SSJ
V. 2.2.

umontreal.iro.lecuyer.probdist
Class BetaDist

java.lang.Object
  extended by umontreal.iro.lecuyer.probdist.ContinuousDistribution
      extended by umontreal.iro.lecuyer.probdist.BetaDist
All Implemented Interfaces:
Distribution
Direct Known Subclasses:
BetaSymmetricalDist

public class BetaDist
extends ContinuousDistribution

Extends the class ContinuousDistribution for the beta distribution with shape parameters α > 0 and β > 0, over the interval (a, b), where a < b. It has density

f (x) = (x - a)α-1(b - x)β-1/[B(α, β)(b - a)α+β-1]

for a < x < b, and 0 elsewhere. It has distribution function

F(x) = Iα, β(x) = ∫ax(ξ - a)α-1(b - ξ)β-1/[B(α, β)(b - a)α+β-1],        for a < x < b,

where B(α, β) is the beta function defined by

B(α, β) = Γ(α)Γ(β)/Γ(α + β)

and Γ(x) is the gamma function defined in GammaDist.


Field Summary
 
Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
decPrec
 
Constructor Summary
BetaDist(double alpha, double beta)
          Constructs a BetaDist object with parameters α = alpha and β = beta and default domain (0, 1).
BetaDist(double alpha, double beta, double a, double b)
          Constructs a BetaDist object with parameters α = alpha and β = beta, and domain (a, b).
BetaDist(double alpha, double beta, double a, double b, int d)
          Constructs a BetaDist object with parameters α = alpha and β = beta, and approximations of roughly d decimal digits of precision when computing distribution, complementary distribution, and inverse functions.
BetaDist(double alpha, double beta, int d)
          Constructs a BetaDist object with parameters α = alpha and β = beta, and approximations of roughly d decimal digits of precision when computing the distribution, complementary distribution, and inverse functions.
 
Method Summary
static double barF(double alpha, double beta, double a, double b, int d, double x)
          Computes the complementary distribution function.
static double barF(double alpha, double beta, int d, double x)
          Same as barF (alpha, beta, 0, 1, d, x).
 double cdf(double x)
          Returns the distribution function F(x).
static double cdf(double alpha, double beta, double a, double b, int d, double x)
          Computes an approximation of the distribution function, with roughly d decimal digits of precision.
static double cdf(double alpha, double beta, int d, double x)
          Same as cdf (alpha, beta, 0, 1, d, x).
 double density(double x)
          Returns f (x), the density evaluated at x.
static double density(double alpha, double beta, double x)
          Same as density (alpha, beta, 0, 1, x).
static double density(double alpha, double beta, double a, double b, double x)
          Computes the density function of the beta distribution.
 double getA()
          Returns the parameter a of this object.
 double getAlpha()
          Returns the parameter α of this object.
 double getB()
          Returns the parameter b of this object.
 double getBeta()
          Returns the parameter β of this object.
static BetaDist getInstanceFromMLE(double[] x, int n)
          Creates a new instance of a beta distribution with parameters α and β over the interval [0, 1] estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.
static double[] getMaximumLikelihoodEstimate(double[] x, int n)
          Deprecated. 
 double getMean()
          Returns the mean.
static double getMean(double alpha, double beta)
          Computes and returns the mean E[X] = α/(α + β) of the beta distribution with parameters α and β, over the interval [0, 1].
static double getMean(double alpha, double beta, double a, double b)
          Computes and returns the mean E[X] = ( + )/(α + β) of the beta distribution with parameters α and β over the interval [a, b].
static double[] getMLE(double[] x, int n)
          Estimates the parameters (α, β) of the beta distribution over the interval [0, 1] 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)
          Computes the standard deviation of the beta distribution with parameters α and β, over the interval [0, 1].
static double getStandardDeviation(double alpha, double beta, double a, double b)
          Computes the standard deviation of the beta distribution with parameters α and β, over the interval [a, b].
 double getVariance()
          Returns the variance.
static double getVariance(double alpha, double beta)
          .
static double getVariance(double alpha, double beta, double a, double b)
          .
 double inverseF(double u)
          Returns the inverse distribution function x = F-1(u).
static double inverseF(double alpha, double beta, double a, double b, int d, double u)
          Returns the inverse beta distribution function using the algorithm implemented in the Cephes math library.
static double inverseF(double alpha, double beta, int d, double u)
          Same as inverseF (alpha, beta, 0, 1, d, u).
 void setParams(double alpha, double beta, double a, double b, int d)
           
 String toString()
           
 
Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
barF, getXinf, getXsup, inverseBisection, inverseBrent, setXinf, setXsup
 
Methods inherited from class java.lang.Object
equals, getClass, hashCode, notify, notifyAll, wait, wait, wait
 

Constructor Detail

BetaDist

public BetaDist(double alpha,
                double beta)
Constructs a BetaDist object with parameters α = alpha and β = beta and default domain (0, 1).


BetaDist

public BetaDist(double alpha,
                double beta,
                double a,
                double b)
Constructs a BetaDist object with parameters α = alpha and β = beta, and domain (a, b).


BetaDist

public BetaDist(double alpha,
                double beta,
                int d)
Constructs a BetaDist object with parameters α = alpha and β = beta, and approximations of roughly d decimal digits of precision when computing the distribution, complementary distribution, and inverse functions. The default domain (0, 1) is used.


BetaDist

public BetaDist(double alpha,
                double beta,
                double a,
                double b,
                int d)
Constructs a BetaDist object with parameters α = alpha and β = beta, and approximations of roughly d decimal digits of precision when computing distribution, complementary distribution, and inverse functions. The domain (a, b) is used.

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

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)
Same as density (alpha, beta, 0, 1, x).


density

public static double density(double alpha,
                             double beta,
                             double a,
                             double b,
                             double x)
Computes the density function of the beta distribution.


cdf

public static double cdf(double alpha,
                         double beta,
                         int d,
                         double x)
Same as cdf (alpha, beta, 0, 1, d, x).


cdf

public static double cdf(double alpha,
                         double beta,
                         double a,
                         double b,
                         int d,
                         double x)
Computes an approximation of the distribution function, with roughly d decimal digits of precision.


barF

public static double barF(double alpha,
                          double beta,
                          int d,
                          double x)
Same as barF (alpha, beta, 0, 1, d, x).


barF

public static double barF(double alpha,
                          double beta,
                          double a,
                          double b,
                          int d,
                          double x)
Computes the complementary distribution function.


inverseF

public static double inverseF(double alpha,
                              double beta,
                              int d,
                              double u)
Same as inverseF (alpha, beta, 0, 1, d, u).


inverseF

public static double inverseF(double alpha,
                              double beta,
                              double a,
                              double b,
                              int d,
                              double u)
Returns the inverse beta distribution function using the algorithm implemented in the Cephes math library. The method performs interval halving or Newton iterations to compute the inverse. The precision depends on the accuracy of the cdf method. The argument d gives a good idea of the precision attained.


getMLE

public static double[] getMLE(double[] x,
                              int n)
Estimates the parameters (α, β) of the beta distribution over the interval [0, 1] 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)]

getMaximumLikelihoodEstimate

@Deprecated
public static double[] getMaximumLikelihoodEstimate(double[] x,
                                                               int n)
Deprecated. 

Same as getMLE.


getInstanceFromMLE

public static BetaDist getInstanceFromMLE(double[] x,
                                          int n)
Creates a new instance of a beta distribution with parameters α and β over the interval [0, 1] 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] = α/(α + β) of the beta distribution with parameters α and β, over the interval [0, 1].

Returns:
the mean of the Beta distribution

getMean

public static double getMean(double alpha,
                             double beta,
                             double a,
                             double b)
Computes and returns the mean E[X] = ( + )/(α + β) of the beta distribution with parameters α and β over the interval [a, b].

Returns:
the mean of the Beta distribution

getVariance

public static double getVariance(double alpha,
                                 double beta)
. Computes and returns the variance Var[X] = $ {\frac{{\alpha\beta}}{{(\alpha + \beta)^2 (\alpha + \beta + 1)}}}$ of the beta distribution with parameters α and β, over the interval [0, 1].

Returns:
the variance of the beta distribution Var[X] = αβ/[(α + β)2(α + β + 1)].

getVariance

public static double getVariance(double alpha,
                                 double beta,
                                 double a,
                                 double b)
. Computes and returns the variance Var[X] = $ {\frac{{\alpha\beta(b-a)^2}}{{(\alpha + \beta)^2 (\alpha + \beta + 1)}}}$ of the beta distribution with parameters α and β, over the interval [a, b].

Returns:
the variance of the beta distribution Var[X] = αβ/[(α + β)2(α + β + 1)].

getStandardDeviation

public static double getStandardDeviation(double alpha,
                                          double beta)
Computes the standard deviation of the beta distribution with parameters α and β, over the interval [0, 1].

Returns:
the standard deviation of the Beta distribution

getStandardDeviation

public static double getStandardDeviation(double alpha,
                                          double beta,
                                          double a,
                                          double b)
Computes the standard deviation of the beta distribution with parameters α and β, over the interval [a, b].

Returns:
the standard deviation of the Beta distribution

getAlpha

public double getAlpha()
Returns the parameter α of this object.


getBeta

public double getBeta()
Returns the parameter β of this object.


getA

public double getA()
Returns the parameter a of this object.


getB

public double getB()
Returns the parameter b of this object.


setParams

public void setParams(double alpha,
                      double beta,
                      double a,
                      double b,
                      int d)

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

SSJ
V. 2.2.

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