SSJ
V. 1.2.5.

umontreal.iro.lecuyer.probdist
Class ParetoDist

java.lang.Object
  extended by umontreal.iro.lecuyer.probdist.ContinuousDistribution
      extended by 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) = $\displaystyle {\frac{{\alpha\beta^\alpha}}{{x^{\alpha+1}}}}$        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
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
 double barF(double x)
          Returns bar(F)(x) = 1 - F(x).
static double barF(double alpha, double beta, double x)
          Computes the complementary distribution function.
 double cdf(double x)
          Computes and 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 of X 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 in table x[i], i = 0, 1,…, n - 1.
static double[] getMaximumLikelihoodEstimate(double[] x, int n)
          Estimates and returns the parameters [ hat(α), hat(β)] of the Pareto 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 alpha, double beta)
          Computes and returns the mean E[X] = αβ/(α - 1) of the Pareto distribution with parameters α and β.
 double getStandardDeviation()
          Returns the standard deviation of the distribution function.
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 of the distribution function.
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)
          Computes and returns the inverse distribution function F-1(u), defined in.
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.
 
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

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 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 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.


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 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(α), hat(β)] of the Pareto 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(α), hat(β)]

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.


SSJ
V. 1.2.5.

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