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## umontreal.iro.lecuyer.probdist
Class WatsonUDist

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

All Implemented Interfaces:
  Distribution

    public class WatsonUDist

extends ContinuousDistribution

Extends the class `ContinuousDistribution` for the Watson U distribution (see). Given a sample of n independent uniforms ui over [0, 1], the Watson statistic Un2 is defined by

Wn2 = 1/12n + ∑j=1n{u(j) - (j-0.5)/n}2,
Un2 = Wn2 - n(bar(u)n -1/2)2.

where the u(j) are the ui sorted in increasing order, and bar(u)n is the average of the observations ui. The distribution function (the cumulative probabilities) is defined as Fn(x) = P[Un2 <= x].

Field Summary

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

Constructor Summary
`WatsonUDist(int n)` Constructs a Watson U 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), where Fn is the Watson U distribution with parameter n.
`double` | `cdf(double x)` Returns the distribution function F(x).
`static double` | `cdf(int n, double x)` Computes the Watson U distribution function, i.e.
`double` | `density(double x)` Returns f (x), the density evaluated at x.
`static double` | `density(int n, double x)` Computes the density of the Watson U distribution with parameter n.
`double` | `getMean()` Returns the mean.
`static double` | `getMean(int n)` Returns the mean of the Watson U 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 Watson U distribution with parameter n.
`double` | `getVariance()` Returns the variance.
`static double` | `getVariance(int n)` Returns the variance of the Watson U distribution with parameter n.
`double` | `inverseF(double u)` Computes and returns the inverse distribution function x = F-1(u), by calling the method `inverseBrent`.
`static double` | `inverseF(int n, double u)` Computes x = Fn-1(u), where Fn is the Watson U 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
`inverseBisection, inverseBrent`

Methods inherited from class java.lang.Object
`equals, getClass, hashCode, notify, notifyAll, wait, wait, wait`

Constructor Detail

### WatsonUDist

    public WatsonUDist(int n)

  Constructs a Watson U 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), which is not accurate when F(x) -> 1.

  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`
  Computes and returns the inverse distribution function x = F-1(u), by calling the method `inverseBrent`. 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 of the Watson U distribution with parameter n.

### cdf

    public static double cdf(int n,
                             double x)

  Computes the Watson U distribution function, i.e. returns P[Un2 <= x], where Un2 is the Watson statistic defined in. We use the asymptotic distribution for n -> ∞, plus a correction in O(1/n), as given in.

### barF

    public static double barF(int n,
                              double x)

  Computes the complementary distribution function bar(F)n(x), where Fn is the Watson U distribution with parameter n.

### inverseF

    public static double inverseF(int n,
                                  double u)

  Computes x = Fn-1(u), where Fn is the Watson U distribution with parameter n.

### getMean

    public static double getMean(int n)

  Returns the mean of the Watson U distribution with parameter n.

  Returns:
    Returns the mean

### getVariance

    public static double getVariance(int n)

  Returns the variance of the Watson U distribution with parameter n.

  Returns:
    the variance

### getStandardDeviation

    public static double getStandardDeviation(int n)

  Returns the standard deviation of the Watson U 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`

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To submit a bug or ask questions, send an e-mail to Pierre L'Ecuyer.