distr (version 2.9.5)

Cauchy-class: Class "Cauchy"

Description

The Cauchy distribution with location \(l\), by default \(=0\), and scale \(s\) , by default \(=1\),has density $$f(x) = \frac{1}{\pi s} \left( 1 + \left(\frac{x - l}{s}\right)^2 \right)^{-1}% $$ 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

Thomas Stabla ,
Florian Camphausen ,
Peter Ruckdeschel ,
Matthias Kohl

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")