Pareto2: The Pareto II Distribution

Pareto2R Documentation

The Pareto II Distribution


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


dpareto2(x, min, shape, rate = 1, scale = 1/rate,
         log = FALSE)
ppareto2(q, min, shape, rate = 1, scale = 1/rate,
         lower.tail = TRUE, log.p = FALSE)
qpareto2(p, min, shape, rate = 1, scale = 1/rate,
         lower.tail = TRUE, log.p = FALSE)
rpareto2(n, min, shape, rate = 1, scale = 1/rate)
mpareto2(order, min, shape, rate = 1, scale = 1/rate)
levpareto2(limit, min, shape, rate = 1, scale = 1/rate,
           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.


lower bound of the support of the distribution.

shape, scale

parameters. Must be strictly positive.


an alternative way to specify the scale.

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 II (or “type II”) distribution with parameters min = \mu, shape = \alpha and scale = \theta has density:

f(x) = \frac{\alpha}{% \theta [1 + (x - \mu)/\theta]^{\alpha + 1}}

for x > \mu, -\infty < \mu < \infty, \alpha > 0 and \theta > 0.

The Pareto II is the distribution of the random variable

\mu + \theta \left(\frac{X}{1 - X}\right),

where X has a beta distribution with parameters 1 and \alpha. It derives from the Feller-Pareto distribution with \tau = \gamma = 1. Setting \mu = 0 yields the familiar Pareto distribution.

The Pareto I (or Single parameter Pareto) distribution is a special case of the Pareto II with min == scale.

The kth raw moment of the random variable X is E[X^k] for nonnegative integer values of k < \alpha.

The kth limited moment at some limit d is E[\min(X, d)^k] for nonnegative integer values of k and \alpha - j, j = 1, \dots, k not a negative integer.


dpareto2 gives the density, ppareto2 gives the distribution function, qpareto2 gives the quantile function, rpareto2 generates random deviates, mpareto2 gives the kth raw moment, and levpareto2 gives the kth moment of the limited loss variable.

Invalid arguments will result in return value NaN, with a warning.


levpareto2 computes the limited expected value using betaint.

For Pareto distributions, we use the classification of Arnold (2015) with the parametrization of Klugman et al. (2012).

The "distributions" package vignette provides the interrelations between the continuous size distributions in actuar and the complete formulas underlying the above functions.


Vincent Goulet


Arnold, B.C. (2015), Pareto Distributions, Second Edition, CRC Press.

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

dpareto for the Pareto distribution without a location parameter.


exp(dpareto2(1, min = 10, 3, 4, log = TRUE))
p <- (1:10)/10
ppareto2(qpareto2(p, min = 10, 2, 3), min = 10, 2, 3)

## variance
mpareto2(2, min = 10, 4, 1) - mpareto2(1, min = 10, 4, 1)^2

## case with shape - order > 0
levpareto2(10, min = 10, 3, scale = 1, order = 2)

## case with shape - order < 0
levpareto2(10, min = 10, 1.5, scale = 1, order = 2)

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