Density function, distribution function, quantile function, random generation,
raw moments and limited moments for the Pareto distribution with
parameters shape and scale.
number of observations. If length(n) > 1, the length is
taken to be the number required.
shape, scale
parameters. Must be strictly positive.
log, log.p
logical; if TRUE, probabilities/densities
$p$ are returned as $\log(p)$.
lower.tail
logical; if TRUE (default), probabilities are
$P[X \le x]$, otherwise, $P[X > x]$.
order
order of the moment.
limit
limit of the loss variable.
Value
dpareto gives the density,
ppareto gives the distribution function,
qpareto gives the quantile function,
rpareto generates random deviates,
mpareto gives the $k$th raw moment, and
levpareto gives the $k$th moment of the limited loss variable.
Invalid arguments will result in return value NaN, with a warning.
Details
The Pareto distribution with parameters shape $=
\alpha$ and scale $= \theta$ has density:
$$f(x) = \frac{\alpha \theta^\alpha}{(x + \theta)^{\alpha + 1}}$$
for $x > 0$, $\alpha > 0$ and $\theta$.
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. (2004),
Loss Models, From Data to Decisions, Second Edition, Wiley.