Pareto | R Documentation |
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. |
p |
vector of probabilities. |
n |
number of observations. If |
shape, scale |
parameters. Must be strictly positive. |
log, log.p |
logical; if |
lower.tail |
logical; if |
order |
order of the moment. |
limit |
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 k
th raw moment of the random variable X
is
E[X^k]
, -1 < k < \alpha
.
The k
th 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 k
th raw moment, and
levpareto
gives the k
th 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 vincent.goulet@act.ulaval.ca 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.
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)
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