umontreal.iro.lecuyer.probdist
Class CramerVonMisesDist

java.lang.Object
  umontreal.iro.lecuyer.probdist.ContinuousDistribution
      umontreal.iro.lecuyer.probdist.CramerVonMisesDist

All Implemented Interfaces:

Distribution

public class CramerVonMisesDist
extends ContinuousDistribution

Extends the class ContinuousDistribution for the Cramér-von Mises distribution (see). Given a sample of n independent uniforms U_i over [0, 1], the Cramér-von Mises statistic W_n² is defined by

W_n² = 1/12n + ∑_j=1ⁿ(U_(j) - (j-0.5)/n)²,

where the U_(j) are the U_i sorted in increasing order. The distribution function (the cumulative probabilities) is defined as F_n(x) = P[W_n² <= x].

Field Summary

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

decPrec

Constructor Summary

CramerVonMisesDist(int n)
Constructs a Cramér-von Mises distribution for a sample of size n.

Method Summary

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 Cramér-von Mises distribution function 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 Cramér-von Mises distribution with parameter n.
double	getMean() Returns the mean.
static double	getMean(int n) Returns the mean of the 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	getStandardDeviation() Returns the standard deviation.
static double	getStandardDeviation(int n) Returns the standard deviation of the distribution with parameter n.
double	getVariance() Returns the variance.
static double	getVariance(int n) Returns the variance of the distribution with parameter n.
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 Cramér-von Mises distribution with parameter n.
void	setN(int n) Sets the parameter n of this object.
String	toString()

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

getXinf, getXsup, inverseBisection, inverseBrent, setXinf, setXsup

Methods inherited from class java.lang.Object

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

Constructor Detail

CramerVonMisesDist

public CramerVonMisesDist(int n)

Constructs a Cramér-von Mises 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

getMean

public double getMean()

Description copied from class: ContinuousDistribution

Returns the mean.

Specified by:

getMean in interface Distribution

Overrides:

getMean in class ContinuousDistribution

Returns:

the mean

getVariance

public double getVariance()

Description copied from class: ContinuousDistribution

Returns the variance.

Specified by:

getVariance in interface Distribution

Overrides:

getVariance in class ContinuousDistribution

Returns:

the variance

getStandardDeviation

public double getStandardDeviation()

Description copied from class: ContinuousDistribution

Returns the standard deviation.

Specified by:

getStandardDeviation in interface Distribution

Overrides:

getStandardDeviation in class ContinuousDistribution

Returns:

the standard deviation

density

public static double density(int n,
                             double x)

Computes the density function for a Cramér-von Mises distribution with parameter n.

cdf

public static double cdf(int n,
                         double x)

Computes the Cramér-von Mises distribution function with parameter n. Returns an approximation of P[W_n² <= x], where W_n² is the Cramér von Mises statistic (see). The approximation is based on the distribution function of W² = lim_{n -> ∞}W_n², which has the following series expansion derived by Anderson and Darling:

P(W² <= x) = ∑_j=0^∞(- 1)^j(4j+1)^1/2 exp{ - }K_1/4([tex2html_wrap_indisplay324]),

where K_ν is the modified Bessel function of the second kind. To correct for the deviation between P(W_n² <= x) and P(W² <= x), we add a correction in 1/n, obtained empirically by simulation. For n = 10, 20, 40, the error is less than 0.002, 0.001, and 0.0005, respectively, while for n >= 100 it is less than 0.0005. For n -> ∞, we estimate that the method returns at least 6 decimal digits of precision. For n = 1, the method uses the exact distribution: P(W₁² <= x) = 2(x - 1/12)^1/2 for 1/12 <= x <= 1/3.

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 Cramér-von Mises distribution with parameter n.

getMean

public static double getMean(int n)

Returns the mean of the distribution with parameter n.

Returns:

the mean

getVariance

public static double getVariance(int n)

Returns the variance of the distribution with parameter n.

Returns:

variance

getStandardDeviation

public static double getStandardDeviation(int n)

Returns the standard deviation of the distribution with parameter n.

Returns:

the standard deviation

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 String toString()

Overrides:

toString in class Object