Extends the class ContinuousDistribution for the Cramér-von Mises distribution (see [55], [224], [225] ). More...
Public Member Functions | |
| CramerVonMisesDist (int n) | |
| Constructs a Cramér-von Mises 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... | |
| double | getMean () |
| Returns the mean of the distribution function. | |
| double | getVariance () |
| Returns the variance of the distribution function. | |
| double | getStandardDeviation () |
| Returns the standard deviation of the distribution function. | |
| 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 Cramér-von Mises distribution with parameter \(n\). | |
| static double | cdf (int n, double x) |
| Computes the Cramér-von Mises distribution function 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 Cramér-von Mises distribution with parameter \(n\). | |
| static double | getMean (int n) |
| Returns the mean of the distribution with parameter \(n\). More... | |
| static double | getVariance (int n) |
| Returns the variance of the distribution with parameter \(n\). More... | |
| static double | getStandardDeviation (int n) |
| Returns the standard deviation of the distribution with parameter \(n\). More... | |
Protected Attributes | |
| int | n |
| Protected Attributes inherited from ContinuousDistribution | |
| double | supportA = Double.NEGATIVE_INFINITY |
| double | supportB = Double.POSITIVE_INFINITY |
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 Cramér-von Mises distribution (see [55], [224], [225] ).
Given a sample of \(n\) independent uniforms \(U_i\) over \([0,1]\), the Cramér-von Mises statistic \(W_n^2\) is defined by
\[ W_n^2 = \frac{1}{12n} + \sum_{j=1}^n \left(U_{(j)} - \frac{(j-0.5)}{n}\right)^2, \tag{CraMis} \]
where the \(U_{(j)}\) are the \(U_i\) sorted in increasing order. The distribution function (the cumulative probabilities) is defined as \(F_n(x) = P[W_n^2 \le x]\).
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 Cramér-von Mises distribution function with parameter \(n\).
Returns an approximation of \(P[W_n^2 \le x]\), where \(W_n^2\) is the Cramér von Mises statistic (see [224], [225], [6], [110] ). The approximation is based on the distribution function of \(W^2 = \lim_{n\to\infty} W_n^2\), which has the following series expansion derived by Anderson and Darling [6] :
\[ \qquad P(W^2 \le x) = \frac{1}{\pi\sqrt{x}} \sum_{j=0}^{\infty}(-1)^j \binom{-1/2}{j} \sqrt{4j+1}\;\; {exp}\left\{-\frac{(4j+1)^2}{16 x}\right\} K_{1/4}\left(\frac{(4j+1)^2}{16 x}\right), \]
where \(K_{\nu}\) is the modified Bessel function of the second kind. To correct for the deviation between \(P(W_n^2\le x)\) and \(P(W^2\le x)\), 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 \(n \ge100\) it is less than 0.0005. For \(n \to\infty\), we estimate that the method returns at least 6 decimal digits of precision. For \(n = 1\), the method uses the exact distribution: \(P(W_1^2 \le x) = 2 \sqrt{x - 1/12}\) for \(1/12 \le x \le1/3\).
◆ getMean()
|
static |
Returns the mean of the distribution with parameter \(n\).
- Returns
- the mean
◆ getStandardDeviation()
|
static |
Returns the standard deviation of the distribution with parameter \(n\).
- Returns
- the standard deviation
◆ getVariance()
|
static |
Returns the variance of the distribution with parameter \(n\).
- Returns
- variance
◆ 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:
- CramerVonMisesDist.java
