umontreal.iro.lecuyer.probdist
Class CauchyDist

java.lang.Object
  umontreal.iro.lecuyer.probdist.ContinuousDistribution
      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