Extends the class ContinuousDistribution for a distribution from the Pareto family, with shape parameter \(\alpha> 0\) and location parameter \(\beta> 0\) [95] (page 574). More...

Inheritance diagram for umontreal.ssj.probdist.ParetoDist:

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)
	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 \(\alpha\).
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.
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)
	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 \(x[i]\), \(i = 0, 1,…, n-1\).
static ParetoDist	getInstanceFromMLE (double[] x, int n)
	Creates a new instance of a Pareto distribution with parameters.
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.
static double	getVariance (double alpha, double beta)
	Computes and returns the variance.
static double	getStandardDeviation (double alpha, double beta)
	Computes and returns the standard deviation of the Pareto distribution with parameters \(\alpha\) and \(\beta\).

Detailed Description

Extends the class ContinuousDistribution for a distribution from the Pareto family, with shape parameter \(\alpha> 0\) and location parameter \(\beta> 0\) [95] (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

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

Definition at line 47 of file ParetoDist.java.

Constructor & Destructor Documentation

◆ ParetoDist() [1/2]

umontreal.ssj.probdist.ParetoDist.ParetoDist ( double alpha )

Constructs a ParetoDist object with parameters \(\alpha=\) alpha and \(\beta= 1\).

Definition at line 55 of file ParetoDist.java.

◆ ParetoDist() [2/2]

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

Constructs a ParetoDist object with parameters \(\alpha=\) alpha and \(\beta= \) beta.

Definition at line 63 of file ParetoDist.java.

Member Function Documentation

◆ barF() [1/2]

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

static

Computes the complementary distribution function.

Definition at line 123 of file ParetoDist.java.

◆ barF() [2/2]

double umontreal.ssj.probdist.ParetoDist.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.

Definition at line 75 of file ParetoDist.java.

◆ cdf() [1/2]

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

static

Computes the distribution function.

Definition at line 110 of file ParetoDist.java.

◆ cdf() [2/2]

double umontreal.ssj.probdist.ParetoDist.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.

Definition at line 71 of file ParetoDist.java.

◆ density() [1/2]

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

static

Computes the density function.

Definition at line 98 of file ParetoDist.java.

◆ density() [2/2]

double umontreal.ssj.probdist.ParetoDist.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 67 of file ParetoDist.java.

◆ getAlpha()

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

Returns the parameter \(\alpha\).

Definition at line 254 of file ParetoDist.java.

◆ getBeta()

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

Returns the parameter \(\beta\).

Definition at line 261 of file ParetoDist.java.

◆ getInstanceFromMLE()

ParetoDist umontreal.ssj.probdist.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

Definition at line 204 of file ParetoDist.java.

◆ getMean() [1/2]

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

Returns the mean.

Returns: the mean

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 83 of file ParetoDist.java.

◆ getMean() [2/2]

double umontreal.ssj.probdist.ParetoDist.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

Definition at line 215 of file ParetoDist.java.

◆ getMLE()

double[] umontreal.ssj.probdist.ParetoDist.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 [95] (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 [ \(\hat{\alpha}\), \(\hat{\beta}\)]

Definition at line 173 of file ParetoDist.java.

◆ getParams()

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

Return a table containing the parameters of the current distribution.

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

Implements umontreal.ssj.probdist.Distribution.

Definition at line 283 of file ParetoDist.java.

◆ getStandardDeviation() [1/2]

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

Returns the standard deviation.

Returns: the standard deviation

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 91 of file ParetoDist.java.

◆ getStandardDeviation() [2/2]

double umontreal.ssj.probdist.ParetoDist.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

Definition at line 247 of file ParetoDist.java.

◆ getVariance() [1/2]

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

Returns the variance.

Returns: the variance

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 87 of file ParetoDist.java.

◆ getVariance() [2/2]

double umontreal.ssj.probdist.ParetoDist.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 \(\mbox{Var}[X] = \alpha\beta^2 / [(\alpha- 2)(\alpha- 1)]\)

Definition at line 232 of file ParetoDist.java.

◆ inverseF() [1/2]

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

static

Computes the inverse of the distribution function.

Definition at line 136 of file ParetoDist.java.

◆ inverseF() [2/2]

double umontreal.ssj.probdist.ParetoDist.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.

Definition at line 79 of file ParetoDist.java.

◆ setParams()

void umontreal.ssj.probdist.ParetoDist.setParams	(	double	alpha,
		double	beta )

Sets the parameter \(\alpha\) and \(\beta\) for this object.

Definition at line 268 of file ParetoDist.java.

◆ toString()

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

Returns a String containing information about the current distribution.

Definition at line 291 of file ParetoDist.java.

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

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

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ ParetoDist() [1/2]

◆ ParetoDist() [2/2]

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ density() [1/2]

◆ density() [2/2]

◆ getAlpha()

◆ getBeta()

◆ getInstanceFromMLE()

◆ getMean() [1/2]

◆ getMean() [2/2]

◆ getMLE()

◆ getParams()

◆ getStandardDeviation() [1/2]

◆ getStandardDeviation() [2/2]

◆ getVariance() [1/2]

◆ getVariance() [2/2]

◆ inverseF() [1/2]

◆ inverseF() [2/2]

◆ setParams()

◆ toString()