Pareto: The Pareto Distribution

ParetoR Documentation

The Pareto Distribution


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


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)


x, q

vector of quantiles.


vector of probabilities.


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).


logical; if TRUE (default), probabilities are P[X \le x], otherwise, P[X > x].


order of the moment.


limit of the loss variable.


The Pareto distribution with parameters shape = \alpha and scale = \theta has density:

f(x) = \frac{\alpha \theta^\alpha}{(x + \theta)^{\alpha + 1}}

for x > 0, \alpha > 0 and \theta.

There are many different definitions of the Pareto distribution in the literature; see Arnold (2015) or Kleiber and Kotz (2003). In the nomenclature of actuar, The “Pareto distribution” does not have a location parameter. The version with a location parameter is the Pareto II.

The kth raw moment of the random variable X is E[X^k], -1 < k < \alpha.

The kth limited moment at some limit d is E[\min(X, d)^k], k > -1 and \alpha - k not a negative integer.


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.


levpareto computes the limited expected value using betaint.

The version of the Pareto defined for x > \theta is named Single Parameter Pareto, or Pareto I, in actuar.


Vincent Goulet and Mathieu Pigeon


Kleiber, C. and Kotz, S. (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley.

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

See Also

dpareto2 for an equivalent distribution with location parameter.

dpareto1 for the Single Parameter Pareto distribution.

"distributions" package vignette for details on the interrelations between the continuous size distributions in actuar and complete formulas underlying the above functions.


exp(dpareto(2, 3, 4, log = TRUE))
p <- (1:10)/10
ppareto(qpareto(p, 2, 3), 2, 3)

## variance
mpareto(2, 4, 1) - mpareto(1, 4, 1)^2

## case with shape - order > 0
levpareto(10, 3, scale = 1, order = 2)

## case with shape - order < 0
levpareto(10, 1.5, scale = 1, order = 2)

actuar documentation built on Nov. 8, 2023, 9:06 a.m.