FellerPareto | R Documentation |
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
.
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)
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. |
The Feller-Pareto distribution with parameters min
= \mu
,
shape1
= \alpha
, shape2
= \gamma
,
shape3
= \tau
and scale
= \theta
, has
density:
f(x) = \frac{\Gamma(\alpha + \tau)}{\Gamma(\alpha)\Gamma(\tau)}
\frac{\gamma ((x - \mu)/\theta)^{\gamma \tau - 1}}{%
\theta [1 + ((x - \mu)/\theta)^\gamma]^{\alpha + \tau}}
for x > \mu
, -\infty < \mu < \infty
,
\alpha > 0
, \gamma > 0
,
\tau > 0
and \theta > 0
.
(Here \Gamma(\alpha)
is the function implemented
by R's gamma()
and defined in its help.)
The Feller-Pareto is the distribution of the random variable
\mu + \theta \left(\frac{1 - X}{X}\right)^{1/\gamma},
where X
has a beta distribution with parameters \alpha
and \tau
.
The Feller-Pareto defines a large family of distributions encompassing
the transformed beta family and many variants of the Pareto
distribution. Setting \mu = 0
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
and min = scale
.
The k
th raw moment of the random variable X
is
E[X^k]
for nonnegative integer values of k <
\alpha\gamma
.
The k
th limited moment at some limit d
is E[\min(X,
d)^k]
for nonnegative integer values of k
and \alpha - j/\gamma
, j = 1, \dots, k
not a negative integer.
Note that the range of admissible values for k
in raw and
limited moments is larger when \mu = 0
.
dfpareto
gives the density,
pfpareto
gives the distribution function,
qfpareto
gives the quantile function,
rfpareto
generates random deviates,
mfpareto
gives the k
th raw moment, and
levfpareto
gives the k
th moment of the limited loss
variable.
Invalid arguments will result in return value NaN
, with a warning.
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.
Vincent Goulet vincent.goulet@act.ulaval.ca and Nicholas Langevin
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. \Sexpr[results=rd]{tools:::Rd_expr_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.
dtrbeta
for the transformed beta distribution.
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)
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