Paralogistic: The Paralogistic Distribution

ParalogisticR Documentation

The Paralogistic Distribution


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


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)


x, q

vector of quantiles.


vector of probabilities.


number of observations. If length(n) > 1, the length is taken to be the number required.

shape, scale

parameters. Must be strictly positive.


an alternative way to specify the scale.

log, log.p

logical; if TRUE, probabilities/densities p are returned as log(p).


logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].


order of the moment.


limit of the loss variable.


The paralogistic distribution with parameters shape = a and scale = s has density:

f(x) = a^2 (x/s)^a / (x [1 + (x/s)^a]^(a + 1))

for x > 0, a > 0 and b > 0.

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

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


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.


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.


Vincent Goulet and Mathieu Pigeon


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.


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 July 16, 2022, 9:05 a.m.