umontreal.iro.lecuyer.probdist
Class BetaSymmetricalDist

java.lang.Object
  umontreal.iro.lecuyer.probdist.ContinuousDistribution
      umontreal.iro.lecuyer.probdist.BetaDist
          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.
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, 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.

setParams

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

Overrides:

setParams in class BetaDist