SSJ
V. 2.6.

umontreal.iro.lecuyer.probdist
Class BetaSymmetricalDist

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

public class BetaSymmetricalDist
extends BetaDist

Specializes the class BetaDist to the case of a symmetrical beta distribution over the interval [0, 1], with shape parameters α = β. Faster methods are implemented here for this special case. Because of the symmetry around 1/2, four series are used to compute the cdf, two around x = 0 and two around x = 1/2.


Field Summary
 
Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
decPrec
 
Constructor Summary
BetaSymmetricalDist(double alpha)
          Constructs a BetaSymmetricalDist object with parameters α = β = alpha, over the unit interval (0, 1).
BetaSymmetricalDist(double alpha, int d)
          Same as BetaSymmetricalDist (alpha), but using approximations of roughly d decimal digits of precision when computing the distribution, complementary distribution, and inverse functions.
 
Method Summary
 double barF(double x)
          Returns the complementary distribution function.
static double barF(double alpha, int d, double x)
          Returns the complementary distribution function.
 double cdf(double x)
          Returns the distribution function F(x).
static double cdf(double alpha, int d, double x)
          Same as cdf (alpha, alpha, d, x).
static double density(double alpha, double x)
          Returns the density evaluated at x.
static BetaSymmetricalDist getInstanceFromMLE(double[] x, int n)
          Creates a new instance of a symmetrical beta distribution with parameter α 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)
          Computes and returns the mean E[X] = 1/2 of the symmetrical beta distribution with parameter α.
static double[] getMLE(double[] x, int n)
          Estimates the parameter α of the symmetrical 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 the parameter of the current distribution.
 double getStandardDeviation()
          Returns the standard deviation.
static double getStandardDeviation(double alpha)
          Computes and returns the standard deviation of the symmetrical beta distribution with parameter α.
 double getVariance()
          Returns the variance.
static double getVariance(double alpha)
          Computes and returns the variance, Var[X] = 1/(8α + 4), of the symmetrical beta distribution with parameter α.
 double inverseF(double u)
          Returns the inverse distribution function x = F-1(u).
static double inverseF(double alpha, double u)
          Returns the inverse distribution function evaluated at u, for the symmetrical beta distribution over the interval [0, 1], with shape parameters 0 < α = β = alpha.
 void setParams(double alpha, double beta, double a, double b, int d)
           
 String toString()
           
 
Methods inherited from class umontreal.iro.lecuyer.probdist.BetaDist
barF, barF, barF, barF, cdf, cdf, cdf, cdf, density, density, density, getA, getAlpha, getB, getBeta, getMean, getMean, getStandardDeviation, getStandardDeviation, getVariance, getVariance, inverseF, inverseF, inverseF, inverseF, setParams
 
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

BetaSymmetricalDist

public BetaSymmetricalDist(double alpha)
Constructs a BetaSymmetricalDist object with parameters α = β = alpha, over the unit interval (0, 1).


BetaSymmetricalDist

public BetaSymmetricalDist(double alpha,
                           int d)
Same as BetaSymmetricalDist (alpha), but using approximations of roughly d decimal digits of precision when computing the distribution, complementary distribution, and inverse functions.

Method Detail

cdf

public double cdf(double x)
Description copied from interface: Distribution
Returns the distribution function F(x).

Specified by:
cdf in interface Distribution
Overrides:
cdf in class BetaDist
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 BetaDist
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 BetaDist
Parameters:
u - value at which the inverse distribution function is evaluated
Returns:
the inverse distribution function evaluated at u

density

public static double density(double alpha,
                             double x)
Returns the density evaluated at x.


cdf

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


barF

public static double barF(double alpha,
                          int d,
                          double x)
Returns the complementary distribution function.


inverseF

public static double inverseF(double alpha,
                              double u)
Returns the inverse distribution function evaluated at u, for the symmetrical beta distribution over the interval [0, 1], with shape parameters 0 < α = β = alpha. Uses four different hypergeometric series to compute the distribution u = F(x) (for the four cases x close to 0 and α < 1, x close to 0 and α > 1, x close to 1/2 and α < 1, and x close to 1/2 and α > 1), which are then solved by Newton's method for the solution of equations. For α > 100000, uses a normal approximation given in.


getMean

public double getMean()
Description copied from class: ContinuousDistribution
Returns the mean.

Specified by:
getMean in interface Distribution
Overrides:
getMean in class BetaDist
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 BetaDist
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 BetaDist
Returns:
the standard deviation

getMLE

public static double[] getMLE(double[] x,
                              int n)
Estimates the parameter α of the symmetrical beta distribution over the interval [0, 1] using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1. The estimate is returned in element 0 of the returned array.

Parameters:
x - the list of observations to use to evaluate parameters
n - the number of observations to use to evaluate parameters
Returns:
returns the parameter [ hat(α)]

getInstanceFromMLE

public static BetaSymmetricalDist getInstanceFromMLE(double[] x,
                                                     int n)
Creates a new instance of a symmetrical beta distribution with parameter α 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)
Computes and returns the mean E[X] = 1/2 of the symmetrical beta distribution with parameter α.

Returns:
the mean of the symmetrical beta distribution E[X] = 1/2

getVariance

public static double getVariance(double alpha)
Computes and returns the variance, Var[X] = 1/(8α + 4), of the symmetrical beta distribution with parameter α.

Returns:
the variance of the symmetrical beta distribution Var[X] = 1/[4(2α + 1)]

getStandardDeviation

public static double getStandardDeviation(double alpha)
Computes and returns the standard deviation of the symmetrical beta distribution with parameter α.

Returns:
the standard deviation of the symmetrical beta distribution

setParams

public void setParams(double alpha,
                      double beta,
                      double a,
                      double b,
                      int d)
Overrides:
setParams in class BetaDist

getParams

public double[] getParams()
Return a table containing the parameter of the current distribution.

Specified by:
getParams in interface Distribution
Overrides:
getParams in class BetaDist

toString

public String toString()
Overrides:
toString in class BetaDist

SSJ
V. 2.6.

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