SingleParameterPareto {actuar}R Documentation

The Single-parameter Pareto Distribution


Density function, distribution function, quantile function, random generation, raw moments, and limited moments for the Single-parameter Pareto distribution with parameter shape.


dpareto1(x, shape, min, log = FALSE)
ppareto1(q, shape, min, lower.tail = TRUE, log.p = FALSE)
qpareto1(p, shape, min, lower.tail = TRUE, log.p = FALSE)
rpareto1(n, shape, min)
mpareto1(order, shape, min)
levpareto1(limit, shape, min, 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.


parameter. Must be strictly positive.


lower bound of the support of the distribution.

log, log.p

logical; if TRUE, probabilities/densities p are returned as log(p).


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


order of the moment.


limit of the loss variable.


The single-parameter Pareto, or Pareto I, distribution with parameter shape = a has density:

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

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

Although there appears to be two parameters, only shape is a true parameter. The value of min = b must be set in advance.

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


dpareto1 gives the density, ppareto1 gives the distribution function, qpareto1 gives the quantile function, rpareto1 generates random deviates, mpareto1 gives the kth raw moment, and levpareto1 gives the kth moment of the limited loss variable.

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


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 and Mathieu Pigeon


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

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 two-parameter Pareto distribution.


exp(dpareto1(5, 3, 4, log = TRUE))
p <- (1:10)/10
ppareto1(qpareto1(p, 2, 3), 2, 3)
mpareto1(2, 3, 4) - mpareto(1, 3, 4) ^ 2
levpareto(10, 3, 4, order = 2)

[Package actuar version 3.1-4 Index]