Extends the class ContinuousDistribution for the Watson \(G\) distribution (see [40], [228] ). More...

Inheritance diagram for umontreal.ssj.probdist.WatsonGDist:

Public Member Functions
	WatsonGDist (int n)
	Constructs a Watson distribution for a sample of size \(n\).
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 \(x = F^{-1}(u)\).
int	getN ()
	Returns the parameter \(n\) of this object.
void	setN (int n)
	Sets the parameter \(n\) of this object.
double[]	getParams ()
	Return an array containing the parameter \(n\) of this object.
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	getMean ()
	Returns the mean.
double	getVariance ()
	Returns the variance.
double	getStandardDeviation ()
	Returns the standard deviation.
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 (int n, double x)
	Computes the density function for a Watson \(G\) distribution with parameter \(n\).
static double	cdf (int n, double x)
	Computes the Watson \(G\) distribution function \(F_n(x)\), with parameter \(n\).
static double	barF (int n, double x)
	Computes the complementary distribution function \(\bar{F}_n(x)\) with parameter \(n\).
static double	inverseF (int n, double u)
	Computes \(x = F_n^{-1}(u)\), where \(F_n\) is the Watson.

Detailed Description

Extends the class ContinuousDistribution for the Watson \(G\) distribution (see [40], [228] ).

Given a sample of

\(n\) independent uniforms \(U_i\) over \([0,1]\), the \(G\) statistic is defined by

\begin{align} G_n & = \sqrt{n} \max_{\Rule{0.0pt}{7.0pt}{0.0pt} 1\le j \le n} \left\{ j/n - U_{(j)} + \bar{U}_n - 1/2 \right\} \tag{WatsonG} \\ & = \sqrt{n}\left(D_n^+ + \bar{U}_n - 1/2\right), \nonumber \end{align}

where the \(U_{(j)}\) are the \(U_i\) sorted in increasing order, \(\bar{U}_n\) is the average of the observations \(U_i\), and \(D_n^+\) is the Kolmogorov-Smirnov+ statistic. The distribution function (the cumulative probabilities) is defined as \(F_n(x) = P[G_n \le x]\).

Definition at line 50 of file WatsonGDist.java.

Constructor & Destructor Documentation

◆ WatsonGDist()

umontreal.ssj.probdist.WatsonGDist.WatsonGDist ( int n )

Constructs a Watson distribution for a sample of size \(n\).

Definition at line 70 of file WatsonGDist.java.

Member Function Documentation

◆ barF() [1/2]

double umontreal.ssj.probdist.WatsonGDist.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 82 of file WatsonGDist.java.

◆ barF() [2/2]

double umontreal.ssj.probdist.WatsonGDist.barF	(	int	n,
		double	x )

static

Computes the complementary distribution function \(\bar{F}_n(x)\) with parameter \(n\).

Definition at line 627 of file WatsonGDist.java.

◆ cdf() [1/2]

double umontreal.ssj.probdist.WatsonGDist.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 78 of file WatsonGDist.java.

◆ cdf() [2/2]

double umontreal.ssj.probdist.WatsonGDist.cdf	(	int	n,
		double	x )

static

Computes the Watson \(G\) distribution function \(F_n(x)\), with parameter \(n\).

A cubic spline interpolation is used for the asymptotic distribution when \(n\to\infty\), and an empirical correction of order

\(1/\sqrt{n}\), obtained empirically from \(10^7\) simulation runs with \(n = 256\) is then added. The absolute error is estimated to be less than 0.01, 0.005, 0.002, 0.0008, 0.0005, 0.0005, 0.0005 for \(n = 16\), 32, 64, 128, 256, 512, 1024, respectively.

Definition at line 564 of file WatsonGDist.java.

◆ density() [1/2]

double umontreal.ssj.probdist.WatsonGDist.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 74 of file WatsonGDist.java.

◆ density() [2/2]

double umontreal.ssj.probdist.WatsonGDist.density	(	int	n,
		double	x )

static

Computes the density function for a Watson \(G\) distribution with parameter \(n\).

Definition at line 94 of file WatsonGDist.java.

◆ getN()

int umontreal.ssj.probdist.WatsonGDist.getN ( )

Returns the parameter \(n\) of this object.

Definition at line 653 of file WatsonGDist.java.

◆ getParams()

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

Return an array containing the parameter \(n\) of this object.

Implements umontreal.ssj.probdist.Distribution.

Definition at line 671 of file WatsonGDist.java.

◆ inverseF() [1/2]

double umontreal.ssj.probdist.WatsonGDist.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 86 of file WatsonGDist.java.

◆ inverseF() [2/2]

double umontreal.ssj.probdist.WatsonGDist.inverseF	(	int	n,
		double	u )

static

Computes \(x = F_n^{-1}(u)\), where \(F_n\) is the Watson.

\(G\) distribution with parameter \(n\).

Definition at line 636 of file WatsonGDist.java.

◆ setN()

void umontreal.ssj.probdist.WatsonGDist.setN ( int n )

Sets the parameter \(n\) of this object.

Definition at line 660 of file WatsonGDist.java.

◆ toString()

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

Returns a String containing information about the current distribution.

Definition at line 679 of file WatsonGDist.java.

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

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

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ WatsonGDist()

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ density() [1/2]

◆ density() [2/2]

◆ getN()

◆ getParams()

◆ inverseF() [1/2]

◆ inverseF() [2/2]

◆ setN()

◆ toString()