R: The Inverse Pareto Distribution

InversePareto {actuar} | 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 <= x], otherwise, P[X > x].
`order` | order of the moment.
`limit` | limit of the loss variable.

### Details

The inverse Pareto distribution with parameters `shape` = a and `scale` = s has density:

f(x) = a s x^(a - 1)/(x + s)^(a + 1)

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

The kth raw moment of the random variable X is E[X^k], -shape < k < 1.

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

### 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)

[Package actuar version 3.1-2 Index]