = 2,
    lty = ltys,
    bty = "n"
)
graphics::title(sub = ~"All with" ~ mu == 0, adj = 1)

ArcadeAntics/essentials documentation built on Nov. 7, 2024, 4:33 p.m.