actuar (version 3.3-4)

Pareto2: The Pareto II Distribution

Description

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

Usage

dpareto2(x, min, shape, rate = 1, scale = 1/rate,
         log = FALSE)
ppareto2(q, min, shape, rate = 1, scale = 1/rate,
         lower.tail = TRUE, log.p = FALSE)
qpareto2(p, min, shape, rate = 1, scale = 1/rate,
         lower.tail = TRUE, log.p = FALSE)
rpareto2(n, min, shape, rate = 1, scale = 1/rate)
mpareto2(order, min, shape, rate = 1, scale = 1/rate)
levpareto2(limit, min, shape, rate = 1, scale = 1/rate,
           order = 1)

Value

dpareto2 gives the density,

ppareto2 gives the distribution function,

qpareto2 gives the quantile function,

rpareto2 generates random deviates,

mpareto2 gives the \(k\)th raw moment, and

levpareto2 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 II (or “type II”) distribution with parameters min \(= \mu\), shape \(= \alpha\) and scale \(= \theta\) has density: $$f(x) = \frac{\alpha}{% \theta [1 + (x - \mu)/\theta]^{\alpha + 1}}$$ for \(x > \mu\), \(-\infty < \mu < \infty\), \(\alpha > 0\) and \(\theta > 0\).

The Pareto II is the distribution of the random variable $$\mu + \theta \left(\frac{X}{1 - X}\right),$$ where \(X\) has a beta distribution with parameters \(1\) and \(\alpha\). It derives from the Feller-Pareto distribution with \(\tau = \gamma = 1\). Setting \(\mu = 0\) yields the familiar Pareto distribution.

The Pareto I (or Single parameter Pareto) distribution is a special case of the Pareto II with min == scale.

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

The \(k\)th limited moment at some limit \(d\) is \(E[\min(X, d)^k]\) for nonnegative integer values of \(k\) and \(\alpha - j\), \(j = 1, \dots, k\) 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

dpareto for the Pareto distribution without a location parameter.

Examples

exp(dpareto2(1, min = 10, 3, 4, log = TRUE))
p <- (1:10)/10
ppareto2(qpareto2(p, min = 10, 2, 3), min = 10, 2, 3)

## variance
mpareto2(2, min = 10, 4, 1) - mpareto2(1, min = 10, 4, 1)^2

## case with shape - order > 0
levpareto2(10, min = 10, 3, scale = 1, order = 2)

## case with shape - order < 0
levpareto2(10, min = 10, 1.5, scale = 1, order = 2)