actuar (version 3.3-4)

Pareto4: The Pareto IV Distribution

Description

Density function, distribution function, quantile function, random generation, raw moments and limited moments for the Pareto IV distribution with parameters min, shape1, shape2 and scale.

Usage

dpareto4(x, min, shape1, shape2, rate = 1, scale = 1/rate,
         log = FALSE)
ppareto4(q, min, shape1, shape2, rate = 1, scale = 1/rate,
         lower.tail = TRUE, log.p = FALSE)
qpareto4(p, min, shape1, shape2, rate = 1, scale = 1/rate,
         lower.tail = TRUE, log.p = FALSE)
rpareto4(n, min, shape1, shape2, rate = 1, scale = 1/rate)
mpareto4(order, min, shape1, shape2, rate = 1, scale = 1/rate)
levpareto4(limit, min, shape1, shape2, rate = 1, scale = 1/rate,
           order = 1)

Value

dpareto4 gives the density,

ppareto4 gives the distribution function,

qpareto4 gives the quantile function,

rpareto4 generates random deviates,

mpareto4 gives the \(k\)th raw moment, and

levpareto4 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.

shape1, shape2, 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 IV (or “type IV”) distribution with parameters min \(= \mu\), shape1 \(= \alpha\), shape2 \(= \gamma\) and scale \(= \theta\) has density: $$f(x) = \frac{\alpha \gamma ((x - \mu)/\theta)^{\gamma - 1}}{% \theta [1 + ((x - \mu)/\theta)^\gamma]^{\alpha + 1}}$$ for \(x > \mu\), \(-\infty < \mu < \infty\), \(\alpha > 0\), \(\gamma > 0\) and \(\theta > 0\).

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

The Pareto IV distribution also has the following direct special cases:

• A Pareto III distribution when shape1 == 1;

• A Pareto II distribution when shape1 == 1.

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

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

dburr for the Burr distribution.

Examples

exp(dpareto4(1, min = 10, 2, 3, log = TRUE))
p <- (1:10)/10
ppareto4(qpareto4(p, min = 10, 2, 3, 2), min = 10, 2, 3, 2)

## variance
mpareto4(2, min = 10, 2, 3, 1) - mpareto4(1, min = 10, 2, 3, 1) ^ 2

## case with shape1 - order/shape2 > 0
levpareto4(10, min = 10, 2, 3, 1, order = 2)

## case with shape1 - order/shape2 < 0
levpareto4(10, min = 10, 1.5, 0.5, 1, order = 2)