FellerPareto {actuar} | R Documentation |
The Feller Pareto Distribution
Description
Density function, distribution function, quantile function, random generation,
raw moments and limited moments for the Feller Pareto distribution
with parameters min
, shape1
, shape2
, shape3
and
scale
.
Usage
dfpareto(x, min, shape1, shape2, shape3, rate = 1, scale = 1/rate,
log = FALSE)
pfpareto(q, min, shape1, shape2, shape3, rate = 1, scale = 1/rate,
lower.tail = TRUE, log.p = FALSE)
qfpareto(p, min, shape1, shape2, shape3, rate = 1, scale = 1/rate,
lower.tail = TRUE, log.p = FALSE)
rfpareto(n, min, shape1, shape2, shape3, rate = 1, scale = 1/rate)
mfpareto(order, min, shape1, shape2, shape3, rate = 1, scale = 1/rate)
levfpareto(limit, min, shape1, shape2, shape3, 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. |
shape1 , shape2 , shape3 , 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 Feller-Pareto distribution with parameters min
,
shape1
,
shape2
,
shape3
and
scale
, has
density:
for ,
,
,
,
and
.
(Here
is the function implemented
by R's
gamma()
and defined in its help.)
The Feller-Pareto is the distribution of the random variable
where has a beta distribution with parameters
and
.
The Feller-Pareto defines a large family of distributions encompassing
the transformed beta family and many variants of the Pareto
distribution. Setting yields the
transformed beta distribution.
The Feller-Pareto distribution also has the following direct special cases:
A Pareto IV distribution when
shape3 == 1
;A Pareto III distribution when
shape1 shape3 == 1
;A Pareto II distribution when
shape1 shape2 == 1
;A Pareto I distribution when
shape1 shape2 == 1
andmin = 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.
Note that the range of admissible values for in raw and
limited moments is larger when
.
Value
dfpareto
gives the density,
pfpareto
gives the distribution function,
qfpareto
gives the quantile function,
rfpareto
generates random deviates,
mfpareto
gives the th raw moment, and
levfpareto
gives the th moment of the limited loss
variable.
Invalid arguments will result in return value NaN
, with a warning.
Note
levfpareto
computes the limited expected value using
betaint
.
For the Feller-Pareto and other 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 and Nicholas Langevin
References
Dutang, C., Goulet, V., Langevin, N. (2022). Feller-Pareto and Related Distributions: Numerical Implementation and Actuarial Applications. Journal of Statistical Software, 103(6), 1–22. doi:10.18637/jss.v103.i06.
Abramowitz, M. and Stegun, I. A. (1972), Handbook of Mathematical Functions, Dover.
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
dtrbeta
for the transformed beta distribution.
Examples
exp(dfpareto(2, 1, 2, 3, 4, 5, log = TRUE))
p <- (1:10)/10
pfpareto(qfpareto(p, 1, 2, 3, 4, 5), 1, 2, 3, 4, 5)
## variance
mfpareto(2, 1, 2, 3, 4, 5) - mfpareto(1, 1, 2, 3, 4, 5)^2
## case with shape1 - order/shape2 > 0
levfpareto(10, 1, 2, 3, 4, scale = 1, order = 2)
## case with shape1 - order/shape2 < 0
levfpareto(20, 10, 0.1, 14, 2, scale = 1.5, order = 2)