Extends the class ContinuousDistribution for the Watson U distribution (see [54], [215], [216] ). More...

Inheritance diagram for umontreal.ssj.probdist.WatsonUDist:

Public Member Functions
	WatsonUDist (int n)
	Constructs a Watson U 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 \(x = F^{-1}(u)\).
double	getMean ()
	Returns the mean.
double	getVariance ()
	Returns the variance.
double	getStandardDeviation ()
	Returns the standard deviation.
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	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 of the Watson U distribution with parameter \(n\).
static double	cdf (int n, double x)
	Computes the Watson \(U\) distribution function, i.e.
static double	barF (int n, double x)
	Computes the complementary distribution function \(\bar{F}_n(x)\), where \(F_n\) is the Watson \(U\) distribution with parameter \(n\).
static double	inverseF (int n, double u)
	Computes \(x = F_n^{-1}(u)\), where \(F_n\) is the Watson.
static double	getMean (int n)
	Returns the mean of the Watson \(U\) distribution with parameter.
static double	getVariance (int n)
	Returns the variance of the Watson \(U\) distribution with parameter \(n\).
static double	getStandardDeviation (int n)
	Returns the standard deviation of the Watson \(U\) distribution with parameter \(n\).

Detailed Description

Extends the class ContinuousDistribution for the Watson U distribution (see [54], [215], [216] ).

Given a sample of \(n\) independent uniforms \(u_i\) over \([0,1]\), the Watson statistic \(U_n^2\) is defined by

\begin{align*} W_n^2 & = \frac{1}{12n} + \sum_{j=1}^n \left\{u_{(j)} - \frac{(j- 1/2)}{n}\right\}^2, \\ U_n^2 & = W_n^2 - n\left(\bar{u}_n - 1/2\right)^2. \tag{WatsonU} \end{align*}

where the \(u_{(j)}\) are the \(u_i\) sorted in increasing order, and \(\bar{u}_n\) is the average of the observations \(u_i\). The distribution function (the cumulative probabilities) is defined as \(F_n(x) = P[U_n^2 \le x]\).

Definition at line 49 of file WatsonUDist.java.

Constructor & Destructor Documentation

◆ WatsonUDist()

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

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

Definition at line 95 of file WatsonUDist.java.

Member Function Documentation

◆ barF() [1/2]

double umontreal.ssj.probdist.WatsonUDist.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 107 of file WatsonUDist.java.

◆ barF() [2/2]

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

static

Computes the complementary distribution function \(\bar{F}_n(x)\), where \(F_n\) is the Watson \(U\) distribution with parameter \(n\).

Definition at line 215 of file WatsonUDist.java.

◆ cdf() [1/2]

double umontreal.ssj.probdist.WatsonUDist.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 103 of file WatsonUDist.java.

◆ cdf() [2/2]

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

static

Computes the Watson \(U\) distribution function, i.e.

returns \(P[U_n^2 \le x]\), where \(U_n^2\) is the Watson statistic defined in ( WatsonU ). We use the asymptotic distribution for \(n \to\infty\), plus a correction in \(O(1/n)\), as given in [37] .

Definition at line 168 of file WatsonUDist.java.

◆ density() [1/2]

double umontreal.ssj.probdist.WatsonUDist.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 99 of file WatsonUDist.java.

◆ density() [2/2]

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

static

Computes the density of the Watson U distribution with parameter \(n\).

Definition at line 130 of file WatsonUDist.java.

◆ getMean() [1/2]

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

Returns the mean.

Returns: the mean

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 115 of file WatsonUDist.java.

◆ getMean() [2/2]

double umontreal.ssj.probdist.WatsonUDist.getMean ( int n )

static

Returns the mean of the Watson \(U\) distribution with parameter.

\(n\).

Returns: Returns the mean

Definition at line 285 of file WatsonUDist.java.

◆ getN()

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

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

Definition at line 312 of file WatsonUDist.java.

◆ getParams()

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

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

Implements umontreal.ssj.probdist.Distribution.

Definition at line 330 of file WatsonUDist.java.

◆ getStandardDeviation() [1/2]

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

Returns the standard deviation.

Returns: the standard deviation

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 123 of file WatsonUDist.java.

◆ getStandardDeviation() [2/2]

double umontreal.ssj.probdist.WatsonUDist.getStandardDeviation ( int n )

static

Returns the standard deviation of the Watson \(U\) distribution with parameter \(n\).

Returns: the standard deviation

Definition at line 305 of file WatsonUDist.java.

◆ getVariance() [1/2]

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

Returns the variance.

Returns: the variance

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 119 of file WatsonUDist.java.

◆ getVariance() [2/2]

double umontreal.ssj.probdist.WatsonUDist.getVariance ( int n )

static

Returns the variance of the Watson \(U\) distribution with parameter \(n\).

Returns: the variance

Definition at line 295 of file WatsonUDist.java.

◆ inverseF() [1/2]

double umontreal.ssj.probdist.WatsonUDist.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 111 of file WatsonUDist.java.

◆ inverseF() [2/2]

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

static

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

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

Definition at line 263 of file WatsonUDist.java.

◆ setN()

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

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

Definition at line 319 of file WatsonUDist.java.

◆ toString()

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

Returns a String containing information about the current distribution.

Definition at line 338 of file WatsonUDist.java.

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

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

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ WatsonUDist()

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ density() [1/2]

◆ density() [2/2]

◆ getMean() [1/2]

◆ getMean() [2/2]

◆ getN()

◆ getParams()

◆ getStandardDeviation() [1/2]

◆ getStandardDeviation() [2/2]

◆ getVariance() [1/2]

◆ getVariance() [2/2]

◆ inverseF() [1/2]

◆ inverseF() [2/2]

◆ setN()

◆ toString()