SingleParameterPareto: The Single-parameter Pareto Distribution
Description
Density function, distribution function, quantile function, random generation,
raw moments, and limited moments for the Single-parameter Pareto
distribution with parameter shape.
number of observations. If length(n) > 1, the length
is taken to be the number required.
shape
parameter. Must be strictly positive.
min
lower bound of the support of the distribution.
log, log.p
logical; if TRUE, probabilities/densities
$p$ are returned as $log(p)$.
lower.tail
logical; if TRUE (default), probabilities are
$P[X <= x]$,="" otherwise,="" $p[x=""> x]$.
order
order of the moment.
limit
limit of the loss variable.
Value
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.
Details
The Single-parameter Pareto distribution with parameter shape
$= a$ has density:
$$f(x) = \frac{\alpha \theta^\alpha}{x^{\alpha + 1}}$$
for $x > b$, $a > 0$ and $b > 0$.
Although there appears to be two parameters, only shape is a true
parameter. The value of min $= b$ must be set in
advance.
The $k$th raw moment of the random variable $X$ is
$E[X^k]$ and the $k$th limited moment at some limit
$d$ is $E[min(X, d)^k]$.
References
Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2008),
Loss Models, From Data to Decisions, Third Edition, Wiley.