The Pareto Distribution

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.

Value

rpareto generates random deviates.

dpareto gives the density.

ppareto gives the distribution function.

qpareto gives the quantile function.

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 \(k < \xi\).

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)