umontreal.iro.lecuyer.probdist
Class LaplaceDist

java.lang.Object
  umontreal.iro.lecuyer.probdist.ContinuousDistribution
      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 μ and scale parameter β > 0. The density function is given by

f (x) = e^{-| x-μ|/β}/(2β) for - ∞ < x < ∞.

The distribution function is

F(x) =	(1/2)e(x-μ)/β	if x <= μ,
F(x) =	1 - (1/2)e(μ-x)/β	otherwise,

and its inverse is

F-1(u) =	βlog(2u) + μ	if 0 <= u <= 1/2,
F-1(u) =	μ - βlog(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 μ = 0 and β = 1.

LaplaceDist(double mu, double beta)
Constructs a LaplaceDist object with parameters μ = mu and β = beta.

Method Summary

double	barF(double x) Returns the complementary distribution function.
static double	barF(double mu, double beta, double x) Computes the Laplace complementary distribution function.
double	cdf(double x) Returns the distribution function F(x).
static double	cdf(double mu, double beta, double x) Computes the Laplace distribution function.
double	density(double x) Returns f (x), the density evaluated at x.
static double	density(double mu, double beta, double x) Computes the Laplace density function.
double	getBeta() Returns the parameter β.
static LaplaceDist	getInstanceFromMLE(double[] x, int n) Creates a new instance of a Laplace distribution with parameters μ 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 mu, double beta) Computes and returns the mean E[X] = μ of the Laplace distribution with parameters μ and β.
static double[]	getMLE(double[] x, int n) Estimates the parameters (μ, β) of the Laplace distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
double	getMu() Returns the parameter μ.
double[]	getParams() Return a table containing the parameters of the current distribution.
double	getStandardDeviation() Returns the standard deviation.
static double	getStandardDeviation(double mu, double beta) Computes and returns the standard deviation of the Laplace distribution with parameters μ and β.
double	getVariance() Returns the variance.
static double	getVariance(double mu, double beta) Computes and returns the variance Var[X] = 2β2 of the Laplace distribution with parameters μ and β.
double	inverseF(double u) Returns the inverse distribution function x = F-1(u).
static double	inverseF(double mu, double beta, double u) Computes the inverse Laplace distribution function.
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

LaplaceDist

public LaplaceDist()

Constructs a LaplaceDist object with default parameters μ = 0 and β = 1.

LaplaceDist

public LaplaceDist(double mu,
                   double beta)

Constructs a LaplaceDist object with parameters μ = mu 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 mu,
                             double beta,
                             double x)

Computes the Laplace density function.

cdf

public static double cdf(double mu,
                         double beta,
                         double x)

Computes the Laplace distribution function.

barF

public static double barF(double mu,
                          double beta,
                          double x)

Computes the Laplace complementary distribution function.

inverseF

public static double inverseF(double mu,
                              double beta,
                              double u)

Computes the inverse Laplace distribution function.

getMLE

public static double[] getMLE(double[] x,
                              int n)

Estimates the parameters (μ, β) of the Laplace 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: [μ, β].

Parameters:

x - the list of observations used to evaluate parameters

n - the number of observations used to evaluate parameters

Returns:

returns the parameters [hat(μ), hat(β)]

getInstanceFromMLE

public static LaplaceDist getInstanceFromMLE(double[] x,
                                             int n)

Creates a new instance of a Laplace distribution with parameters μ 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 mu,
                             double beta)

Computes and returns the mean E[X] = μ of the Laplace distribution with parameters μ and β.

Returns:

the mean of the Laplace distribution E[X] = μ

getVariance

public static double getVariance(double mu,
                                 double beta)

Computes and returns the variance Var[X] = 2β² of the Laplace distribution with parameters μ and β.

Returns:

the variance of the Laplace distribution Var[X] = 2β²

getStandardDeviation

public static double getStandardDeviation(double mu,
                                          double beta)

Computes and returns the standard deviation of the Laplace distribution with parameters μ and β.

Returns:

the standard deviation of the Laplace distribution

getMu

public double getMu()

Returns the parameter μ.

getBeta

public double getBeta()

Returns the parameter β.

getParams

public double[] getParams()

Return a table containing the parameters of the current distribution. This table is put in regular order: [μ, β].

toString

public String toString()

Overrides:

toString in class Object