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 3.1-4 Index]