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)
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.
vector of quantiles.
vector of probabilities.
number of observations. If length(n) > 1
, the length
is taken to be the number required.
parameter. Must be strictly positive.
lower bound of the support of the distribution.
logical; if TRUE
, probabilities/densities
\(p\) are returned as \(\log(p)\).
logical; if TRUE
(default), probabilities are
\(P[X \le x]\), otherwise, \(P[X > x]\).
order of the moment.
limit of the loss variable.
Vincent Goulet vincent.goulet@act.ulaval.ca and Mathieu Pigeon
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 \(E[\min(X, d)^k]\), \(x \ge \theta\).
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)
Run the code above in your browser using DataLab