SSJ
V. 2.6.

## 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 WatsonUDistextends 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)`
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 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
`getXinf, getXsup, inverseBisection, inverseBrent, setXinf, setXsup`

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).

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 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`

SSJ
V. 2.6.

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