# Pareto distribution

### Description

[borrowed from actuar]:

The (Single-parameter) Pareto distribution with parameter `shape`

*= a* has density:

*
f(x) = a b^a/x^(a + 1)*

for *x > b*, *a > 0* and *b > 0*.

Although there appears to be two parameters, only `shape`

is a true
parameter. The value of `min`

*= b* must be set in
advance.

### Objects from the Class

Objects can be created by calls of the form `new("Pareto", shape, Min)`

.
More frequently they are created via the generating function
`Pareto`

.

### Slots

`img`

Object of class

`"Reals"`

.`param`

Object of class

`"ParetoParameter"`

.`r`

`rpareto1`

`d`

`dpareto1`

`p`

`ppareto1`

`q`

`qpareto1`

`gaps`

(numeric) matrix or

`NULL`

`.withArith`

logical: used internally to issue warnings as to interpretation of arithmetics

`.withSim`

logical: used internally to issue warnings as to accuracy

`.logExact`

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

`.lowerExact`

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

### Extends

Class `"AbscontDistribution"`

, directly.

Class `"UnivariateDistribution"`

, by class `"AbscontDistribution"`

.

Class `"Distribution"`

, by class `"AbscontDistribution"`

.

### Methods

- initialize
`signature(.Object = "Pareto")`

: initialize method.- shape
`signature(object = "Pareto")`

: wrapped access method for slot`shape`

of slot`param`

.- Min
`signature(x = "Pareto")`

: wrapped access method for slot`Min`

of slot`param`

.- shape<-
`signature(object = "Pareto")`

: wrapped replace method for slot`shape`

of slot`param`

.- Min<-
`signature(x = "Pareto")`

: wrapped replace method for slot`Min`

of slot`param`

.- E
`signature(object = "Pareto", fun = "missing", cond = "missing")`

: exact evaluation using explicit expressions.- var
`signature(signature(x = "Pareto")`

: exact evaluation using explicit expressions.- median
`signature(signature(x = "Pareto")`

: exact evaluation using explicit expressions.- IQR
`signature(signature(x = "Pareto")`

: exact evaluation using explicit expressions.- skewness
`signature(signature(x = "Pareto")`

: exact evaluation using explicit expressions.- kurtosis
`signature(signature(x = "Pareto")`

: exact evaluation using explicit expressions.

### Note

This class is based on the code provided by the package actuar by Vincent Goulet and Mathieu Pigeon.

### Author(s)

Nataliya Horbenko Nataliya.Horbenko@itwm.fraunhofer.de

### References

Johnson et al. (1995) *Continuous Univariate Distributions. Vol. 2. 2nd ed.*
New York: Wiley.

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2004),
*Loss Models, From Data to Decisions, Second Edition*, Wiley.

### See Also

`dpareto1`

, `AbscontDistribution-class`

### Examples

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