Extends the class ContinuousDistribution for a distribution from the Pareto family, with shape parameter \(\alpha> 0\) and location parameter \(\beta> 0\) [99] (page 574). More...

Inheritance diagram for ParetoDist:

[legend]

Collaboration diagram for ParetoDist:

[legend]

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

	ParetoDist (double alpha, double beta)
	Constructs a `ParetoDist` object with parameters \(\alpha=\) `alpha` 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	getAlpha ()
	Returns the parameter \(\alpha\).

double	getBeta ()
	Returns the parameter \(\beta\).

void	setParams (double alpha, double beta)
	Sets the parameter \(\alpha\) and \(\beta\) for 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 alpha, double beta, double x)
	Computes the density function.

static double	cdf (double alpha, double beta, double x)
	Computes the distribution function.

static double	barF (double alpha, double beta, double x)
	Computes the complementary distribution function.

static double	inverseF (double alpha, double beta, double u)
	Computes the inverse of the distribution function.

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

static ParetoDist	getInstanceFromMLE (double[] x, int n)
	Creates a new instance of a Pareto distribution with parameters \(\alpha\) 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 alpha, double beta)
	Computes and returns the mean \(E[X] = \alpha\beta/(\alpha- 1)\) of the Pareto distribution with parameters \(\alpha\) and \(\beta\). More...

static double	getVariance (double alpha, double beta)
	Computes and returns the variance \(\mbox{Var}[X] = \frac{\alpha\beta^2}{(\alpha- 2)(\alpha- 1)}\) of the Pareto distribution with parameters \(\alpha\) and \(\beta\). More...

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

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 a distribution from the Pareto family, with shape parameter \(\alpha> 0\) and location parameter \(\beta> 0\) [99] (page 574).

The density for this type of Pareto distribution is

\[ f(x) = \frac{\alpha\beta^{\alpha}}{x^{\alpha+1}} \qquad\mbox{for }x \ge\beta, \tag{fpareto} \]

and 0 otherwise. The distribution function is

\[ F(x) = 1 - \left(\beta/x\right)^{\alpha}\qquad\mbox{for }x\ge\beta, \tag{Fpareto} \]

and the inverse distribution function is

\[ F^{-1}(u) = \beta(1 - u)^{-1/\alpha} \qquad\mbox{for } 0 \le u < 1. \]

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 ParetoDist getInstanceFromMLE	(	double []	x,
		int	n
	)

static

Creates a new instance of a Pareto distribution with parameters \(\alpha\) 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

◆ getMean()

static double getMean	(	double	alpha,
		double	beta
	)

static

Computes and returns the mean \(E[X] = \alpha\beta/(\alpha- 1)\) of the Pareto distribution with parameters \(\alpha\) and \(\beta\).

Returns: the mean of the Pareto distribution

◆ getMLE()

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

static

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

\begin{align*} \hat{\beta} & = \min_i \{x_i\} \\ \hat{\alpha} & = \frac{n}{\sum_{i=1}^n \ln\left(\frac{x_i}{\hat{\beta}\Rule{0.0pt}{5.5pt}{0.0pt}}\right)}. \end{align*}

See [99] (page 581).

Parameters

x	the list of observations used to evaluate parameters
n	the number of observations used to evaluate parameters

Returns: returns the parameters [ \(\hat{\alpha}\), \(\hat{\beta}\)]

◆ getParams()

double [] getParams ( )

Return a table containing the parameters of the current distribution.

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

Implements Distribution.

◆ getStandardDeviation()

static double getStandardDeviation	(	double	alpha,
		double	beta
	)

static

Computes and returns the standard deviation of the Pareto distribution with parameters \(\alpha\) and \(\beta\).

Returns: the standard deviation of the Pareto distribution

◆ getVariance()

static double getVariance	(	double	alpha,
		double	beta
	)

static

Computes and returns the variance \(\mbox{Var}[X] = \frac{\alpha\beta^2}{(\alpha- 2)(\alpha- 1)}\) of the Pareto distribution with parameters \(\alpha\) and \(\beta\).

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

◆ 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:

ParetoDist.java

Public Member Functions

Static Public Member Functions

Additional Inherited Members

Detailed Description

Member Function Documentation

◆ barF()

◆ cdf()

◆ getInstanceFromMLE()

◆ getMean()

◆ getMLE()

◆ getParams()

◆ getStandardDeviation()

◆ getVariance()

◆ inverseF()