Pareto {actuar}R Documentation

The Pareto Distribution

Description

Density function, distribution function, quantile function, random generation, raw moments and limited moments for the Pareto distribution with parameters shape and scale.

Usage

dpareto(x, shape, scale, log = FALSE)
ppareto(q, shape, scale, lower.tail = TRUE, log.p = FALSE)
qpareto(p, shape, scale, lower.tail = TRUE, log.p = FALSE)
rpareto(n, shape, scale)
mpareto(order, shape, scale)
levpareto(limit, shape, scale, order = 1)

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.
shape, scale parameters. Must be strictly positive.
log, log.p logical; if TRUE, probabilities/densities p are returned as log(p).
lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].
order order of the moment.
limit limit of the loss variable.

Details

The Pareto distribution with parameters shape = a and scale = s has density:

f(x) = a s^a / (x + s)^(a + 1)

for x > 0, a > 0 and s > 0.

The kth raw moment of the random variable X is E[X^k] and the kth limited moment at some limit d is E[min(X, d)^k].

Value

dpareto gives the density, ppareto gives the distribution function, qpareto gives the quantile function, rpareto generates random deviates, mpareto gives the kth raw moment, and levpareto gives the kth moment of the limited loss variable.
Invalid arguments will result in return value NaN, with a warning.

Note

Distribution also known as the Pareto Type II or Lomax distribution.

Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca and Mathieu Pigeon

References

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2004), Loss Models, From Data to Decisions, Second Edition, Wiley.

Examples

exp(dpareto(2, 3, 4, log = TRUE))
p <- (1:10)/10
ppareto(qpareto(p, 2, 3), 2, 3)
mpareto(1, 2, 3)
levpareto(10, 2, 3, order = 1)
[Package actuar version 1.0-2 Index]