umontreal.iro.lecuyer.probdist

Class ParetoDist

java.lang.Object

umontreal.iro.lecuyer.probdist.ContinuousDistribution

umontreal.iro.lecuyer.probdist.ParetoDist

All Implemented Interfaces:

Distribution

public class ParetoDist
extends ContinuousDistribution

Extends the class ContinuousDistribution for a distribution from the Pareto family, with shape parameter α > 0 and location parameter β > 0. The density for this type of Pareto distribution is

f (x) = αβ^α/x^α+1 for x >= β,

and 0 otherwise. The distribution function is

F(x) = 1 - (β/x)^α for x >= β,

and the inverse distribution function is

F^-1(u) = β(1 - u)^-1/α for 0 <= u < 1.

Field Summary

Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

decPrec

Constructor Summary

Constructors

Constructor and Description
ParetoDist(double alpha) Constructs a ParetoDist object with parameters α = alpha and β = 1.
ParetoDist(double alpha, double beta) Constructs a ParetoDist object with parameters α = alpha and β = beta.

Method Summary

Modifier and Type	Method and Description
double	barF(double x) Returns the complementary distribution function.
static double	barF(double alpha, double beta, double x) Computes the complementary distribution function.
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 parameter α.
double	getBeta() Returns the parameter β.
static ParetoDist	getInstanceFromMLE(double[] x, int n) Creates a new instance of a Pareto 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) Computes and returns the mean E[X] = αβ/(α - 1) of the Pareto distribution with parameters α and β.
static double[]	getMLE(double[] x, int n) Estimates the parameters (α, β) of the Pareto distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
double[]	getParams() Return a table containing the parameters of the current distribution.
double	getStandardDeviation() Returns the standard deviation.
static double	getStandardDeviation(double alpha, double beta) Computes and returns the standard deviation of the Pareto distribution with parameters α and β.
double	getVariance() Returns the variance.
static double	getVariance(double alpha, double beta) Computes and returns the variance of the Pareto distribution with parameters α and β.
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 function.
void	setParams(double alpha, double beta) Sets the parameter α and β for this object.
java.lang.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

ParetoDist

public ParetoDist(double alpha)

Constructs a ParetoDist object with parameters α = alpha and β = 1.

ParetoDist

public ParetoDist(double alpha,
                  double beta)

Constructs a ParetoDist 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 function.

inverseF

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

Computes the inverse of the distribution function.

getMLE

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

Estimates the parameters (α, β) of the Pareto 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 used to evaluate parameters

n - the number of observations used to evaluate parameters

Returns:

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

getInstanceFromMLE

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

Creates a new instance of a Pareto 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)

Computes and returns the mean E[X] = αβ/(α - 1) of the Pareto distribution with parameters α and β.

Returns:

the mean of the Pareto distribution

getVariance

public static double getVariance(double alpha,
                                 double beta)

Computes and returns the variance of the Pareto distribution with parameters α and β.

Returns:

the variance of the Pareto distribution Var[X] = αβ²/[(α -2)(α - 1)]

getStandardDeviation

public static double getStandardDeviation(double alpha,
                                          double beta)

Computes and returns the standard deviation of the Pareto distribution with parameters α and β.

Returns:

the standard deviation of the Pareto distribution

getAlpha

public double getAlpha()

Returns the parameter α.

getBeta

public double getBeta()

Returns the parameter β.

setParams

public void setParams(double alpha,
                      double beta)

Sets the parameter α and β for this object.

getParams

public double[] getParams()

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

toString

public java.lang.String toString()

Overrides:

toString in class java.lang.Object