InverseParalogistic: The Inverse Paralogistic Distribution

In actuar: Actuarial Functions and Heavy Tailed Distributions

The Inverse Paralogistic Distribution

Description

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

Usage

dinvparalogis(x, shape, rate = 1, scale = 1/rate, log = FALSE)
pinvparalogis(q, shape, rate = 1, scale = 1/rate,
              lower.tail = TRUE, log.p = FALSE)
qinvparalogis(p, shape, rate = 1, scale = 1/rate,
              lower.tail = TRUE, log.p = FALSE)
rinvparalogis(n, shape, rate = 1, scale = 1/rate)
minvparalogis(order, shape, rate = 1, scale = 1/rate)
levinvparalogis(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 inverse paralogistic distribution with parameters shape = \tau and scale = \theta has density:

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

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

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

The kth limited moment at some limit d is E[\min(X, d)^k], k > -\tau^2 and 1 - k/\tau not a negative integer.

Value

dinvparalogis gives the density, pinvparalogis gives the distribution function, qinvparalogis gives the quantile function, rinvparalogis generates random deviates, minvparalogis gives the kth raw moment, and levinvparalogis gives the kth moment of the limited loss variable.

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

Note

levinvparalogis computes 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(dinvparalogis(2, 3, 4, log = TRUE))
p <- (1:10)/10
pinvparalogis(qinvparalogis(p, 2, 3), 2, 3)

## first negative moment
minvparalogis(-1, 2, 2)

## case with 1 - order/shape > 0
levinvparalogis(10, 2, 2, order = 1)

## case with 1 - order/shape < 0
levinvparalogis(10, 2/3, 2, order = 1)

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