SingleParameterPareto | R Documentation |
Density function, distribution function, quantile function, random generation,
raw moments, and limited moments for the Single-parameter Pareto
distribution with parameter shape
.
dpareto1(x, shape, min, log = FALSE)
ppareto1(q, shape, min, lower.tail = TRUE, log.p = FALSE)
qpareto1(p, shape, min, lower.tail = TRUE, log.p = FALSE)
rpareto1(n, shape, min)
mpareto1(order, shape, min)
levpareto1(limit, shape, min, order = 1)
x, q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If |
shape |
parameter. Must be strictly positive. |
min |
lower bound of the support of the distribution. |
log, log.p |
logical; if |
lower.tail |
logical; if |
order |
order of the moment. |
limit |
limit of the loss variable. |
The single-parameter Pareto, or Pareto I, distribution with parameter
shape
= \alpha
has density:
f(x) = \frac{\alpha \theta^\alpha}{x^{\alpha + 1}}
for x > \theta
, \alpha > 0
and \theta >
0
.
Although there appears to be two parameters, only shape
is a true
parameter. The value of min
= \theta
must be set in
advance.
The k
th raw moment of the random variable X
is
E[X^k]
, k < \alpha
and the k
th
limited moment at some limit d
is E[\min(X, d)^k]
, x \ge \theta
.
dpareto1
gives the density,
ppareto1
gives the distribution function,
qpareto1
gives the quantile function,
rpareto1
generates random deviates,
mpareto1
gives the k
th raw moment, and
levpareto1
gives the k
th moment of the limited loss
variable.
Invalid arguments will result in return value NaN
, with a warning.
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 and Mathieu Pigeon
Arnold, B.C. (2015), Pareto Distributions, Second Edition, CRC Press.
Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.
dpareto
for the two-parameter Pareto distribution.
exp(dpareto1(5, 3, 4, log = TRUE))
p <- (1:10)/10
ppareto1(qpareto1(p, 2, 3), 2, 3)
mpareto1(2, 3, 4) - mpareto(1, 3, 4) ^ 2
levpareto(10, 3, 4, order = 2)
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