Extends the class ContinuousDistribution for the Pearson type VI distribution with shape parameters \(\alpha_1 > 0\) and \(\alpha_2 > 0\), and scale parameter \(\beta> 0\). More...

Inheritance diagram for Pearson6Dist:

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Collaboration diagram for Pearson6Dist:

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Public Member Functions
	Pearson6Dist (double alpha1, double alpha2, double beta)
	Constructs a `Pearson6Dist` object with parameters \(\alpha_1\) = `alpha1`, \(\alpha_2\) = `alpha2` and \(\beta\) = `beta`.

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	getAlpha1 ()
	Returns the \(\alpha_1\) parameter of this object.

double	getAlpha2 ()
	Returns the \(\alpha_2\) parameter of this object.

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

void	setParam (double alpha1, double alpha2, double beta)
	Sets the parameters \(\alpha_1\), \(\alpha_2\) and \(\beta\) of this object.

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 alpha1, double alpha2, double beta, double x)
	Computes the density function of a Pearson VI distribution with shape parameters \(\alpha_1\) and \(\alpha_2\), and scale parameter \(\beta\).

static double	cdf (double alpha1, double alpha2, double beta, double x)
	Computes the distribution function of a Pearson VI distribution with shape parameters \(\alpha_1\) and \(\alpha_2\), and scale parameter \(\beta\).

static double	barF (double alpha1, double alpha2, double beta, double x)
	Computes the complementary distribution function of a Pearson VI distribution with shape parameters \(\alpha_1\) and \(\alpha_2\), and scale parameter \(\beta\).

static double	inverseF (double alpha1, double alpha2, double beta, double u)
	Computes the inverse distribution function of a Pearson VI distribution with shape parameters \(\alpha_1\) and \(\alpha_2\), and scale parameter \(\beta\).

static double []	getMLE (double[] x, int n)
	Estimates the parameters \((\alpha_1,\alpha_2,\beta)\) of the Pearson VI distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\). More...

static Pearson6Dist	getInstanceFromMLE (double[] x, int n)
	Creates a new instance of a Pearson VI distribution with parameters \(\alpha_1\), \(\alpha_2\) and \(\beta\), estimated using the maximum likelihood method based on the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\). More...

static double	getMean (double alpha1, double alpha2, double beta)
	Computes and returns the mean \(E[X] = (\beta\alpha_1) / (\alpha_2 - 1)\) of a Pearson VI distribution with shape parameters \(\alpha_1\) and \(\alpha_2\), and scale parameter \(\beta\).

static double	getVariance (double alpha1, double alpha2, double beta)
	Computes and returns the variance \(\mbox{Var}[X] = [\beta^2 \alpha_1 (\alpha_1 + \alpha_2 - 1)] / [(\alpha_2 - 1)^2(\alpha_2 - 2)]\) of a Pearson VI distribution with shape parameters \(\alpha_1\) and \(\alpha_2\), and scale parameter \(\beta\).

static double	getStandardDeviation (double alpha1, double alpha2, double beta)
	Computes and returns the standard deviation of a Pearson VI distribution with shape parameters \(\alpha_1\) and \(\alpha_2\), and scale parameter \(\beta\).

Protected Attributes
double	alpha1

double	alpha2

double	beta

double	logBeta

Protected Attributes inherited from ContinuousDistribution
double	supportA = Double.NEGATIVE_INFINITY

double	supportB = Double.POSITIVE_INFINITY

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 Pearson type VI distribution with shape parameters \(\alpha_1 > 0\) and \(\alpha_2 > 0\), and scale parameter \(\beta> 0\).

The density function is given by

\[ f(x) =\left\{\begin{array}{ll} \displaystyle\frac{\left(x/{\beta}\right)^{\alpha_1 - 1}}{\beta\mathcal{B}(\alpha_1, \alpha_2)(1 + x/{\beta})^{\alpha_1 + \alpha_2}} & \quad\mbox{for } x > 0, \\ 0 & \quad\mbox{otherwise,} \end{array} \right. \tag{fpearson6} \]

where \(\mathcal{B}\) is the beta function. The distribution function is given by

\[ F(x) = F_B\left(\frac{x}{x + \beta}\right) \qquad\mbox{for } x > 0, \tag{Fpearson6} \]

and \(F(x) = 0\) otherwise, where \(F_B(x)\) is the distribution function of a beta distribution with shape parameters \(\alpha_1\) and \(\alpha_2\).

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()

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.

◆ getInstanceFromMLE()

static Pearson6Dist getInstanceFromMLE	(	double []	x,
		int	n
	)

static

Creates a new instance of a Pearson VI distribution with parameters \(\alpha_1\), \(\alpha_2\) and \(\beta\), 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

◆ getMLE()

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

static

Estimates the parameters \((\alpha_1,\alpha_2,\beta)\) of the Pearson VI distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\).

The estimates are returned in a three-element array, in regular order: [ \(\alpha_1, \alpha_2\), \(\beta\)]. The estimate of the parameters is given by maximizing numerically the log-likelihood function, using the Uncmin package [211], [233] .

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 [ \(\hat{\alpha_1}, \hat{\alpha_2}, \hat{\beta}\)]

◆ getParams()

double [] getParams ( )

Return a table containing the parameters of the current distribution.

This table is put in regular order: [ \(\alpha_1\), \(\alpha_2\), \(\beta\)].

Implements Distribution.

◆ 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:

Pearson6Dist.java

Public Member Functions

Static Public Member Functions

Protected Attributes

Additional Inherited Members

Detailed Description

Member Function Documentation

◆ barF()

◆ cdf()

◆ getInstanceFromMLE()

◆ getMLE()

◆ getParams()

◆ inverseF()