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.

Usage

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)

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.

Arguments

x, q: vector of quantiles.
p: vector of probabilities.
n: 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 $P[X \le x]$, otherwise, $P[X > x]$.
order: order of the moment.
limit: limit of the loss variable.

Author

Vincent Goulet vincent.goulet@act.ulaval.ca and Mathieu Pigeon

Details

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 $E[\min(X, d)^k]$, $x \ge \theta$.

References

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.

Examples

Run this code

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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