Extends the class ContinuousDistribution for the beta distribution. More...

Inheritance diagram for umontreal.ssj.probdist.BetaDist:

Public Member Functions
	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	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	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.
double[]	getParams ()
	Return an array containing the parameters of the current distribution as [ \(\alpha\), \(\beta\), \(a\), \(b\)].
String	toString ()
	Returns a String containing information about 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 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[]	getMLE (double[] x, int n)
	Estimates the parameters \((\alpha,\beta)\) of the beta distribution over the interval \([0,1]\) using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\).
static BetaDist	getInstanceFromMLE (double[] x, int n)
	Creates a new instance of a beta distribution with parameters.
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

Extends the class ContinuousDistribution for the beta distribution.

[96] (page 210) with shape parameters \(\alpha> 0\) and \(\beta> 0\), over the interval \([a,b]\), where \(a < b\). This distribution has density

\[ f(x) = \frac{ (x-a)^{\alpha- 1}(b - x)^{\beta- 1}}{\mathcal{B} (\alpha, \beta)(b - a)^{\alpha+ \beta- 1}}, \qquad\mbox{for } a\le x\le b, \mbox{ and }0\mbox{ elsewhere}, \]

and distribution function

\[ F(x) = I_{\alpha,\beta}(x) = \int_a^x \frac{(\xi- a)^{\alpha-1} (b - \xi)^{\beta-1}}{\mathcal{B} (\alpha, \beta)(b - a)^{\alpha+ \beta- 1}} d\xi, \qquad\mbox{for } a\le x\le b, \tag{Fbeta} \]

where \(\mathcal{B}(\alpha,\beta)\) is the beta function defined by

\[ \mathcal{B} (\alpha,\beta) = \frac{\Gamma(\alpha) \Gamma(\beta)}{ \Gamma(\alpha+\beta)},\tag{betadef} \]

and \(\Gamma(x)\) is the gamma function defined in ( Gamma ).

Definition at line 54 of file BetaDist.java.

Constructor & Destructor Documentation

◆ BetaDist() [1/2]

umontreal.ssj.probdist.BetaDist.BetaDist	(	double	alpha,
		double	beta )

Constructs a BetaDist object with parameters \(\alpha=\) alpha, \(\beta=\) beta and default domain \([0,1]\).

Definition at line 104 of file BetaDist.java.

◆ BetaDist() [2/2]

umontreal.ssj.probdist.BetaDist.BetaDist	(	double	alpha,
		double	beta,
		double	a,
		double	b )

Constructs a BetaDist object with parameters \(\alpha=\) alpha, \(\beta=\) beta and domain.

\([\)a \(,\) b \(]\).

Definition at line 114 of file BetaDist.java.

Member Function Documentation

◆ barF() [1/3]

double umontreal.ssj.probdist.BetaDist.barF	(	double	alpha,
		double	beta,
		double	a,
		double	b,
		double	x )

static

Computes the complementary distribution function.

Definition at line 368 of file BetaDist.java.

◆ barF() [2/3]

double umontreal.ssj.probdist.BetaDist.barF	(	double	alpha,
		double	beta,
		double	x )

static

Same as barF(alpha, beta, 0, 1, x).

Definition at line 361 of file BetaDist.java.

◆ barF() [3/3]

double umontreal.ssj.probdist.BetaDist.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.ContinuousDistribution.

Reimplemented in umontreal.ssj.probdist.BetaSymmetricalDist.

Definition at line 132 of file BetaDist.java.

◆ cdf() [1/3]

double umontreal.ssj.probdist.BetaDist.cdf	(	double	alpha,
		double	beta,
		double	a,
		double	b,
		double	x )

static

Computes the distribution function.

If \(\max(\alpha, \beta) \le10^4\), uses a continuous fraction in \(\alpha\) and

\(\beta\) given in [51], [38] . Otherwise, if \(\min(\alpha, \beta) \le30\), uses an approximation due to Bol’shev [168] , else uses a normal approximation [195] .

Definition at line 353 of file BetaDist.java.

◆ cdf() [2/3]

double umontreal.ssj.probdist.BetaDist.cdf	(	double	alpha,
		double	beta,
		double	x )

static

Same as cdf(alpha, beta, 0, 1, x).

Definition at line 317 of file BetaDist.java.

◆ cdf() [3/3]

double umontreal.ssj.probdist.BetaDist.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

Implements umontreal.ssj.probdist.Distribution.

Reimplemented in umontreal.ssj.probdist.BetaSymmetricalDist.

Definition at line 127 of file BetaDist.java.

◆ density() [1/3]

double umontreal.ssj.probdist.BetaDist.density	(	double	alpha,
		double	beta,
		double	a,
		double	b,
		double	x )

static

Computes the density function of the beta distribution.

Definition at line 167 of file BetaDist.java.

◆ density() [2/3]

double umontreal.ssj.probdist.BetaDist.density	(	double	alpha,
		double	beta,
		double	x )

static

Same as density(alpha, beta, 0, 1, x).

Definition at line 160 of file BetaDist.java.

◆ density() [3/3]

double umontreal.ssj.probdist.BetaDist.density ( double x )

Returns \(f(x)\), the density evaluated at \(x\).

Parameters

x	value at which the density is evaluated

Returns: density function evaluated at x

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 119 of file BetaDist.java.

◆ getA()

double umontreal.ssj.probdist.BetaDist.getA ( )

Returns the parameter \(a\) of this object.

Definition at line 818 of file BetaDist.java.

◆ getAlpha()

double umontreal.ssj.probdist.BetaDist.getAlpha ( )

Returns the parameter \(\alpha\) of this object.

Definition at line 804 of file BetaDist.java.

◆ getB()

double umontreal.ssj.probdist.BetaDist.getB ( )

Returns the parameter \(b\) of this object.

Definition at line 825 of file BetaDist.java.

◆ getBeta()

double umontreal.ssj.probdist.BetaDist.getBeta ( )

Returns the parameter \(\beta\) of this object.

Definition at line 811 of file BetaDist.java.

◆ getInstanceFromMLE()

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

static

Creates a new instance of a beta distribution with parameters.

\(\alpha\) and \(\beta\) over the interval \([0,1]\) 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 in umontreal.ssj.probdist.BetaSymmetricalDist.

Definition at line 718 of file BetaDist.java.

◆ getMean() [1/3]

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

Returns the mean.

Returns: the mean

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Reimplemented in umontreal.ssj.probdist.BetaSymmetricalDist.

Definition at line 142 of file BetaDist.java.

◆ getMean() [2/3]

double umontreal.ssj.probdist.BetaDist.getMean	(	double	alpha,
		double	beta )

static

Computes and returns the mean \(E[X] = \alpha/ (\alpha+ \beta)\) of the beta distribution with parameters \(\alpha\) and \(\beta\), over the interval \([0, 1]\).

Returns: the mean of the Beta distribution

Definition at line 730 of file BetaDist.java.

◆ getMean() [3/3]

double umontreal.ssj.probdist.BetaDist.getMean	(	double	alpha,
		double	beta,
		double	a,
		double	b )

static

Computes and returns the mean \(E[X] = (b\alpha+ a\beta)/ (\alpha+ \beta)\) of the beta distribution with parameters.

\(\alpha\) and \(\beta\) over the interval \([a, b]\).

Returns: the mean of the Beta distribution

Definition at line 741 of file BetaDist.java.

◆ getMLE()

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

static

Estimates the parameters \((\alpha,\beta)\) of the beta distribution over the interval \([0,1]\) using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\).

The estimates are returned in a two-element array, in regular order: [ \(\alpha\), \(\beta\)]. The maximum likelihood estimators are the values \((\hat{\alpha}, \hat{\beta})\) that satisfy the equations:

\begin{align*} \psi(\alpha) - \psi(\alpha+ \beta) & = \frac{1}{n} \sum_{i=1}^n \ln(x_i) \\ \psi(\beta) - \psi(\alpha+ \beta) & = \frac{1}{n} \sum_{i=1}^n \ln(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 parameters [ \(\hat{\alpha}\), \(\hat{\beta}\)]

Reimplemented in umontreal.ssj.probdist.BetaSymmetricalDist.

Definition at line 665 of file BetaDist.java.

◆ getParams()

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

Return an array containing the parameters of the current distribution as [ \(\alpha\), \(\beta\), \(a\), \(b\)].

Implements umontreal.ssj.probdist.Distribution.

Reimplemented in umontreal.ssj.probdist.BetaSymmetricalDist.

Definition at line 859 of file BetaDist.java.

◆ getStandardDeviation() [1/3]

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

Returns the standard deviation.

Returns: the standard deviation

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Reimplemented in umontreal.ssj.probdist.BetaSymmetricalDist.

Definition at line 152 of file BetaDist.java.

◆ getStandardDeviation() [2/3]

double umontreal.ssj.probdist.BetaDist.getStandardDeviation	(	double	alpha,
		double	beta )

static

Computes the standard deviation of the beta distribution with parameters \(\alpha\) and \(\beta\), over the interval \([0, 1]\).

Returns: the standard deviation of the Beta distribution

Definition at line 787 of file BetaDist.java.

◆ getStandardDeviation() [3/3]

double umontreal.ssj.probdist.BetaDist.getStandardDeviation	(	double	alpha,
		double	beta,
		double	a,
		double	b )

static

Computes the standard deviation of the beta distribution with parameters \(\alpha\) and \(\beta\), over the interval \([a, b]\).

Returns: the standard deviation of the Beta distribution

Definition at line 797 of file BetaDist.java.

◆ getVariance() [1/3]

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

Returns the variance.

Returns: the variance

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Reimplemented in umontreal.ssj.probdist.BetaSymmetricalDist.

Definition at line 147 of file BetaDist.java.

◆ getVariance() [2/3]

double umontreal.ssj.probdist.BetaDist.getVariance	(	double	alpha,
		double	beta )

static

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.

\(\beta\), over the interval \([0, 1]\).

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

Definition at line 759 of file BetaDist.java.

◆ getVariance() [3/3]

double umontreal.ssj.probdist.BetaDist.getVariance	(	double	alpha,
		double	beta,
		double	a,
		double	b )

static

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.

\(\beta\), over the interval \([a, b]\).

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

Definition at line 772 of file BetaDist.java.

◆ inverseF() [1/3]

double umontreal.ssj.probdist.BetaDist.inverseF	(	double	alpha,
		double	beta,
		double	a,
		double	b,
		double	u )

static

Returns the inverse beta distribution function using the algorithm implemented in [178] .

The method performs interval halving or Newton iterations to compute the inverse.

Definition at line 640 of file BetaDist.java.

◆ inverseF() [2/3]

double umontreal.ssj.probdist.BetaDist.inverseF	(	double	alpha,
		double	beta,
		double	u )

static

Same as inverseF(alpha, beta, 0, 1, u).

Definition at line 624 of file BetaDist.java.

◆ inverseF() [3/3]

double umontreal.ssj.probdist.BetaDist.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.ContinuousDistribution.

Reimplemented in umontreal.ssj.probdist.BetaSymmetricalDist.

Definition at line 137 of file BetaDist.java.

◆ setParams()

void umontreal.ssj.probdist.BetaDist.setParams	(	double	alpha,
		double	beta,
		double	a,
		double	b )

Sets the parameters of the current distribution.

See the constructor.

Definition at line 832 of file BetaDist.java.

◆ toString()

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

Returns a String containing information about the current distribution.

Reimplemented in umontreal.ssj.probdist.BetaSymmetricalDist.

Definition at line 868 of file BetaDist.java.

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

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

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ BetaDist() [1/2]

◆ BetaDist() [2/2]

Member Function Documentation

◆ barF() [1/3]

◆ barF() [2/3]

◆ barF() [3/3]

◆ cdf() [1/3]

◆ cdf() [2/3]

◆ cdf() [3/3]

◆ density() [1/3]

◆ density() [2/3]

◆ density() [3/3]

◆ getA()

◆ getAlpha()

◆ getB()

◆ getBeta()

◆ getInstanceFromMLE()

◆ getMean() [1/3]

◆ getMean() [2/3]

◆ getMean() [3/3]

◆ getMLE()

◆ getParams()

◆ getStandardDeviation() [1/3]

◆ getStandardDeviation() [2/3]

◆ getStandardDeviation() [3/3]

◆ getVariance() [1/3]

◆ getVariance() [2/3]

◆ getVariance() [3/3]

◆ inverseF() [1/3]

◆ inverseF() [2/3]

◆ inverseF() [3/3]

◆ setParams()

◆ toString()