Paralogistic: The Paralogistic Distribution

In actuar: Actuarial Functions and Heavy Tailed Distributions

The Paralogistic Distribution

Description

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

Usage

dparalogis(x, shape, rate = 1, scale = 1/rate, log = FALSE)
pparalogis(q, shape, rate = 1, scale = 1/rate,
           lower.tail = TRUE, log.p = FALSE)
qparalogis(p, shape, rate = 1, scale = 1/rate,
           lower.tail = TRUE, log.p = FALSE)
rparalogis(n, shape, rate = 1, scale = 1/rate)
mparalogis(order, shape, rate = 1, scale = 1/rate)
levparalogis(limit, 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.
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 paralogistic distribution with parameters shape = \alpha and scale = \theta has density:

f(x) = \frac{\alpha^2 (x/\theta)^\alpha}{% x [1 + (x/\theta)^\alpha)^{\alpha + 1}}

for x > 0, \alpha > 0 and \theta > 0.

The kth raw moment of the random variable X is E[X^k], -\alpha < k < \alpha^2.

The kth limited moment at some limit d is E[\min(X, d)^k], k > -\alpha and \alpha - k/\alpha not a negative integer.

Value

dparalogis gives the density, pparalogis gives the distribution function, qparalogis gives the quantile function, rparalogis generates random deviates, mparalogis gives the kth raw moment, and levparalogis gives the kth moment of the limited loss variable.

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

Note

levparalogis computes the limited expected value using betaint.

See Kleiber and Kotz (2003) for alternative names and parametrizations.

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] and Mathieu Pigeon

References

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.

Examples

exp(dparalogis(2, 3, 4, log = TRUE))
p <- (1:10)/10
pparalogis(qparalogis(p, 2, 3), 2, 3)

## variance
mparalogis(2, 2, 3) - mparalogis(1, 2, 3)^2

## case with shape - order/shape > 0
levparalogis(10, 2, 3, order = 2)

## case with shape - order/shape < 0
levparalogis(10, 1.25, 3, order = 2)

actuar documentation built on Aug. 9, 2025, 1:06 a.m.