The Pareto II Distribution
Density function, distribution function, quantile function, random generation,
raw moments and limited moments for the Pareto II distribution with
parameters min, shape and scale.
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)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. |
The Pareto II (or “type II”) distribution with parameters
min = m,
shape = a and
scale = s has density:
f(x) = a/(s [1 + (x - m)/s]^(a + 1))
for x > m, -Inf < m < Inf, a > 0 and s > 0.
The Pareto II is the distribution of the random variable
m + s X/(1 - X),
where X has a beta distribution with parameters 1 and a. It derives from the Feller-Pareto distribution with shape2 = shape3 = 1. Setting min = 0 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 kth raw moment of the random variable X is E[X^k] for nonnegative integer values of k < shape.
The kth limited moment at some limit d is E[min(X, d)^k] for nonnegative integer values of k and shape1 - j, j = 1, …, k not a negative integer.
dpareto2 gives the density,
ppareto2 gives the distribution function,
qpareto2 gives the quantile function,
rpareto2 generates random deviates,
mpareto2 gives the kth raw moment, and
levpareto2 gives the kth moment of the limited loss
variable.
Invalid arguments will result in return value NaN, with a warning.
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.
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.
dpareto for the Pareto distribution without a location
parameter.
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)
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