Pareto3: The Pareto III Distribution

In actuar: Actuarial Functions and Heavy Tailed Distributions

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)

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.

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

The kth 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.

Value

dpareto3 gives the density, ppareto3 gives the distribution function, qpareto3 gives the quantile function, rpareto3 generates random deviates, mpareto3 gives the kth raw moment, and levpareto3 gives the kth moment of the limited loss variable.

Invalid arguments will result in return value NaN, with a warning.

Note

levpareto3 computes the limited expected value using betaint.

For Pareto distributions, we use the classification of Arnold (2015) with the parametrization of Klugman et al. (2012).

The "distributions" package vignette provides the interrelations between the continuous size distributions in actuar and the complete formulas underlying the above functions.

Author(s)

Vincent Goulet [e-mail]

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)

actuar documentation built on Nov. 8, 2023, 9:06 a.m.