umontreal.iro.lecuyer.probdist

Class WatsonGDist

java.lang.Object

umontreal.iro.lecuyer.probdist.ContinuousDistribution

umontreal.iro.lecuyer.probdist.WatsonGDist

All Implemented Interfaces:

Distribution

public class WatsonGDist
extends ContinuousDistribution

Extends the class ContinuousDistribution for the Watson G distribution (see). Given a sample of n independent uniforms U_i over [0, 1], the G statistic is defined by

Gn	=	(n)1/2max1 <= j <= n{j/n - U(j) + bar(U)n -1/2}
	=	(n)1/2(Dn+ + bar(U)n - 1/2),

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 <= x].

Field Summary

Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

decPrec

Constructor Summary

Constructors

Constructor and Description
WatsonGDist(int n) Constructs a Watson distribution for a sample of size n.

Method Summary

Modifier and Type	Method and Description
double	barF(double x) Returns the complementary distribution function.
static double	barF(int n, double x) Computes the complementary distribution function bar(F)n(x) with parameter n.
double	cdf(double x) Returns the distribution function F(x).
static double	cdf(int n, double x) Computes the Watson G distribution function Fn(x), with parameter n.
double	density(double x) Returns f (x), the density evaluated at x.
static double	density(int n, double x) Computes the density function for a Watson G distribution with parameter n.
int	getN() Returns the parameter n of this object.
double[]	getParams() Return an array containing the parameter n of this object.
double	inverseF(double u) Returns the inverse distribution function x = F-1(u).
static double	inverseF(int n, double u) Computes x = Fn-1(u), where Fn is the Watson G distribution with parameter n.
void	setN(int n) Sets the parameter n of this object.
java.lang.String	toString()

Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

getMean, getStandardDeviation, getVariance, getXinf, getXsup, inverseBisection, inverseBrent, setXinf, setXsup

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, wait, wait, wait

Constructor Detail

WatsonGDist

public WatsonGDist(int n)

Constructs a Watson distribution for a sample of size n.

Method Detail

density

public double density(double x)

Description copied from class: ContinuousDistribution

Returns f (x), the density evaluated at x.

Specified by:

density in class ContinuousDistribution

Parameters:

x - value at which the density is evaluated

Returns:

density function evaluated at x

cdf

public double cdf(double x)

Description copied from interface: Distribution

Returns the distribution function F(x).

Parameters:

x - value at which the distribution function is evaluated

Returns:

distribution function evaluated at x

barF

public double barF(double x)

Description copied from class: ContinuousDistribution

Returns the complementary distribution function. The default implementation computes bar(F)(x) = 1 - F(x).

Specified by:

barF in interface Distribution

Overrides:

barF in class ContinuousDistribution

Parameters:

x - value at which the complementary distribution function is evaluated

Returns:

complementary distribution function evaluated at x

inverseF

public double inverseF(double u)

Description copied from class: ContinuousDistribution

Returns the inverse distribution function x = F^-1(u). Restrictions: u∈[0, 1].

Specified by:

inverseF in interface Distribution

Overrides:

inverseF in class ContinuousDistribution

Parameters:

u - value at which the inverse distribution function is evaluated

Returns:

the inverse distribution function evaluated at u

density

public static double density(int n,
                             double x)

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

cdf

public static double cdf(int n,
                         double x)

Computes the Watson G distribution function F_n(x), with parameter n. A cubic spline interpolation is used for the asymptotic distribution when n -> ∞, and an empirical correction of order 1/(n)^1/2, obtained empirically from 10⁷ 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.

barF

public static double barF(int n,
                          double x)

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

inverseF

public static double inverseF(int n,
                              double u)

Computes x = F_n^-1(u), where F_n is the Watson G distribution with parameter n.

getN

public int getN()

Returns the parameter n of this object.

setN

public void setN(int n)

Sets the parameter n of this object.

getParams

public double[] getParams()

Return an array containing the parameter n of this object.

toString

public java.lang.String toString()

Overrides:

toString in class java.lang.Object