SSJ
V. 1.2.5.

umontreal.iro.lecuyer.probdist
Class LaplaceDist

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

public class LaplaceDist
extends ContinuousDistribution

Extends the class ContinuousDistribution for the Laplace distribution. It has location parameter θ and scale parameter φ > 0. The density function is given by

f (x) = e-| x-θ|/φ/(2φ)     for - ∞ < x < ∞.

The distribution function is

F(x) = (1/2)e(x-θ)/φ          if x <= θ,
F(x) = 1 - (1/2)e(θ-x)/φ          otherwise,

and its inverse is

F-1(u) = φlog(2u) + θ          if 0 <= u <= 1/2,
F-1(u) = θ - φlog(2(1 - u))          otherwise.


Field Summary
 
Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
decPrec
 
Constructor Summary
LaplaceDist()
          Constructs a LaplaceDist object with default parameters θ = 0 and φ = 1.
LaplaceDist(double theta, double phi)
          Constructs a LaplaceDist object with parameters θ = theta and φ = phi.
 
Method Summary
 double barF(double x)
          Returns bar(F)(x) = 1 - F(x).
static double barF(double theta, double phi, double x)
          Computes the Laplace complementary distribution function.
 double cdf(double x)
          Computes and returns the distribution function F(x).
static double cdf(double theta, double phi, double x)
          Computes the Laplace distribution function.
 double density(double x)
          Returns f (x), the density of X evaluated at x.
static double density(double theta, double phi, double x)
          Computes the Laplace density function.
static LaplaceDist getInstanceFromMLE(double[] x, int n)
          Creates a new instance of a Laplace distribution with parameters θ and φ estimated using the maximum likelihood method based on the n observations in table x[i], i = 0, 1,…, n - 1.
static double[] getMaximumLikelihoodEstimate(double[] x, int n)
          Estimates and returns the parameters [ hat(&thetas;), hat(&phis;)] of the Laplace distribution using the maximum likelihood method based on the n observations in table x[i], i = 0, 1,…, n - 1.
 double getMean()
          Returns the mean of the distribution function.
static double getMean(double theta, double phi)
          Computes and returns the mean E[X] = θ of the Laplace distribution with parameters θ and φ.
 double getPhi()
          Returns the parameter φ.
 double getStandardDeviation()
          Returns the standard deviation of the distribution function.
static double getStandardDeviation(double theta, double phi)
          Computes and returns the standard deviation of the Laplace distribution with parameters θ and φ.
 double getTheta()
          Returns the parameter θ.
 double getVariance()
          Returns the variance of the distribution function.
static double getVariance(double theta, double phi)
          Computes and returns the variance Var[X] = 2φ2 of the Laplace distribution with parameters θ and φ.
 double inverseF(double u)
          Computes and returns the inverse distribution function F-1(u), defined in.
static double inverseF(double theta, double phi, double u)
          Computes the inverse Laplace distribution function.
 
Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
inverseBisection, inverseBrent
 
Methods inherited from class java.lang.Object
equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
 

Constructor Detail

LaplaceDist

public LaplaceDist()
Constructs a LaplaceDist object with default parameters θ = 0 and φ = 1.


LaplaceDist

public LaplaceDist(double theta,
                   double phi)
Constructs a LaplaceDist object with parameters θ = theta and φ = phi.

Method Detail

density

public double density(double x)
Description copied from class: ContinuousDistribution
Returns f (x), the density of X 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
Computes and 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 interface: Distribution
Returns 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 interface: Distribution
Computes and returns the inverse distribution function F-1(u), defined in.

Specified by:
inverseF in interface Distribution
Overrides:
inverseF in class ContinuousDistribution
Parameters:
u - value in the interval (0, 1) for which the inverse distribution function is evaluated
Returns:
the inverse distribution function evaluated at u

getMean

public double getMean()
Description copied from interface: Distribution
Returns the mean of the distribution function.


getVariance

public double getVariance()
Description copied from interface: Distribution
Returns the variance of the distribution function.


getStandardDeviation

public double getStandardDeviation()
Description copied from interface: Distribution
Returns the standard deviation of the distribution function.


density

public static double density(double theta,
                             double phi,
                             double x)
Computes the Laplace density function.


cdf

public static double cdf(double theta,
                         double phi,
                         double x)
Computes the Laplace distribution function.


barF

public static double barF(double theta,
                          double phi,
                          double x)
Computes the Laplace complementary distribution function.


inverseF

public static double inverseF(double theta,
                              double phi,
                              double u)
Computes the inverse Laplace distribution function.


getInstanceFromMLE

public static LaplaceDist getInstanceFromMLE(double[] x,
                                             int n)
Creates a new instance of a Laplace distribution with parameters θ and φ estimated using the maximum likelihood method based on the n observations in table 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

getMaximumLikelihoodEstimate

public static double[] getMaximumLikelihoodEstimate(double[] x,
                                                    int n)
Estimates and returns the parameters [ hat(&thetas;), hat(&phis;)] of the Laplace distribution using the maximum likelihood method based on the n observations in table x[i], i = 0, 1,…, n - 1.

Parameters:
x - the list of observations used to evaluate parameters
n - the number of observations used to evaluate parameters
Returns:
returns the parameters [ hat(&thetas;), hat(&phis;)]

getMean

public static double getMean(double theta,
                             double phi)
Computes and returns the mean E[X] = θ of the Laplace distribution with parameters θ and φ.

Returns:
the mean of the Laplace distribution E[X] = θ

getVariance

public static double getVariance(double theta,
                                 double phi)
Computes and returns the variance Var[X] = 2φ2 of the Laplace distribution with parameters θ and φ.

Returns:
the variance of the Laplace distribution Var[X] = 2φ2

getStandardDeviation

public static double getStandardDeviation(double theta,
                                          double phi)
Computes and returns the standard deviation of the Laplace distribution with parameters θ and φ.

Returns:
the standard deviation of the Laplace distribution

getTheta

public double getTheta()
Returns the parameter θ.


getPhi

public double getPhi()
Returns the parameter φ.


SSJ
V. 1.2.5.

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