InverseWeibull: The Inverse Weibull Distribution

In actuar: Actuarial Functions and Heavy Tailed Distributions

The Inverse Weibull Distribution

Description

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

Usage

dinvweibull(x, shape, rate = 1, scale = 1/rate, log = FALSE)
pinvweibull(q, shape, rate = 1, scale = 1/rate,
            lower.tail = TRUE, log.p = FALSE)
qinvweibull(p, shape, rate = 1, scale = 1/rate,
            lower.tail = TRUE, log.p = FALSE)
rinvweibull(n, shape, rate = 1, scale = 1/rate)
minvweibull(order, shape, rate = 1, scale = 1/rate)
levinvweibull(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 Weibull distribution with parameters shape = \tau and scale = \theta has density:

f(x) = \frac{\tau (\theta/x)^\tau e^{-(\theta/x)^\tau}}{x}

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

The special case shape == 1 is an Inverse Exponential distribution.

The kth raw moment of the random variable X is E[X^k], k < \tau, and the kth limited moment at some limit d is E[\min(X, d)^k], all k.

Value

dinvweibull gives the density, pinvweibull gives the distribution function, qinvweibull gives the quantile function, rinvweibull generates random deviates, minvweibull gives the kth raw moment, and levinvweibull gives the kth moment of the limited loss variable.

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

Note

levinvweibull computes the limited expected value using gammainc from package expint.

Distribution also knonw as the log-Gompertz. See also 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(dinvweibull(2, 3, 4, log = TRUE))
p <- (1:10)/10
pinvweibull(qinvweibull(p, 2, 3), 2, 3)
mlgompertz(-1, 3, 3)
levinvweibull(10, 2, 3, order = 1)

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