SSJ
V. 1.2.5.

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. A faster inversion method is 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. Given u, one then solves each series for x by using the Newton-Raphson method which shows quadratic convergence when the starting iterate is close enough to the solution x.


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
static double barF(double alpha, int d, double x)
          Same as barF (alpha, beta, d, x).
 double cdf(double x)
          Computes and 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 BetaDist 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 in table x[i], i = 0, 1,…, n - 1.
static double[] getMaximumLikelihoodEstimate(double[] x, int n)
          Estimates and returns the parameter [ hat(α)] of the symmetrical beta 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)
          Computes and returns the mean E[X] = 1/2 of the symmetrical beta distribution with parameter α.
 double getStandardDeviation()
          Returns the standard deviation of the distribution function.
static double getStandardDeviation(double alpha)
          Computes and returns the standard deviation of the symmetrical beta distribution with parameter α.
 double getVariance()
          Returns the variance of the distribution function.
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)
          Computes and returns the inverse distribution function F-1(u), defined in.
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)
           
 
Methods inherited from class umontreal.iro.lecuyer.probdist.BetaDist
barF, barF, cdf, cdf, density, density, density, getA, getAlpha, getB, getBeta, getMean, getStandardDeviation, getVariance, inverseF, inverseF
 
Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
barF, inverseBisection, inverseBrent
 
Methods inherited from class java.lang.Object
equals, getClass, hashCode, notify, notifyAll, toString, 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
Computes and 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

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 BetaDist
Parameters:
u - value in the interval (0, 1) for 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)
Same as barF (alpha, beta, d, x).


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 interface: Distribution
Returns the mean of the distribution function.

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

getVariance

public double getVariance()
Description copied from interface: Distribution
Returns the variance of the distribution function.

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

getStandardDeviation

public double getStandardDeviation()
Description copied from interface: Distribution
Returns the standard deviation of the distribution function.

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

getInstanceFromMLE

public static BetaDist 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 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 parameter [ hat(α)] of the symmetrical beta 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 to use to evaluate parameters
n - the number of observations to use to evaluate parameters
Returns:
returns the parameter [ hat(α)]

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

SSJ
V. 1.2.5.

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