Pareto2 {actuar} | R Documentation |
The Pareto II Distribution
Description
Density function, distribution function, quantile function, random generation,
raw moments and limited moments for the Pareto II distribution with
parameters min
, shape
and scale
.
Usage
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)
Arguments
x , q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If |
min |
lower bound of the support of the distribution. |
shape , scale |
parameters. Must be strictly positive. |
rate |
an alternative way to specify the scale. |
log , log.p |
logical; if |
lower.tail |
logical; if |
order |
order of the moment. |
limit |
limit of the loss variable. |
Details
The Pareto II (or “type II”) distribution with parameters
min
,
shape
and
scale
has density:
for ,
,
and
.
The Pareto II is the distribution of the random variable
where has a beta distribution with parameters
and
. It derives from the Feller-Pareto
distribution with
.
Setting
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 th raw moment of the random variable
is
for nonnegative integer values of
.
The th limited moment at some limit
is
for nonnegative integer values of
and
,
not a negative integer.
Value
dpareto2
gives the density,
ppareto2
gives the distribution function,
qpareto2
gives the quantile function,
rpareto2
generates random deviates,
mpareto2
gives the th raw moment, and
levpareto2
gives the th moment of the limited loss
variable.
Invalid arguments will result in return value NaN
, with a warning.
Note
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.
Author(s)
Vincent Goulet vincent.goulet@act.ulaval.ca
References
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.
Examples
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)