Extends the class ContinuousDistribution for the gamma distribution [99] (page 337) with shape parameter \(\alpha> 0\) and scale parameter \(\lambda> 0\). More...

Inheritance diagram for GammaDist:

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Collaboration diagram for GammaDist:

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Public Member Functions
	GammaDist (double alpha)
	Constructs a `GammaDist` object with parameters \(\alpha\) = `alpha` and \(\lambda=1\).

	GammaDist (double alpha, double lambda)
	Constructs a `GammaDist` object with parameters \(\alpha\) = `alpha` and \(\lambda\) = `lambda`.

	GammaDist (double alpha, double lambda, int d)
	Constructs a `GammaDist` object with parameters \(\alpha\) = `alpha` and \(\lambda\) = `lambda`, and approximations of roughly `d` decimal digits of precision when computing functions.

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 ()
	Return the parameter \(\alpha\) for this object.

double	getLambda ()
	Return the parameter \(\lambda\) for this object.

void	setParams (double alpha, double lambda, int d)

double []	getParams ()
	Return a table containing the parameters of the current distribution. More...

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 lambda, double x)
	Computes the density function ( fgamma ) at \(x\).

static double	cdf (double alpha, double lambda, int d, double x)
	Returns an approximation of the gamma distribution function with parameters \(\alpha\) = `alpha` and \(\lambda\) = `lambda`, whose density is given by ( fgamma ). More...

static double	cdf (double alpha, int d, double x)
	Equivalent to `cdf (alpha, 1.0, d, x)`.

static double	barF (double alpha, double lambda, int d, double x)
	Computes the complementary distribution function.

static double	barF (double alpha, int d, double x)
	Same as barF(alpha, 1.0, d, x).

static double	inverseF (double alpha, double lambda, int d, double u)
	Computes the inverse distribution function.

static double	inverseF (double alpha, int d, double u)
	Same as inverseF(alpha, 1, d, u).

static double []	getMLE (double[] x, int n)
	Estimates the parameters \((\alpha,\lambda)\) of the gamma distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\). More...

static GammaDist	getInstanceFromMLE (double[] x, int n)
	Creates a new instance of a gamma distribution with parameters \(\alpha\) and \(\lambda\) 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 lambda)
	Computes and returns the mean \(E[X] = \alpha/\lambda\) of the gamma distribution with parameters \(\alpha\) and \(\lambda\). More...

static double	getVariance (double alpha, double lambda)
	Computes and returns the variance \(\mbox{Var}[X] = \alpha/\lambda^2\) of the gamma distribution with parameters \(\alpha\) and \(\lambda\). More...

static double	getStandardDeviation (double alpha, double lambda)
	Computes and returns the standard deviation of the gamma distribution with parameters \(\alpha\) and \(\lambda\). More...

Static Package Functions
static double	mybelog (double x)

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

Protected Attributes inherited from ContinuousDistribution
double	supportA = Double.NEGATIVE_INFINITY

double	supportB = Double.POSITIVE_INFINITY

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 gamma distribution [99] (page 337) with shape parameter \(\alpha> 0\) and scale parameter \(\lambda> 0\).

The density is

\[ f(x) = \frac{\lambda^{\alpha}x^{\alpha- 1}e^{-\lambda x}}{\Gamma(\alpha)}, \qquad\mbox{for } x > 0,\tag{fgamma} \]

where \(\Gamma\) is the gamma function, defined by

\[ \Gamma(\alpha) = \int_0^{\infty}x^{\alpha-1} e^{-x} dx. \tag{Gamma} \]

In particular, \(\Gamma(n) = (n-1)!\) when \(n\) is a positive integer.

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	lambda,
		int	d,
		double	x
	)

static

Returns an approximation of the gamma distribution function with parameters \(\alpha\) = alpha and \(\lambda\) = lambda, whose density is given by ( fgamma ).

The approximation is an improved version of the algorithm in [19] . The function tries to return \(d\) decimals digits of precision. For \(\alpha\) not too large (e.g., \(\alpha\le1000\)), \(d\) gives a good idea of the precision attained.

◆ getInstanceFromMLE()

static GammaDist getInstanceFromMLE	(	double []	x,
		int	n
	)

static

Creates a new instance of a gamma distribution with parameters \(\alpha\) and \(\lambda\) 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()

static double getMean	(	double	alpha,
		double	lambda
	)

static

Computes and returns the mean \(E[X] = \alpha/\lambda\) of the gamma distribution with parameters \(\alpha\) and \(\lambda\).

Returns: the mean of the gamma distribution \(E[X] = \alpha/ \lambda\)

◆ getMLE()

static double [] getMLE	(	double []	x,
		int	n
	)

static

Estimates the parameters \((\alpha,\lambda)\) of the gamma distribution 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: [ \(\alpha\), \(\lambda\)]. The maximum likelihood estimators are the values \((\hat{\alpha}, \hat{\lambda})\) that satisfy the equations:

\begin{align*} \frac{1}{n} \sum_{i=1}^n \ln(x_i) - \ln(\bar{x}_n) & = \psi(\hat{\alpha}) - \ln(\hat{\alpha}) \\ \hat{\lambda} \bar{x}_n & = \hat{\alpha} \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 \(\psi(x) = \Gamma’(x) / \Gamma(x)\) ([99] (page 361)).

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

◆ getParams()

double [] getParams ( )

Return a table containing the parameters of the current distribution.

This table is put in regular order: [ \(\alpha\), \(\lambda\)].

Implements Distribution.

◆ getStandardDeviation()

static double getStandardDeviation	(	double	alpha,
		double	lambda
	)

static

Computes and returns the standard deviation of the gamma distribution with parameters \(\alpha\) and \(\lambda\).

Returns: the standard deviation of the gamma distribution

◆ getVariance()

static double getVariance	(	double	alpha,
		double	lambda
	)

static

Computes and returns the variance \(\mbox{Var}[X] = \alpha/\lambda^2\) of the gamma distribution with parameters \(\alpha\) and \(\lambda\).

Returns: the variance of the gamma distribution \(\mbox{Var}[X] = \alpha/ \lambda^2\)

◆ inverseF()

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.

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

GammaDist.java

Public Member Functions

Static Public Member Functions

Static Package Functions

Additional Inherited Members

Detailed Description

Member Function Documentation

◆ barF()

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ getInstanceFromMLE()

◆ getMean()

◆ getMLE()

◆ getParams()

◆ getStandardDeviation()

◆ getVariance()

◆ inverseF()