Extends the class ContinuousDistribution for the Watson U distribution (see [55], [224], [225] ). More...

Inheritance diagram for WatsonUDist:

Collaboration diagram for WatsonUDist:

Public Member Functions
	WatsonUDist (int n)
	Constructs a Watson U distribution for a sample of size \(n\).

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.

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 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 (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. More...

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

static double	getMean (int n)
	Returns the mean of the Watson \(U\) distribution with parameter \(n\). More...

static double	getVariance (int n)
	Returns the variance of the Watson \(U\) distribution with parameter \(n\). More...

static double	getStandardDeviation (int n)
	Returns the standard deviation of the Watson \(U\) distribution with parameter \(n\). More...

Protected Attributes
int	n

Protected Attributes inherited from ContinuousDistribution
double	supportA = Double.NEGATIVE_INFINITY

double	supportB = Double.POSITIVE_INFINITY

Additional Inherited Members
Public Attributes inherited from ContinuousDistribution
int	decPrec = 15

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 Watson U distribution (see [55], [224], [225] ).

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 \(F_n(x) = P[U_n^2 \le x]\).

Member Function Documentation

◆ barF()

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.

◆ 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	(	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 [38] .

◆ getMean()

static double getMean ( int n )

static

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

Returns: Returns the mean

◆ getStandardDeviation()

static double getStandardDeviation ( int n )

static

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

Returns: the standard deviation

◆ getVariance()

static double getVariance ( int n )

static

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

Returns: the variance

◆ inverseF()

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.

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

WatsonUDist.java

Public Member Functions

Static Public Member Functions

Protected Attributes

Additional Inherited Members

Detailed Description

Member Function Documentation

◆ barF()

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ getMean()

◆ getStandardDeviation()

◆ getVariance()

◆ inverseF()