actuar (version 3.3-4)

InversePareto: The Inverse Pareto Distribution

Description

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

Usage

dinvpareto(x, shape, scale, log = FALSE)
pinvpareto(q, shape, scale, lower.tail = TRUE, log.p = FALSE)
qinvpareto(p, shape, scale, lower.tail = TRUE, log.p = FALSE)
rinvpareto(n, shape, scale)
minvpareto(order, shape, scale)
levinvpareto(limit, shape, scale, order = 1)

Value

dinvpareto gives the density,

pinvpareto gives the distribution function,

qinvpareto gives the quantile function,

rinvpareto generates random deviates,

minvpareto gives the \(k\)th raw moment, and

levinvpareto calculates the \(k\)th limited moment.

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

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.

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.

Author

Vincent Goulet and Mathieu Pigeon

Details

The inverse Pareto distribution with parameters shape \(= \tau\) and scale \(= \theta\) has density: $$f(x) = \frac{\tau \theta x^{\tau - 1}}{% (x + \theta)^{\tau + 1}}$$ for \(x > 0\), \(\tau > 0\) and \(\theta > 0\).

The \(k\)th raw moment of the random variable \(X\) is \(E[X^k]\), \(-\tau < k < 1\).

The \(k\)th limited moment at some limit \(d\) is \(E[\min(X, d)^k]\), \(k > -\tau\).

References

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

Examples

exp(dinvpareto(2, 3, 4, log = TRUE))
p <- (1:10)/10
pinvpareto(qinvpareto(p, 2, 3), 2, 3)
minvpareto(0.5, 1, 2)