SSJ
V. 2.6.

umontreal.iro.lecuyer.probdist
Class CauchyDist

java.lang.Object
  extended by umontreal.iro.lecuyer.probdist.ContinuousDistribution
      extended by umontreal.iro.lecuyer.probdist.CauchyDist
All Implemented Interfaces:
Distribution

public class CauchyDist
extends ContinuousDistribution

Extends the class ContinuousDistribution for the Cauchy distribution with location parameter α and scale parameter β > 0. The density function is given by

f (x) = β/(π[(x - α)2 + β2]) for - ∞ < x < ∞.

The distribution function is

F(x) = 1/2 + arctan((x - α)/β)/π,                for - ∞ < x < ∞,

and its inverse is

F-1(u) = α + βtan(π(u - 1/2)).        for 0 < u < 1.


Field Summary
 
Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
decPrec
 
Constructor Summary
CauchyDist()
          Constructs a CauchyDist object with parameters α = 0 and β = 1.
CauchyDist(double alpha, double beta)
          Constructs a CauchyDist object with parameters α = alpha and β = beta.
 
Method Summary
 double barF(double x)
          Returns the complementary distribution function.
static double barF(double alpha, double beta, double x)
          Computes the complementary distribution.
 double cdf(double x)
          Returns the distribution function F(x).
static double cdf(double alpha, double beta, double x)
          Computes the distribution function.
 double density(double x)
          Returns f (x), the density evaluated at x.
static double density(double alpha, double beta, double x)
          Computes the density function.
 double getAlpha()
          Returns the value of α for this object.
 double getBeta()
          Returns the value of β for this object.
static CauchyDist getInstanceFromMLE(double[] x, int n)
          Creates a new instance of a Cauchy distribution with parameters α and β estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.
 double getMean()
          Returns the mean.
static double getMean(double alpha, double beta)
          Throws an exception since the mean does not exist.
static double[] getMLE(double[] x, int n)
          Estimates the parameters (α, β) of the Cauchy distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
 double[] getParams()
          Return a table containing parameters of the current distribution.
 double getStandardDeviation()
          Returns the standard deviation.
static double getStandardDeviation(double alpha, double beta)
          Returns since the standard deviation does not exist.
 double getVariance()
          Returns the variance.
static double getVariance(double alpha, double beta)
          Returns since the variance does not exist.
 double inverseF(double u)
          Returns the inverse distribution function x = F-1(u).
static double inverseF(double alpha, double beta, double u)
          Computes the inverse of the distribution.
 void setParams(double alpha, double beta)
          Sets the value of the parameters α and β for this object.
 String toString()
           
 
Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
getXinf, getXsup, inverseBisection, inverseBrent, setXinf, setXsup
 
Methods inherited from class java.lang.Object
equals, getClass, hashCode, notify, notifyAll, wait, wait, wait
 

Constructor Detail

CauchyDist

public CauchyDist()
Constructs a CauchyDist object with parameters α = 0 and β = 1.


CauchyDist

public CauchyDist(double alpha,
                  double beta)
Constructs a CauchyDist object with parameters α = alpha and β = beta.

Method Detail

density

public double density(double x)
Description copied from class: ContinuousDistribution
Returns f (x), the density evaluated at x.

Specified by:
density in class ContinuousDistribution
Parameters:
x - value at which the density is evaluated
Returns:
density function evaluated at x

cdf

public double cdf(double x)
Description copied from interface: Distribution
Returns the distribution function F(x).

Parameters:
x - value at which the distribution function is evaluated
Returns:
distribution function evaluated at x

barF

public double barF(double x)
Description copied from class: ContinuousDistribution
Returns the complementary distribution function. The default implementation computes bar(F)(x) = 1 - F(x).

Specified by:
barF in interface Distribution
Overrides:
barF in class ContinuousDistribution
Parameters:
x - value at which the complementary distribution function is evaluated
Returns:
complementary distribution function evaluated at x

inverseF

public double inverseF(double u)
Description copied from class: ContinuousDistribution
Returns the inverse distribution function x = F-1(u). Restrictions: u∈[0, 1].

Specified by:
inverseF in interface Distribution
Overrides:
inverseF in class ContinuousDistribution
Parameters:
u - value at which the inverse distribution function is evaluated
Returns:
the inverse distribution function evaluated at u

getMean

public double getMean()
Description copied from class: ContinuousDistribution
Returns the mean.

Specified by:
getMean in interface Distribution
Overrides:
getMean in class ContinuousDistribution
Returns:
the mean

getVariance

public double getVariance()
Description copied from class: ContinuousDistribution
Returns the variance.

Specified by:
getVariance in interface Distribution
Overrides:
getVariance in class ContinuousDistribution
Returns:
the variance

getStandardDeviation

public double getStandardDeviation()
Description copied from class: ContinuousDistribution
Returns the standard deviation.

Specified by:
getStandardDeviation in interface Distribution
Overrides:
getStandardDeviation in class ContinuousDistribution
Returns:
the standard deviation

density

public static double density(double alpha,
                             double beta,
                             double x)
Computes the density function.


cdf

public static double cdf(double alpha,
                         double beta,
                         double x)
Computes the distribution function.


barF

public static double barF(double alpha,
                          double beta,
                          double x)
Computes the complementary distribution.


inverseF

public static double inverseF(double alpha,
                              double beta,
                              double u)
Computes the inverse of the distribution.


getMLE

public static double[] getMLE(double[] x,
                              int n)
Estimates the parameters (α, β) of the Cauchy distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1. The estimates are returned in a two-element array, in regular order: [α, β].

Parameters:
x - the list of observations to use to evaluate parameters
n - the number of observations to use to evaluate parameters
Returns:
returns the parameters [ hat(α), hat(β)]

getInstanceFromMLE

public static CauchyDist getInstanceFromMLE(double[] x,
                                            int n)
Creates a new instance of a Cauchy distribution with parameters α and β estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.

Parameters:
x - the list of observations to use to evaluate parameters
n - the number of observations to use to evaluate parameters

getMean

public static double getMean(double alpha,
                             double beta)
Throws an exception since the mean does not exist.

Throws:
UnsupportedOperationException - the mean of the Cauchy distribution is undefined.

getVariance

public static double getVariance(double alpha,
                                 double beta)
Returns since the variance does not exist.

Returns:
.

getStandardDeviation

public static double getStandardDeviation(double alpha,
                                          double beta)
Returns since the standard deviation does not exist.

Returns:

getAlpha

public double getAlpha()
Returns the value of α for this object.


getBeta

public double getBeta()
Returns the value of β for this object.


setParams

public void setParams(double alpha,
                      double beta)
Sets the value of the parameters α and β for this object.


getParams

public double[] getParams()
Return a table containing parameters of the current distribution. This table is put in regular order: [α, β].


toString

public String toString()
Overrides:
toString in class Object

SSJ
V. 2.6.

To submit a bug or ask questions, send an e-mail to Pierre L'Ecuyer.