The Pareto IV Distribution
Density function, distribution function, quantile function, random generation,
raw moments and limited moments for the Pareto IV distribution with
parameters min, shape1, shape2 and scale.
dpareto4(x, min, shape1, shape2, rate = 1, scale = 1/rate,
log = FALSE)
ppareto4(q, min, shape1, shape2, rate = 1, scale = 1/rate,
lower.tail = TRUE, log.p = FALSE)
qpareto4(p, min, shape1, shape2, rate = 1, scale = 1/rate,
lower.tail = TRUE, log.p = FALSE)
rpareto4(n, min, shape1, shape2, rate = 1, scale = 1/rate)
mpareto4(order, min, shape1, shape2, rate = 1, scale = 1/rate)
levpareto4(limit, min, shape1, shape2, 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, 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 Pareto IV (or “type IV”) distribution with parameters
min = m,
shape1 = a,
shape2 = b and
scale = s has density:
f(x) = (a b ((x - m)/s)^(b - 1))/(s [1 + ((x - m)/s)^b]^(a + 1))
for x > m, -Inf < m < Inf, a > 0, b > 0 and s > 0.
The Pareto IV is the distribution of the random variable
m + s (X/(1 - X))^(1/b),
where X has a beta distribution with parameters 1 and a. It derives from the Feller-Pareto distribution with shape3 = 1. Setting min = 0 yields the Burr distribution.
The Pareto IV distribution also has the following direct special cases:
A Pareto III distribution when shape1
== 1;
A Pareto II distribution when shape1
== 1.
The kth raw moment of the random variable X is E[X^k] for nonnegative integer values of k < shape1 * shape2.
The kth limited moment at some limit d is E[min(X, d)^k] for nonnegative integer values of k and shape1 - j/shape2, j = 1, …, k not a negative integer.
dpareto4 gives the density,
ppareto4 gives the distribution function,
qpareto4 gives the quantile function,
rpareto4 generates random deviates,
mpareto4 gives the kth raw moment, and
levpareto4 gives the kth moment of the limited loss
variable.
Invalid arguments will result in return value NaN, with a warning.
levpareto4 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.
Vincent Goulet vincent.goulet@act.ulaval.ca
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.
dburr for the Burr distribution.
exp(dpareto4(1, min = 10, 2, 3, log = TRUE)) p <- (1:10)/10 ppareto4(qpareto4(p, min = 10, 2, 3, 2), min = 10, 2, 3, 2) ## variance mpareto4(2, min = 10, 2, 3, 1) - mpareto4(1, min = 10, 2, 3, 1) ^ 2 ## case with shape1 - order/shape2 > 0 levpareto4(10, min = 10, 2, 3, 1, order = 2) ## case with shape1 - order/shape2 < 0 levpareto4(10, min = 10, 1.5, 0.5, 1, order = 2)
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