Extends the class ContinuousDistribution for the Watson \(G\) distribution (see [41], [238] ). More...
Public Member Functions | |
| WatsonGDist (int n) | |
| Constructs a Watson distribution for a sample of size \(n\). | |
| double | density (double x) |
| double | cdf (double x) |
| Returns the distribution function \(F(x)\). More... | |
| double | barF (double x) |
| Returns \(\bar{F}(x) = 1 - F(x)\). More... | |
| double | inverseF (double u) |
| Returns the inverse distribution function \(F^{-1}(u)\), defined in ( inverseF ). More... | |
| int | getN () |
| Returns the parameter \(n\) of this object. | |
| void | setN (int n) |
| Sets the parameter \(n\) of this object. | |
| double [] | getParams () |
| Return an array containing the parameter \(n\) of this object. | |
| String | toString () |
Returns a String containing information about the current distribution. | |
 Public Member Functions inherited from ContinuousDistribution | |
| abstract double | density (double x) |
| Returns \(f(x)\), the density evaluated at \(x\). More... | |
| double | barF (double x) |
| Returns the complementary distribution function. More... | |
| double | inverseBrent (double a, double b, double u, double tol) |
| Computes the inverse distribution function \(x = F^{-1}(u)\), using the Brent-Dekker method. More... | |
| double | inverseBisection (double u) |
| Computes and returns the inverse distribution function \(x = F^{-1}(u)\), using bisection. More... | |
| double | inverseF (double u) |
| Returns the inverse distribution function \(x = F^{-1}(u)\). More... | |
| double | getMean () |
| Returns the mean. More... | |
| double | getVariance () |
| Returns the variance. More... | |
| double | getStandardDeviation () |
| Returns the standard deviation. More... | |
| double | getXinf () |
| Returns \(x_a\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More... | |
| double | getXsup () |
| Returns \(x_b\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More... | |
| void | setXinf (double xa) |
Sets the value \(x_a=\) xa, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More... | |
| void | setXsup (double xb) |
Sets the value \(x_b=\) xb, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More... | |
Static Public Member Functions | |
| static double | density (int n, double x) |
| Computes the density function for a Watson \(G\) distribution with parameter \(n\). | |
| static double | cdf (int n, double x) |
| Computes the Watson \(G\) distribution function \(F_n(x)\), with parameter \(n\). More... | |
| static double | barF (int n, double x) |
| Computes the complementary distribution function \(\bar{F}_n(x)\) with parameter \(n\). | |
| static double | inverseF (int n, double u) |
| Computes \(x = F_n^{-1}(u)\), where \(F_n\) is the Watson \(G\) distribution with parameter \(n\). | |
Protected Attributes | |
| int | n |
| Protected Attributes inherited from ContinuousDistribution | |
| double | supportA = Double.NEGATIVE_INFINITY |
| double | supportB = Double.POSITIVE_INFINITY |
Static Package Functions | |
| [static initializer] | |
Additional Inherited Members | |
| Public Attributes inherited from ContinuousDistribution | |
| int | decPrec = 15 |
| Static Protected Attributes inherited from ContinuousDistribution | |
| static final double | XBIG = 100.0 |
| static final double | XBIGM = 1000.0 |
| static final double [] | EPSARRAY |
Detailed Description
Extends the class ContinuousDistribution for the Watson \(G\) distribution (see [41], [238] ).
Given a sample of \(n\) independent uniforms \(U_i\) over \([0,1]\), the \(G\) statistic is defined by
\begin{align} G_n & = \sqrt{n} \max_{\Rule{0.0pt}{7.0pt}{0.0pt} 1\le j \le n} \left\{ j/n - U_{(j)} + \bar{U}_n - 1/2 \right\} \tag{WatsonG} \\ & = \sqrt{n}\left(D_n^+ + \bar{U}_n - 1/2\right), \nonumber \end{align}
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 \(F_n(x) = P[G_n \le x]\).
Member Function Documentation
◆ barF()
| double barF | ( | double | x | ) |
Returns \(\bar{F}(x) = 1 - F(x)\).
- Parameters
-
x value at which the complementary distribution function is evaluated
- Returns
- complementary distribution function evaluated at
x
Implements Distribution.
◆ cdf() [1/2]
| double cdf | ( | double | x | ) |
Returns the distribution function \(F(x)\).
- Parameters
-
x value at which the distribution function is evaluated
- Returns
- distribution function evaluated at
x
Implements Distribution.
◆ cdf() [2/2]
|
static |
Computes the Watson \(G\) distribution function \(F_n(x)\), with parameter \(n\).
A cubic spline interpolation is used for the asymptotic distribution when \(n\to\infty\), and an empirical correction of order \(1/\sqrt{n}\), obtained empirically from \(10^7\) 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.
◆ inverseF()
| double inverseF | ( | double | u | ) |
Returns the inverse distribution function \(F^{-1}(u)\), defined in ( inverseF ).
- Parameters
-
u value in the interval \((0,1)\) for which the inverse distribution function is evaluated
- Returns
- the inverse distribution function evaluated at
u
Implements Distribution.
The documentation for this class was generated from the following file:
- WatsonGDist.java
