Pareto: The Pareto Distribution

In reservr: Fit Distributions and Neural Networks to Censored and Truncated Data

The Pareto Distribution

Description

These functions provide information about the Pareto distribution. dpareto gives the density, ppareto gives the distribution function, qpareto gives the quantile function and rpareto generates random deviates.

Usage

rpareto(n = 1L, shape = 0, scale = 1)

dpareto(x, shape = 1, scale = 1, log = FALSE)

ppareto(q, shape = 1, scale = 1, lower.tail = TRUE, log.p = FALSE)

qpareto(p, shape = 1, scale = 1, lower.tail = TRUE, log.p = FALSE)

Arguments

n	integer number of observations.
shape	shape parameter (must be positive).
scale	scale parameter (must be positive).
x, q	vector of quantiles.
log, log.p	logical; if TRUE, probabilities/densities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are P(X \le x), otherwise P(X > x).
p	vector of probabilities.

Details

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

The Pareto distribution with scale \theta and shape \xi has density

f(x) = \xi \theta^\xi / (x + \theta)^(\xi + 1)

The support is x \ge 0.

The Expected value exists if \xi > 1 and is equal to

E(X) = \theta / (\xi - 1)

k-th moments exist in general for k < \xi.

Value

rpareto generates random deviates.

dpareto gives the density.

ppareto gives the distribution function.

qpareto gives the quantile function.

References

https://en.wikipedia.org/wiki/Pareto_distribution - named Lomax therein.

Examples

x <- rpareto(1000, shape = 10, scale = 5)
xx <- seq(-1, 10, 0.01)
hist(x, breaks = 100, freq = FALSE, xlim = c(-1, 10))
lines(xx, dpareto(xx, shape = 10, scale = 5))

plot(xx, dpareto(xx, shape = 10, scale = 5), type = "l")
lines(xx, dpareto(xx, shape = 3, scale = 5), col = "red", lwd = 2)

plot(xx, dpareto(xx, shape = 10, scale = 10), type = "l")
lines(xx, dpareto(xx, shape = 10, scale = 5), col = "blue", lwd = 2)
lines(xx, dpareto(xx, shape = 10, scale = 20), col = "red", lwd = 2)

reservr documentation built on June 24, 2024, 5:10 p.m.