R: Class "Cauchy"

Cauchy-class {distr} | R Documentation

## Class "Cauchy"

### Description

The Cauchy distribution with location l, by default =0, and scale s , by default =1,has density

f(x) = 1 / (pi s (1 + ((x-l)/s)^2))

for all x. C.f. `rcauchy`

### Objects from the Class

Objects can be created by calls of the form `Cauchy(location, scale)`. This object is a Cauchy distribution.

### Slots

`img`

  Object of class `"Reals"`: The domain of this distribution has got dimension 1 and the name "Real Space".

`param`

  Object of class `"CauchyParameter"`: the parameter of this distribution (location and scale), declared at its instantiation

`r`

  Object of class `"function"`: generates random numbers (calls function `rcauchy`)

`d`

  Object of class `"function"`: density function (calls function `dcauchy`)

`p`

  Object of class `"function"`: cumulative function (calls function `pcauchy`)

`q`

  Object of class `"function"`: inverse of the cumulative function (calls function `qcauchy`)

`.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

`Symmetry`

  object of class `"DistributionSymmetry"`; used internally to avoid unnecessary calculations.

### Extends

Class `"AbscontDistribution"`, directly.
Class `"UnivariateDistribution"`, by class `"AbscontDistribution"`.
Class `"Distribution"`, by class `"AbscontDistribution"`.

### Is-Relations

By means of `setIs`, R “knows” that a distribution object `obj` of class `"Cauchy"` with location 0 and scale 1 also is a T distribution with parameters `df = 1, ncp = 0`.

### Methods

initialize

  `signature(.Object = "Cauchy")`: initialize method

location

  `signature(object = "Cauchy")`: returns the slot `location` of the parameter of the distribution

location<-

  `signature(object = "Cauchy")`: modifies the slot `location` of the parameter of the distribution

scale

  `signature(object = "Cauchy")`: returns the slot `scale` of the parameter of the distribution

scale<-

  `signature(object = "Cauchy")`: modifies the slot `scale` of the parameter of the distribution

+

  `signature(e1 = "Cauchy", e2 = "Cauchy")`: For the Cauchy distribution the exact convolution formula is implemented thereby improving the general numerical approximation.

*

  `signature(e1 = "Cauchy", e2 = "numeric")`

+

  `signature(e1 = "Cauchy", e2 = "numeric")`: For the Cauchy location scale family we use its closedness under affine linear transformations.

further arithmetic methods see operators-methods

### Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

### See Also

`CauchyParameter-class` `AbscontDistribution-class` `Reals-class` `rcauchy`

### Examples

    C <- Cauchy(location = 1, scale = 1) # C is a Cauchy distribution with location=1 and scale=1.
    r(C)(1) # one random number generated from this distribution, e.g. 4.104603
    d(C)(1) # Density of this distribution is 0.3183099 for x=1.
    p(C)(1) # Probability that x<1 is 0.5.
    q(C)(.1) # Probability that x<-2.077684 is 0.1.
    ## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
    location(C) # location of this distribution is 1.
    location(C) <- 2 # location of this distribution is now 2.
    is(C,"Td") # no
    C0 <- Cauchy() # standard, i.e. location = 0, scale = 1
    is(C0,"Td") # yes
    as(C0,"Td")

[Package distr version 2.8.0 Index]