CramerVonMisesDist (Java Libraries for Stochastic Simulation)

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

Constructors
Constructor and Description

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

Constructors
Constructor and Description
`CramerVonMisesDist(int n)` Constructs a Cramér-von Mises distribution for a sample of size n.

Method Summary

Methods
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 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 = F_n^-1(u), where F_n is the Cramér-von Mises 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
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) = $\displaystyle {\frac{1}{{\pi\sqrt x}}}$ ∑_j=0^∞(- 1)^j-1/2 $\displaystyle \choosej$ (4j+1)^1/2 exp{ - $\displaystyle {\frac{{(4j+1)^2}}{{16 x}}}$ }K_1/4([tex2html_wrap_indisplay245]),
    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 java.lang.String toString()
```
    Overrides:
    
    toString in class java.lang.Object

Class CramerVonMisesDist

Field Summary

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

Constructor Summary

Method Summary

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

Methods inherited from class java.lang.Object

Constructor Detail

CramerVonMisesDist

Method Detail

density

cdf

barF

inverseF

getMean

getVariance

getStandardDeviation

density

cdf

barF

inverseF

getMean

getVariance

getStandardDeviation

getN

setN

getParams

toString