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

Constructors

Constructor and Description
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

Modifier and Type	Method and Description
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)
java.lang.String	toString()

Methods inherited from class umontreal.iro.lecuyer.probdist.BetaDist

barF, barF, cdf, cdf, density, density, density, getA, getAlpha, getB, getBeta, getMean, getMean, getStandardDeviation, getStandardDeviation, getVariance, getVariance, inverseF, inverseF

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 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 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 java.lang.String toString()

Overrides:

toString in class BetaDist