# InversePareto: The Inverse Pareto Distribution In actuar: Actuarial Functions and Heavy Tailed Distributions

 InversePareto R Documentation

## 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)


### 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.

### 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 kth raw moment of the random variable X is E[X^k], -\tau < k < 1.

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

### Value

dinvpareto gives the density, pinvpareto gives the distribution function, qinvpareto gives the quantile function, rinvpareto generates random deviates, minvpareto gives the kth raw moment, and levinvpareto calculates the kth limited moment.

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

### Note

Evaluation of levinvpareto is done using numerical integration.

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 vincent.goulet@act.ulaval.ca and Mathieu Pigeon

### 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)


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