Specializes the class BetaDist to the case of a symmetrical beta distribution over the interval \([0,1]\), with shape parameters. More...

Inheritance diagram for umontreal.ssj.probdist.BetaSymmetricalDist:

Public Member Functions
	BetaSymmetricalDist (double alpha)
	Constructs a BetaSymmetricalDist object with parameters.
	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.
double	cdf (double x)
	Returns the distribution function \(F(x)\).
double	barF (double x)
	Returns the complementary distribution function.
double	inverseF (double u)
	Returns the inverse distribution function \(x = F^{-1}(u)\).
double	getMean ()
	Returns the mean.
double	getVariance ()
	Returns the variance.
double	getStandardDeviation ()
	Returns the standard deviation.
double[]	getParams ()
	Return a table containing the parameter of the current distribution.
String	toString ()
	Returns a String containing information about the current distribution.
Public Member Functions inherited from umontreal.ssj.probdist.BetaDist
	BetaDist (double alpha, double beta)
	Constructs a BetaDist object with parameters \(\alpha=\) alpha, \(\beta=\) beta and default domain \([0,1]\).
	BetaDist (double alpha, double beta, double a, double b)
	Constructs a BetaDist object with parameters \(\alpha=\) alpha, \(\beta=\) beta and domain.
double	density (double x)
	Returns \(f(x)\), the density evaluated at \(x\).
double	getAlpha ()
	Returns the parameter \(\alpha\) of this object.
double	getBeta ()
	Returns the parameter \(\beta\) of this object.
double	getA ()
	Returns the parameter \(a\) of this object.
double	getB ()
	Returns the parameter \(b\) of this object.
void	setParams (double alpha, double beta, double a, double b)
	Sets the parameters of the current distribution.
Public Member Functions inherited from umontreal.ssj.probdist.ContinuousDistribution
double	inverseBrent (double a, double b, double u, double tol)
	Computes the inverse distribution function \(x = F^{-1}(u)\), using the Brent-Dekker method.
double	inverseBisection (double u)
	Computes and returns the inverse distribution function \(x = F^{-1}(u)\), using bisection.
double	getXinf ()
	Returns \(x_a\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
double	getXsup ()
	Returns \(x_b\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
void	setXinf (double xa)
	Sets the value \(x_a=\) xa, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
void	setXsup (double xb)
	Sets the value \(x_b=\) xb, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).

Static Public Member Functions
static double	density (double alpha, double x)
	Returns the density evaluated at \(x\).
static double	cdf (double alpha, int d, double x)
	Same as `cdf(alpha, alpha, d, x)`.
static double	barF (double alpha, int d, double x)
	Returns the complementary distribution function.
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 \(0 < \alpha= \beta\) = alpha.
static double[]	getMLE (double[] x, int n)
	Estimates the parameter \(\alpha\) of the symmetrical beta distribution over the interval [0, 1] using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\).
static BetaSymmetricalDist	getInstanceFromMLE (double[] x, int n)
	Creates a new instance of a symmetrical beta distribution with parameter \(\alpha\) estimated using the maximum likelihood method based on the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\).
static double	getMean (double alpha)
	Computes and returns the mean \(E[X] = 1/2\) of the symmetrical beta distribution with parameter \(\alpha\).
static double	getVariance (double alpha)
	Computes and returns the variance, \(\mbox{Var}[X] = 1/(8\alpha+ 4)\), of the symmetrical beta distribution with parameter.
static double	getStandardDeviation (double alpha)
	Computes and returns the standard deviation of the symmetrical beta distribution with parameter \(\alpha\).
Static Public Member Functions inherited from umontreal.ssj.probdist.BetaDist
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.
static double	cdf (double alpha, double beta, double x)
	Same as `cdf(alpha, beta, 0, 1, x)`.
static double	cdf (double alpha, double beta, double a, double b, double x)
	Computes the distribution function.
static double	barF (double alpha, double beta, double x)
	Same as `barF(alpha, beta, 0, 1, x)`.
static double	barF (double alpha, double beta, double a, double b, double x)
	Computes the complementary distribution function.
static double	inverseF (double alpha, double beta, double u)
	Same as `inverseF(alpha, beta, 0, 1, u)`.
static double	inverseF (double alpha, double beta, double a, double b, double u)
	Returns the inverse beta distribution function using the algorithm implemented in [178] .
static double	getMean (double alpha, double beta)
	Computes and returns the mean \(E[X] = \alpha/ (\alpha+ \beta)\) of the beta distribution with parameters \(\alpha\) and \(\beta\), over the interval \([0, 1]\).
static double	getMean (double alpha, double beta, double a, double b)
	Computes and returns the mean \(E[X] = (b\alpha+ a\beta)/ (\alpha+ \beta)\) of the beta distribution with parameters.
static double	getVariance (double alpha, double beta)
	Computes and returns the variance \(\mbox{Var}[X] = \frac{\alpha\beta}{(\alpha+ \beta)^2 (\alpha+ \beta+ 1)}\) of the beta distribution with parameters \(\alpha\) and.
static double	getVariance (double alpha, double beta, double a, double b)
	Computes and returns the variance \(\mbox{Var}[X] = \frac{\alpha\beta(b-a)^2}{(\alpha+ \beta)^2 (\alpha+ \beta+ 1)}\) of the beta distribution with parameters \(\alpha\) and.
static double	getStandardDeviation (double alpha, double beta)
	Computes the standard deviation of the beta distribution with parameters \(\alpha\) and \(\beta\), 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 \(\alpha\) and \(\beta\), over the interval \([a, b]\).

Detailed Description

Specializes the class BetaDist to the case of a symmetrical beta distribution over the interval \([0,1]\), with shape parameters.

\(\alpha= \beta\). Faster methods are implemented here for this special case [129] . Because of the symmetry around 1/2, four series are used to compute the cdf, two around \(x = 0\) and two around \(x = 1/2\).

Definition at line 43 of file BetaSymmetricalDist.java.

Constructor & Destructor Documentation

◆ BetaSymmetricalDist() [1/2]

umontreal.ssj.probdist.BetaSymmetricalDist.BetaSymmetricalDist ( double alpha )

Constructs a BetaSymmetricalDist object with parameters.

\(\alpha= \beta=\) alpha, over the unit interval \((0,1)\).

Definition at line 80 of file BetaSymmetricalDist.java.

◆ BetaSymmetricalDist() [2/2]

umontreal.ssj.probdist.BetaSymmetricalDist.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.

Definition at line 90 of file BetaSymmetricalDist.java.

Member Function Documentation

◆ barF() [1/2]

double umontreal.ssj.probdist.BetaSymmetricalDist.barF	(	double	alpha,
		int	d,
		double	x )

static

Returns the complementary distribution function.

Definition at line 124 of file BetaSymmetricalDist.java.

◆ barF() [2/2]

double umontreal.ssj.probdist.BetaSymmetricalDist.barF ( double x )

Returns the complementary distribution function.

The default implementation computes \(\bar{F}(x) = 1 - F(x)\).

Parameters

x	value at which the complementary distribution function is evaluated

Returns: complementary distribution function evaluated at x

Reimplemented from umontreal.ssj.probdist.BetaDist.

Definition at line 99 of file BetaSymmetricalDist.java.

◆ cdf() [1/2]

double umontreal.ssj.probdist.BetaSymmetricalDist.cdf	(	double	alpha,
		int	d,
		double	x )

static

Same as cdf(alpha, alpha, d, x).

Definition at line 117 of file BetaSymmetricalDist.java.

◆ cdf() [2/2]

double umontreal.ssj.probdist.BetaSymmetricalDist.cdf ( double x )

Returns the distribution function \(F(x)\).

Parameters

x	value at which the distribution function is evaluated

Returns: distribution function evaluated at x

Reimplemented from umontreal.ssj.probdist.BetaDist.

Definition at line 95 of file BetaSymmetricalDist.java.

◆ density()

double umontreal.ssj.probdist.BetaSymmetricalDist.density	(	double	alpha,
		double	x )

static

Returns the density evaluated at \(x\).

Definition at line 110 of file BetaSymmetricalDist.java.

◆ getInstanceFromMLE()

BetaSymmetricalDist umontreal.ssj.probdist.BetaSymmetricalDist.getInstanceFromMLE	(	double[]	x,
		int	n )

static

Creates a new instance of a symmetrical beta distribution with parameter \(\alpha\) 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

Reimplemented from umontreal.ssj.probdist.BetaDist.

Definition at line 849 of file BetaSymmetricalDist.java.

◆ getMean() [1/2]

double umontreal.ssj.probdist.BetaSymmetricalDist.getMean ( )

Returns the mean.

Returns: the mean

Reimplemented from umontreal.ssj.probdist.BetaDist.

Definition at line 784 of file BetaSymmetricalDist.java.

◆ getMean() [2/2]

double umontreal.ssj.probdist.BetaSymmetricalDist.getMean ( double alpha )

static

Computes and returns the mean \(E[X] = 1/2\) of the symmetrical beta distribution with parameter \(\alpha\).

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

Definition at line 860 of file BetaSymmetricalDist.java.

◆ getMLE()

double[] umontreal.ssj.probdist.BetaSymmetricalDist.getMLE	(	double[]	x,
		int	n )

static

Estimates the parameter \(\alpha\) 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. The maximum likelihood estimator \(\hat{\alpha}\) satisfies the equation

\begin{align*} \psi(\hat{\alpha}) - \psi(2\hat{\alpha}) = \frac{1}{2n} \sum_{i=1}^n \ln(x_i(1 - x_i)) \end{align*}

where \(\bar{x}_n\) is the average of \(x[0], …, x[n-1]\), and \(\psi\) is the logarithmic derivative of the Gamma function \(\psi(x) = \Gamma’(x) / \Gamma(x)\).

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 [ \(\hat{\alpha}\)]

Reimplemented from umontreal.ssj.probdist.BetaDist.

Definition at line 812 of file BetaSymmetricalDist.java.

◆ getParams()

double[] umontreal.ssj.probdist.BetaSymmetricalDist.getParams ( )

Return a table containing the parameter of the current distribution.

Reimplemented from umontreal.ssj.probdist.BetaDist.

Definition at line 925 of file BetaSymmetricalDist.java.

◆ getStandardDeviation() [1/2]

double umontreal.ssj.probdist.BetaSymmetricalDist.getStandardDeviation ( )

Returns the standard deviation.

Returns: the standard deviation

Reimplemented from umontreal.ssj.probdist.BetaDist.

Definition at line 792 of file BetaSymmetricalDist.java.

◆ getStandardDeviation() [2/2]

double umontreal.ssj.probdist.BetaSymmetricalDist.getStandardDeviation ( double alpha )

static

Computes and returns the standard deviation of the symmetrical beta distribution with parameter \(\alpha\).

Returns: the standard deviation of the symmetrical beta distribution

Definition at line 887 of file BetaSymmetricalDist.java.

◆ getVariance() [1/2]

double umontreal.ssj.probdist.BetaSymmetricalDist.getVariance ( )

Returns the variance.

Returns: the variance

Reimplemented from umontreal.ssj.probdist.BetaDist.

Definition at line 788 of file BetaSymmetricalDist.java.

◆ getVariance() [2/2]

double umontreal.ssj.probdist.BetaSymmetricalDist.getVariance ( double alpha )

static

Computes and returns the variance, \(\mbox{Var}[X] = 1/(8\alpha+ 4)\), of the symmetrical beta distribution with parameter.

\(\alpha\).

Returns: the variance of the symmetrical beta distribution \(\mbox{Var}[X] = 1 / [4 (2\alpha+ 1)]\)

Definition at line 874 of file BetaSymmetricalDist.java.

◆ inverseF() [1/2]

double umontreal.ssj.probdist.BetaSymmetricalDist.inverseF	(	double	alpha,
		double	u )

static

Returns the inverse distribution function evaluated at \(u\), for the symmetrical beta distribution over the interval \([0,1]\), with shape parameters \(0 < \alpha= \beta\) = alpha.

Uses four different hypergeometric series to compute the distribution \(u = F(x)\) (for the four cases \(x\) close to 0 and \(\alpha< 1\),

\(x\) close to 0 and \(\alpha> 1\), \(x\) close to 1/2 and \(\alpha< 1\), and \(x\) close to 1/2 and \(\alpha> 1\)), which are then solved by Newton’s method for the solution of equations. For \(\alpha> 100000\), uses a normal approximation given in [195] .

Definition at line 141 of file BetaSymmetricalDist.java.

◆ inverseF() [2/2]

double umontreal.ssj.probdist.BetaSymmetricalDist.inverseF ( double u )

Returns the inverse distribution function \(x = F^{-1}(u)\).

Restrictions: \(u \in[0,1]\).

Parameters

u	value at which the inverse distribution function is evaluated

Returns: the inverse distribution function evaluated at u

Exceptions

IllegalArgumentException if \(u\) is not in the interval \([0,1]\)

Reimplemented from umontreal.ssj.probdist.BetaDist.

Definition at line 103 of file BetaSymmetricalDist.java.

◆ toString()

String umontreal.ssj.probdist.BetaSymmetricalDist.toString ( )

Returns a String containing information about the current distribution.

Reimplemented from umontreal.ssj.probdist.BetaDist.

Definition at line 933 of file BetaSymmetricalDist.java.

The documentation for this class was generated from the following file:

src/main/java/umontreal/ssj/probdist/BetaSymmetricalDist.java

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ BetaSymmetricalDist() [1/2]

◆ BetaSymmetricalDist() [2/2]

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ density()

◆ getInstanceFromMLE()

◆ getMean() [1/2]

◆ getMean() [2/2]

◆ getMLE()

◆ getParams()

◆ getStandardDeviation() [1/2]

◆ getStandardDeviation() [2/2]

◆ getVariance() [1/2]

◆ getVariance() [2/2]

◆ inverseF() [1/2]

◆ inverseF() [2/2]

◆ toString()