This class extends the class ContinuousDistribution for the Rayleigh distribution [57] with location parameter \(a\), and scale parameter \(\beta> 0\). More...

Inheritance diagram for RayleighDist:

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Collaboration diagram for RayleighDist:

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Public Member Functions
	RayleighDist (double beta)
	Constructs a `RayleighDist` object with parameters \(a = 0\) and \(\beta\) = `beta`.

	RayleighDist (double a, double beta)
	Constructs a `RayleighDist` object with parameters \(a =\) `a`, 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	getA ()
	Returns the parameter \(a\). More...

double	getSigma ()
	Returns the parameter \(\beta\). More...

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

double []	getParams ()
	Return an array containing the parameters of the current distribution in the order: [ \(a\), \(\beta\)]. 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 beta, double x)
	Computes the density function ( frayleigh ). More...

static double	density (double beta, double x)
	Same as `density (0, beta, x)`. More...

static double	cdf (double a, double beta, double x)
	Computes the distribution function ( Frayleigh ). More...

static double	cdf (double beta, double x)
	Same as `cdf (0, beta, x)`. More...

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

static double	barF (double beta, double x)
	Same as `barF (0, beta, x)`. More...

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

static double	inverseF (double beta, double u)
	Same as `inverseF (0, beta, u)`. More...

static double []	getMLE (double[] x, int n, double a)
	Estimates the parameter \(\beta\) of the Rayleigh distribution using the maximum likelihood method, assuming that \(a\) is known, from the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\). More...

static RayleighDist	getInstanceFromMLE (double[] x, int n, double a)
	Creates a new instance of a Rayleigh distribution with parameters \(a\) and \(\hat{\beta}\). More...

static double	getMean (double a, double beta)
	Returns the mean \(a + \beta\sqrt{\pi/2}\) of the Rayleigh distribution with parameters \(a\) and \(\beta\). More...

static double	getVariance (double beta)
	Returns the variance of the Rayleigh distribution with parameter \(\beta\). More...

static double	getStandardDeviation (double beta)
	Returns the standard deviation \(\beta\sqrt{2 - \pi/2}\) of the Rayleigh distribution with parameter \(\beta\). 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

This class extends the class ContinuousDistribution for the Rayleigh distribution [57] with location parameter \(a\), and scale parameter \(\beta> 0\).

The density function is

\[ f(x) = \frac{(x-a)}{\beta^2} e^{-(x-a)^2/(2\beta^2)} \qquad\mbox{for } x \ge a, \tag{frayleigh} \]

and \(f(x) = 0\) for \(x < a\). The distribution function is

\[ F(x) = 1 - e^{-(x - a)^2/(2\beta^2)} \qquad\mbox{for } x \ge a, \tag{Frayleigh} \]

and the inverse distribution function is

\[ F^{-1}(u) = x = a + \beta\sqrt{-2\ln(1-u)} \qquad\mbox{for } 0 \le u \le1. \tag{Invrayleigh} \]

Member Function Documentation

◆ barF() [1/3]

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() [2/3]

static double barF	(	double	a,
		double	beta,
		double	x
	)

static

Computes the complementary distribution function.

Parameters

a	the location parameter
beta	the scale parameter
x	the value at which the complementary distribution is evaluated

Returns: returns the complementary distribution function

◆ barF() [3/3]

static double barF	(	double	beta,
		double	x
	)

static

Same as barF (0, beta, x).

Parameters

beta	the scale parameter
x	the value at which the complementary distribution is evaluated

Returns: returns the complementary distribution function

◆ cdf() [1/3]

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() [2/3]

static double cdf	(	double	a,
		double	beta,
		double	x
	)

static

Computes the distribution function ( Frayleigh ).

Parameters

a	the location parameter
beta	the scale parameter
x	the value at which the distribution is evaluated

Returns: returns the distribution function

◆ cdf() [3/3]

static double cdf	(	double	beta,
		double	x
	)

static

Same as cdf (0, beta, x).

Parameters

beta	the scale parameter
x	the value at which the distribution is evaluated

Returns: returns the distribution function

◆ density() [1/2]

static double density	(	double	a,
		double	beta,
		double	x
	)

static

Computes the density function ( frayleigh ).

Parameters

a	the location parameter
beta	the scale parameter
x	the value at which the density is evaluated

Returns: the density function

◆ density() [2/2]

static double density	(	double	beta,
		double	x
	)

static

Same as density (0, beta, x).

Parameters

beta	the scale parameter
x	the value at which the density is evaluated

Returns: returns the density function

◆ getA()

double getA ( )

Returns the parameter \(a\).

Returns: the location parameter \(a\)

◆ getInstanceFromMLE()

static RayleighDist getInstanceFromMLE	(	double []	x,
		int	n,
		double	a
	)

static

Creates a new instance of a Rayleigh distribution with parameters \(a\) and \(\hat{\beta}\).

This last is 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	the location parameter

◆ getMean()

static double getMean	(	double	a,
		double	beta
	)

static

Returns the mean \(a + \beta\sqrt{\pi/2}\) of the Rayleigh distribution with parameters \(a\) and \(\beta\).

Parameters

a	the location parameter
beta	the scale parameter

Returns: the mean of the Rayleigh distribution

◆ getMLE()

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

static

Estimates the parameter \(\beta\) of the Rayleigh distribution using the maximum likelihood method, assuming that \(a\) is known, from the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\).

The estimate is returned in a one-element array: [ \(\hat{\beta}\)]. The maximum likelihood estimator is the value \(\hat{\beta}\) that satisfies the equation

\[ \hat{\beta} = \sqrt{\frac{1}{2n}\sum_{i=1}^n x_i^2} \]

Parameters

x	the list of observations to use to evaluate parameters
n	the number of observations to use to evaluate parameters
a	the location parameter

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

◆ getParams()

double [] getParams ( )

Return an array containing the parameters of the current distribution in the order: [ \(a\), \(\beta\)].

Returns: [ \(a\), \(\beta\)]

Implements Distribution.

◆ getSigma()

double getSigma ( )

Returns the parameter \(\beta\).

Returns: the scale parameter \(beta\)

◆ getStandardDeviation()

static double getStandardDeviation ( double beta )

static

Returns the standard deviation \(\beta\sqrt{2 - \pi/2}\) of the Rayleigh distribution with parameter \(\beta\).

Parameters

beta	the scale parameter

Returns: the standard deviation of the Rayleigh distribution

◆ getVariance()

static double getVariance ( double beta )

static

Returns the variance of the Rayleigh distribution with parameter \(\beta\).

Parameters

beta	the scale parameter

Returns: the variance of the Rayleigh distribution

◆ inverseF() [1/3]

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() [2/3]

static double inverseF	(	double	a,
		double	beta,
		double	u
	)

static

Computes the inverse of the distribution function ( Invrayleigh ).

Parameters

a	the location parameter
beta	the scale parameter
u	the value at which the inverse distribution is evaluated

Returns: returns the inverse of the distribution function

◆ inverseF() [3/3]

static double inverseF	(	double	beta,
		double	u
	)

static

Same as inverseF (0, beta, u).

Parameters

beta	the scale 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	beta
	)

Sets the parameters \(a\) and \(\beta\) for this object.

Parameters

a	the location parameter
beta	the scale 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:

RayleighDist.java

Public Member Functions

Static Public Member Functions

Additional Inherited Members

Detailed Description

Member Function Documentation

◆ barF() [1/3]

◆ barF() [2/3]

◆ barF() [3/3]

◆ cdf() [1/3]

◆ cdf() [2/3]

◆ cdf() [3/3]

◆ density() [1/2]

◆ density() [2/2]

◆ getA()

◆ getInstanceFromMLE()

◆ getMean()

◆ getMLE()

◆ getParams()

◆ getSigma()

◆ getStandardDeviation()

◆ getVariance()

◆ inverseF() [1/3]

◆ inverseF() [2/3]

◆ inverseF() [3/3]

◆ setParams()

◆ toString()