Extends the class ContinuousDistribution for the beta distribution [100] (page 210) with shape parameters \(\alpha> 0\) and \(\beta> 0\), over the interval \([a,b]\), where \(a < b\). More...

Inheritance diagram for BetaDist:

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Collaboration diagram for BetaDist:

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

	BetaDist (double alpha, double beta, int d)

	BetaDist (double alpha, double beta, double a, double b, int d)

double	density (double x)

double	cdf (double x)
	Returns the distribution function \(F(x)\). More...

double	barF (double x)
	Returns \(\bar{F}(x) = 1 - F(x)\). More...

double	inverseF (double u)
	Returns the inverse distribution function \(F^{-1}(u)\), defined in ( inverseF ). More...

double	getMean ()
	Returns the mean of the distribution function.

double	getVariance ()
	Returns the variance of the distribution function.

double	getStandardDeviation ()
	Returns the standard deviation of the distribution function.

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, int d)

void	setParams (double alpha, double beta, double a, double b)
	Sets the parameters of the current distribution. More...

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 ContinuousDistribution
abstract double	density (double x)
	Returns \(f(x)\), the density evaluated at \(x\). More...

double	barF (double x)
	Returns the complementary distribution function. More...

double	inverseBrent (double a, double b, double u, double tol)
	Computes the inverse distribution function \(x = F^{-1}(u)\), using the Brent-Dekker method. More...

double	inverseBisection (double u)
	Computes and returns the inverse distribution function \(x = F^{-1}(u)\), using bisection. More...

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

double	getMean ()
	Returns the mean. More...

double	getVariance ()
	Returns the variance. More...

double	getStandardDeviation ()
	Returns the standard deviation. More...

double	getXinf ()
	Returns \(x_a\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

double	getXsup ()
	Returns \(x_b\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

void	setXinf (double xa)
	Sets the value \(x_a=\) `xa`, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

void	setXsup (double xb)
	Sets the value \(x_b=\) `xb`, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

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, int d, double x)

static double	cdf (double alpha, double beta, double a, double b, int d, double x)

static double	barF (double alpha, double beta, int d, double x)

static double	barF (double alpha, double beta, double a, double b, int d, double x)

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. More...

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, int d, double u)

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, int d, double 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 [183] . More...

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 \(x[i]\), \(i = 0, 1,…, n-1\). More...

static BetaDist	getInstanceFromMLE (double[] x, int n)
	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 \(x[i]\), \(i = 0, 1, …, n-1\). More...

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 \([0, 1]\). More...

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

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

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

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

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

Protected Attributes
double	alpha

double	beta

double	a

double	bminusa

double	logFactor

double	Beta

double	logBeta

Protected Attributes inherited from ContinuousDistribution
double	supportA = Double.NEGATIVE_INFINITY

double	supportB = Double.POSITIVE_INFINITY

Static Package Functions
static double	beta_g (double x)

Package Attributes
double	b

Additional Inherited Members
Public Attributes inherited from ContinuousDistribution
int	decPrec = 15

Static Protected Attributes inherited from ContinuousDistribution
static final double	XBIG = 100.0

static final double	XBIGM = 1000.0

static final double []	EPSARRAY

Detailed Description

Extends the class ContinuousDistribution for the beta distribution [100] (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 ).

Member Function Documentation

◆ barF()

double barF ( double x )

Returns \(\bar{F}(x) = 1 - F(x)\).

Parameters

x	value at which the complementary distribution function is evaluated

Returns: complementary distribution function evaluated at x

Implements Distribution.

◆ cdf() [1/2]

double 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 Distribution.

◆ cdf() [2/2]

static double 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 [52], [39] . Otherwise, if \(\min(\alpha, \beta) \le30\), uses an approximation due to Bol’shev [172] , else uses a normal approximation [201] .

◆ getInstanceFromMLE()

static 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

◆ getMean() [1/2]

static double 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

◆ getMean() [2/2]

static double 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

◆ getMLE()

static double [] 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 [ \(\hat{\alpha}\), \(\hat{\beta}\)]

◆ getStandardDeviation() [1/2]

static double 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

◆ getStandardDeviation() [2/2]

static double 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

◆ getVariance() [1/2]

static double 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 \(\mbox{Var}[X] = \alpha\beta/ [(\alpha+ \beta)^2 (\alpha+ \beta+ 1)]\).

◆ getVariance() [2/2]

static double 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 \(\mbox{Var}[X] = \alpha\beta/ [(\alpha+ \beta)^2 (\alpha+ \beta+ 1)]\).

◆ inverseF() [1/2]

double inverseF ( double u )

Returns the inverse distribution function \(F^{-1}(u)\), defined in ( inverseF ).

Parameters

u	value in the interval \((0,1)\) for which the inverse distribution function is evaluated

Returns: the inverse distribution function evaluated at u

Implements Distribution.

◆ inverseF() [2/2]

static double inverseF	(	double	alpha,
		double	beta,
		double	a,
		double	b,
		double	u
	)

static

Returns the inverse beta distribution function using the algorithm implemented in [183] .

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

◆ setParams()

void setParams	(	double	alpha,
		double	beta,
		double	a,
		double	b
	)

Sets the parameters of the current distribution.

See the constructor.

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

BetaDist.java

Public Member Functions

Static Public Member Functions

Protected Attributes

Static Package Functions

Package Attributes

Additional Inherited Members

Detailed Description

Member Function Documentation

◆ barF()

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ getInstanceFromMLE()

◆ getMean() [1/2]

◆ getMean() [2/2]

◆ getMLE()

◆ getStandardDeviation() [1/2]

◆ getStandardDeviation() [2/2]

◆ getVariance() [1/2]

◆ getVariance() [2/2]

◆ inverseF() [1/2]

◆ inverseF() [2/2]

◆ setParams()