Extends the class ContinuousDistribution for the power distribution [57] (page 161) with shape parameter \(c > 0\), over the interval \([a,b]\), where \(a < b\). More...

Inheritance diagram for PowerDist:

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Collaboration diagram for PowerDist:

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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)

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	getA ()
	Returns the parameter \(a\). More...

double	getB ()
	Returns the parameter \(b\). More...

double	getC ()
	Returns the parameter \(c\). More...

void	setParams (double a, double b, double c)
	Sets the parameters \(a\), \(b\) and \(c\) for this object. More...

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

String	toString ()
	Returns a `String` containing information about the current distribution. More...

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 a, double b, double c, double x)
	Computes the density function ( fpower ). More...

static double	cdf (double a, double b, double c, double x)
	Computes the distribution function ( Fpower ). More...

static double	barF (double a, double b, double c, double x)
	Computes the complementary distribution function. More...

static double	inverseF (double a, double b, double c, double u)
	Computes the inverse of the distribution function. More...

static double []	getMLE (double[] x, int n, double a, double b)
	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. More...

static PowerDist	getInstanceFromMLE (double[] x, int n, double a, double b)
	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 \(x[i]\), \(i = 0, …, n-1\). More...

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 \(c\). More...

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 \(c\). More...

static double	getStandardDeviation (double a, double b, double c)
	Computes and returns the standard deviation of the power distribution with parameters \(a\), \(b\) and \(c\). 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 the power distribution [57] (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 \(x \ge b\).

Member Function Documentation

◆ barF() [1/2]

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.

◆ barF() [2/2]

static double 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

◆ 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	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

◆ density()

static double 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

◆ getA()

double getA ( )

Returns the parameter \(a\).

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

◆ getB()

double getB ( )

Returns the parameter \(b\).

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

◆ getC()

double getC ( )

Returns the parameter \(c\).

Returns: the shape parameter \(c\)

◆ getInstanceFromMLE()

static 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

◆ getMean()

static double 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

◆ getMLE()

static double [] 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 [ \(\hat{c}\)]

◆ getParams()

double [] 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 Distribution.

◆ getStandardDeviation()

static double 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

◆ getVariance()

static double 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

◆ inverseF() [1/2]

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.

◆ inverseF() [2/2]

static double 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

◆ setParams()

void 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

◆ toString()

String toString ( )

Returns a String containing information about the current distribution.

Returns: a String containing information about the current distribution

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

PowerDist.java

Public Member Functions

Static Public Member Functions

Additional Inherited Members

Detailed Description

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ density()

◆ getA()

◆ getB()

◆ getC()

◆ getInstanceFromMLE()

◆ getMean()

◆ getMLE()

◆ getParams()

◆ getStandardDeviation()

◆ getVariance()

◆ inverseF() [1/2]

◆ inverseF() [2/2]

◆ setParams()

◆ toString()