actuar (version 3.3-4)

Pareto3: The Pareto III Distribution

Description

Density function, distribution function, quantile function, random generation, raw moments and limited moments for the Pareto III distribution with parameters min, shape and scale.

Usage

dpareto3(x, min, shape, rate = 1, scale = 1/rate,
         log = FALSE)
ppareto3(q, min, shape, rate = 1, scale = 1/rate,
         lower.tail = TRUE, log.p = FALSE)
qpareto3(p, min, shape, rate = 1, scale = 1/rate,
         lower.tail = TRUE, log.p = FALSE)
rpareto3(n, min, shape, rate = 1, scale = 1/rate)
mpareto3(order, min, shape, rate = 1, scale = 1/rate)
levpareto3(limit, min, shape, rate = 1, scale = 1/rate,
           order = 1)

Value

dpareto3 gives the density,

ppareto3 gives the distribution function,

qpareto3 gives the quantile function,

rpareto3 generates random deviates,

mpareto3 gives the \(k\)th raw moment, and

levpareto3 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.

min

lower bound of the support of the distribution.

shape, scale

parameters. Must be strictly positive.

rate

an alternative way to specify the scale.

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

Details

The Pareto III (or “type III”) distribution with parameters min \(= \mu\), shape \(= \gamma\) and scale \(= \theta\) has density: $$f(x) = \frac{\gamma ((x - \mu)/\theta)^{\gamma - 1}}{% \theta [1 + ((x - \mu)/\theta)^\gamma]^2}$$ for \(x > \mu\), \(-\infty < \mu < \infty\), \(\gamma > 0\) and \(\theta > 0\).

The Pareto III is the distribution of the random variable $$\mu + \theta \left(\frac{X}{1 - X}\right)^{1/\gamma},$$ where \(X\) has a uniform distribution on \((0, 1)\). It derives from the Feller-Pareto distribution with \(\alpha = \tau = 1\). Setting \(\mu = 0\) yields the loglogistic distribution.

The \(k\)th raw moment of the random variable \(X\) is \(E[X^k]\) for nonnegative integer values of \(k < \gamma\).

The \(k\)th limited moment at some limit \(d\) is \(E[\min(X, d)^k]\) for nonnegative integer values of \(k\) and \(1 - j/\gamma\), \(j = 1, \dots, k\) not a negative integer.

References

Arnold, B.C. (2015), Pareto Distributions, Second Edition, CRC Press.

Kleiber, C. and Kotz, S. (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley.

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

See Also

dllogis for the loglogistic distribution.

Examples

exp(dpareto3(1, min = 10, 3, 4, log = TRUE))
p <- (1:10)/10
ppareto3(qpareto3(p, min = 10, 2, 3), min = 10, 2, 3)

## mean
mpareto3(1, min = 10, 2, 3)

## case with 1 - order/shape > 0
levpareto3(20, min = 10, 2, 3, order = 1)

## case with 1 - order/shape < 0
levpareto3(20, min = 10, 2/3, 3, order = 1)