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# Pareto {actuar}

The Pareto Distribution
Package:
actuar
Version:
1.1-6

### Description

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

### Usage

```dpareto(x, shape, scale, log = FALSE)
ppareto(q, shape, scale, lower.tail = TRUE, log.p = FALSE)
qpareto(p, shape, scale, lower.tail = TRUE, log.p = FALSE)
rpareto(n, shape, scale)
mpareto(order, shape, scale)
levpareto(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 Pareto distribution with parameters `shape` = a and `scale` = s has density: for x > 0, a > 0 and s > 0.

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

### Values

`dpareto` gives the density, `ppareto` gives the distribution function, `qpareto` gives the quantile function, `rpareto` generates random deviates, `mpareto` gives the kth raw moment, and `levpareto` gives the kth moment of the limited loss variable.

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

### References

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2008), Loss Models, From Data to Decisions, Third Edition, Wiley.

### Note

Distribution also known as the Pareto Type II or Lomax distribution.

### Examples

```exp(dpareto(2, 3, 4, log = TRUE))
p <- (1:10)/10
ppareto(qpareto(p, 2, 3), 2, 3)
mpareto(1, 2, 3)
levpareto(10, 2, 3, order = 1)```

### Author(s)

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

Documentation reproduced from package actuar, version 1.1-6. License: GPL (>= 2)