Extends the class ContinuousDistribution for the power distribution [56] (page 161) with shape parameter \(c > 0\), over the interval \([a,b]\), where \(a < b\). More...

Inheritance diagram for umontreal.ssj.probdist.PowerDist:

Public Member Functions
	PowerDist (double a, double b, double c)
	Constructs a PowerDist object with parameters \(a =\) a, \(b =\) b and \(c =\) c.
	PowerDist (double b, double c)
	Constructs a PowerDist object with parameters \(a = 0\), \(b =\) b and \(c =\) c.
	PowerDist (double c)
	Constructs a PowerDist object with parameters \(a = 0\), \(b =1\) and \(c =\) c.
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	getA ()
	Returns the parameter \(a\).
double	getB ()
	Returns the parameter \(b\).
double	getC ()
	Returns the parameter \(c\).
void	setParams (double a, double b, double c)
	Sets the parameters \(a\), \(b\) and \(c\) 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 a, double b, double c, double x)
	Computes the density function ( `fpower` ).
static double	cdf (double a, double b, double c, double x)
	Computes the distribution function ( `Fpower` ).
static double	barF (double a, double b, double c, double x)
	Computes the complementary distribution function.
static double	inverseF (double a, double b, double c, double u)
	Computes the inverse of the distribution function.
static double[]	getMLE (double[] x, int n, double a, double b)
	Estimates the parameter \(c\) of the power distribution from the.
static PowerDist	getInstanceFromMLE (double[] x, int n, double a, double b)
	Creates a new instance of a power distribution with parameters.
static double	getMean (double a, double b, double c)
	Returns the mean \(a + (b-a)c/(c+1)\) of the power distribution with parameters \(a\), \(b\) and \(c\).
static double	getVariance (double a, double b, double c)
	Computes and returns the variance \((b-a)^2 c / [(c+1)^2(c+2)]\) of the power distribution with parameters \(a\), \(b\) and.
static double	getStandardDeviation (double a, double b, double c)
	Computes and returns the standard deviation of the power distribution with parameters \(a\), \(b\) and \(c\).

Detailed Description

Extends the class ContinuousDistribution for the power distribution [56] (page 161) with shape parameter \(c > 0\), over the interval \([a,b]\), where \(a < b\).

This distribution has density

\[ f(x) = \frac{c(x-a)^{c - 1}}{(b - a)^c}, \qquad\mbox{for } a \le x \le b, \tag{fpower} \]

and \(f(x) = 0\) elsewhere. Its distribution function is

\[ F(x) = \frac{(x - a)^c}{(b - a)^c}, \qquad\mbox{for } a \le x \le b, \tag{Fpower} \]

with \(F(x) = 0\) for \(x \le a\) and \(F(x) = 1\) for \(x \ge b\).

Definition at line 46 of file PowerDist.java.

Constructor & Destructor Documentation

◆ PowerDist() [1/3]

umontreal.ssj.probdist.PowerDist.PowerDist	(	double	a,
		double	b,
		double	c )

Constructs a PowerDist object with parameters \(a =\) a, \(b =\) b and \(c =\) c.

Definition at line 55 of file PowerDist.java.

◆ PowerDist() [2/3]

umontreal.ssj.probdist.PowerDist.PowerDist	(	double	b,
		double	c )

Constructs a PowerDist object with parameters \(a = 0\), \(b =\) b and \(c =\) c.

Definition at line 63 of file PowerDist.java.

◆ PowerDist() [3/3]

umontreal.ssj.probdist.PowerDist.PowerDist ( double c )

Constructs a PowerDist object with parameters \(a = 0\), \(b =1\) and \(c =\) c.

Definition at line 71 of file PowerDist.java.

Member Function Documentation

◆ barF() [1/2]

double umontreal.ssj.probdist.PowerDist.barF	(	double	a,
		double	b,
		double	c,
		double	x )

static

Computes the complementary distribution function.

Parameters

a	left limit of interval
b	right limit of interval
c	shape parameter
x	the value at which the complementary distribution is evaluated

Returns: returns the complementary distribution function

Definition at line 153 of file PowerDist.java.

◆ barF() [2/2]

double umontreal.ssj.probdist.PowerDist.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 83 of file PowerDist.java.

◆ cdf() [1/2]

double umontreal.ssj.probdist.PowerDist.cdf	(	double	a,
		double	b,
		double	c,
		double	x )

static

Computes the distribution function ( Fpower ).

Parameters

a	left limit of interval
b	right limit of interval
c	shape parameter
x	the value at which the distribution is evaluated

Returns: returns the distribution function

Definition at line 134 of file PowerDist.java.

◆ cdf() [2/2]

double umontreal.ssj.probdist.PowerDist.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 79 of file PowerDist.java.

◆ density() [1/2]

double umontreal.ssj.probdist.PowerDist.density	(	double	a,
		double	b,
		double	c,
		double	x )

static

Computes the density function ( fpower ).

Parameters

a	left limit of interval
b	right limit of interval
c	shape parameter
x	the value at which the density is evaluated

Returns: returns the density function

Definition at line 113 of file PowerDist.java.

◆ density() [2/2]

double umontreal.ssj.probdist.PowerDist.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 75 of file PowerDist.java.

◆ getA()

double umontreal.ssj.probdist.PowerDist.getA ( )

Returns the parameter \(a\).

Returns: the left limit of interval \(a\)

Definition at line 273 of file PowerDist.java.

◆ getB()

double umontreal.ssj.probdist.PowerDist.getB ( )

Returns the parameter \(b\).

Returns: the right limit of interval \(b\)

Definition at line 282 of file PowerDist.java.

◆ getC()

double umontreal.ssj.probdist.PowerDist.getC ( )

Returns the parameter \(c\).

Returns: the shape parameter \(c\)

Definition at line 291 of file PowerDist.java.

◆ getInstanceFromMLE()

PowerDist umontreal.ssj.probdist.PowerDist.getInstanceFromMLE	(	double[]	x,
		int	n,
		double	a,
		double	b )

static

Creates a new instance of a power distribution with parameters.

\(a\) and \(b\), with \(c\) estimated using the maximum likelihood method based on the \(n\) observations \(x[i]\), \(i = 0, …, n-1\).

Parameters

x	the list of observations to use to evaluate parameters
n	the number of observations to use to evaluate parameters
a	left limit of interval
b	right limit of interval

Definition at line 227 of file PowerDist.java.

◆ getMean() [1/2]

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

Returns the mean.

Returns: the mean

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 91 of file PowerDist.java.

◆ getMean() [2/2]

double umontreal.ssj.probdist.PowerDist.getMean	(	double	a,
		double	b,
		double	c )

static

Returns the mean \(a + (b-a)c/(c+1)\) of the power distribution with parameters \(a\), \(b\) and \(c\).

Parameters

a	left limit of interval
b	right limit of interval
c	shape parameter

Returns: returns the mean

Definition at line 241 of file PowerDist.java.

◆ getMLE()

double[] umontreal.ssj.probdist.PowerDist.getMLE	(	double[]	x,
		int	n,
		double	a,
		double	b )

static

Estimates the parameter \(c\) of the power distribution from the.

\(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\), using the maximum likelihood method and assuming that \(a\) and \(b\) are known. The estimate is returned in a one-element array: [ \(c\)]. The maximum likelihood estimator is the value \(\hat{c}\) that satisfies the equation

\begin{align*} \frac{1}{\hat{c}} = -\frac{1}{n} \sum_{i=1}^n \ln\left(\frac{x_i - a}{b - a} \right) \end{align*}

Parameters

x	the list of observations to use to evaluate parameters
n	the number of observations to use to evaluate parameters
a	left limit of interval
b	right limit of interval

Returns: returns the shape parameter [ \(\hat{c}\)]

Definition at line 202 of file PowerDist.java.

◆ getParams()

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

Return a table containing the parameters of the current distribution.

This table is put in regular order: [ \(a\), \(b\), \(c\)].

Returns: [ \(a\), \(b, \)c]

Implements umontreal.ssj.probdist.Distribution.

Definition at line 314 of file PowerDist.java.

◆ getStandardDeviation() [1/2]

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

Returns the standard deviation.

Returns: the standard deviation

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 99 of file PowerDist.java.

◆ getStandardDeviation() [2/2]

double umontreal.ssj.probdist.PowerDist.getStandardDeviation	(	double	a,
		double	b,
		double	c )

static

Computes and returns the standard deviation of the power distribution with parameters \(a\), \(b\) and \(c\).

Returns: the standard deviation of the power distribution

Definition at line 264 of file PowerDist.java.

◆ getVariance() [1/2]

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

Returns the variance.

Returns: the variance

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 95 of file PowerDist.java.

◆ getVariance() [2/2]

double umontreal.ssj.probdist.PowerDist.getVariance	(	double	a,
		double	b,
		double	c )

static

Computes and returns the variance \((b-a)^2 c / [(c+1)^2(c+2)]\) of the power distribution with parameters \(a\), \(b\) and.

\(c\).

Parameters

a	left limit of interval
b	right limit of interval
c	shape parameter

Returns: returns the variance

Definition at line 254 of file PowerDist.java.

◆ inverseF() [1/2]

double umontreal.ssj.probdist.PowerDist.inverseF	(	double	a,
		double	b,
		double	c,
		double	u )

static

Computes the inverse of the distribution function.

Parameters

a	left limit of interval
b	right limit of interval
c	shape parameter
u	the value at which the inverse distribution is evaluated

Returns: returns the inverse of the distribution function

Definition at line 172 of file PowerDist.java.

◆ inverseF() [2/2]

double umontreal.ssj.probdist.PowerDist.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 87 of file PowerDist.java.

◆ setParams()

void umontreal.ssj.probdist.PowerDist.setParams	(	double	a,
		double	b,
		double	c )

Sets the parameters \(a\), \(b\) and \(c\) for this object.

Parameters

a	left limit of interval
b	right limit of interval
c	shape parameter

Definition at line 302 of file PowerDist.java.

◆ toString()

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

Returns a String containing information about the current distribution.

Returns: a String containing information about the current distribution

Definition at line 324 of file PowerDist.java.

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

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

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ PowerDist() [1/3]

◆ PowerDist() [2/3]

◆ PowerDist() [3/3]

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ density() [1/2]

◆ density() [2/2]

◆ getA()

◆ getB()

◆ getC()

◆ 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()