Pareto4: The Pareto IV Distribution

In actuar: Actuarial Functions and Heavy Tailed Distributions

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)

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.

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

The kth 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.

Value

dpareto4 gives the density, ppareto4 gives the distribution function, qpareto4 gives the quantile function, rpareto4 generates random deviates, mpareto4 gives the kth raw moment, and levpareto4 gives the kth moment of the limited loss variable.

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

Note

levpareto4 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

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)

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