Extends the class ContinuousDistribution for the Kolmogorov–Smirnov+ distribution (see [39], [54],. More...

Inheritance diagram for umontreal.ssj.probdist.KolmogorovSmirnovPlusDist:

Public Member Functions
	KolmogorovSmirnovPlusDist (int n)
	Constructs an Kolmogorov–Smirnov+ distribution for a sample of size \(n\).
double	density (double x)
	Returns \(f(x)\), the density evaluated at \(x\).
double	cdf (double x)
	Returns the distribution function \(F(x)\).
double	barF (double x)
	Returns the complementary distribution function.
double	inverseF (double u)
	Returns the inverse distribution function \(x = F^{-1}(u)\).
int	getN ()
	Returns the parameter \(n\) of this object.
void	setN (int n)
	Sets the parameter \(n\) of this object.
double[]	getParams ()
	Returns 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 umontreal.ssj.probdist.ContinuousDistribution
double	inverseBrent (double a, double b, double u, double tol)
	Computes the inverse distribution function \(x = F^{-1}(u)\), using the Brent-Dekker method.
double	inverseBisection (double u)
	Computes and returns the inverse distribution function \(x = F^{-1}(u)\), using bisection.
double	getMean ()
	Returns the mean.
double	getVariance ()
	Returns the variance.
double	getStandardDeviation ()
	Returns the standard deviation.
double	getXinf ()
	Returns \(x_a\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
double	getXsup ()
	Returns \(x_b\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
void	setXinf (double xa)
	Sets the value \(x_a=\) xa, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
void	setXsup (double xb)
	Sets the value \(x_b=\) xb, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).

Static Public Member Functions
static double	density (int n, double x)
	Computes the density of the Kolmogorov–Smirnov+ distribution with parameter \(n\).
static double	cdf (int n, double x)
	Computes the Kolmogorov–Smirnov+ distribution function.
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 the inverse \(x = F^{-1}(u)\) of the distribution with parameter \(n\).

Detailed Description

Extends the class ContinuousDistribution for the Kolmogorov–Smirnov+ distribution (see [39], [54],.

[26] ). Given a sample of \(n\) independent uniforms \(U_i\) over \([0,1]\), the Kolmogorov–Smirnov+ statistic \(D_n^+\) and the Kolmogorov–Smirnov \(-\) statistic \(D_n^-\), are defined by

\begin{align} D_n^+ & = \max_{1\le j\le n} \left(j/n - U_{(j)}\right), \tag{DNp} \\ D_n^- & = \max_{1\le j\le n} \left(U_{(j)} - (j-1)/n\right), \tag{DNm} \end{align}

where the \(U_{(j)}\) are the \(U_i\) sorted in increasing order. Both statistics follows the same distribution function, i.e. \(F_n(x) = P[D_n^+ \le x] = P[D_n^- \le x]\).

Definition at line 51 of file KolmogorovSmirnovPlusDist.java.

Constructor & Destructor Documentation

◆ KolmogorovSmirnovPlusDist()

umontreal.ssj.probdist.KolmogorovSmirnovPlusDist.KolmogorovSmirnovPlusDist ( int n )

Constructs an Kolmogorov–Smirnov+ distribution for a sample of size \(n\).

Definition at line 72 of file KolmogorovSmirnovPlusDist.java.

Member Function Documentation

◆ barF() [1/2]

double umontreal.ssj.probdist.KolmogorovSmirnovPlusDist.barF ( double x )

Returns the complementary distribution function.

The default implementation computes \(\bar{F}(x) = 1 - F(x)\).

Parameters

x	value at which the complementary distribution function is evaluated

Returns: complementary distribution function evaluated at x

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 84 of file KolmogorovSmirnovPlusDist.java.

◆ barF() [2/2]

double umontreal.ssj.probdist.KolmogorovSmirnovPlusDist.barF	(	int	n,
		double	x )

static

Computes the complementary distribution function \(\bar{F}_n(x)\) with parameter \(n\).

Definition at line 310 of file KolmogorovSmirnovPlusDist.java.

◆ cdf() [1/2]

double umontreal.ssj.probdist.KolmogorovSmirnovPlusDist.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 umontreal.ssj.probdist.Distribution.

Definition at line 80 of file KolmogorovSmirnovPlusDist.java.

◆ cdf() [2/2]

double umontreal.ssj.probdist.KolmogorovSmirnovPlusDist.cdf	(	int	n,
		double	x )

static

Computes the Kolmogorov–Smirnov+ distribution function.

\(F_n(x)\) with parameter \(n\). The distribution function can be approximated via the following expressions:

\begin{align} P[D_n^+ \le x] & = 1 - x \sum_{i=0}^{\lfloor n(1-x)\rfloor} \binom{n}{i} \left(\frac{i}{n} + x \right)^{i-1} \left(1 - \frac{i}{n} - x \right)^{n-i} \tag{DistKS1} \\ & = x \sum_{j=0}^{\lfloor nx \rfloor} \binom{n}{j} \left(\frac{j}{n} - x \right)^j \left(1 - \frac{j}{n} + x \right)^{n-j-1} \tag{DistKS2} \\ & \approx 1 - e^{-2 n x^2} \left[1 - \frac{2x}{3} \left( 1 - x\left(1 - \frac{2 n x^2}{3}\right) \right.\right. \nonumber \\ & \left.\left. - \frac{2}{3n} \left(\frac{1}{5} - \frac{19 n x^2}{15} + \frac{2n^2 x^4}{3}\right) \right) + O(n^{-2}) \right]. \tag{DistKS3} \end{align}

Formula ( DistKS1 ) and ( DistKS2 ) can be found in [54] , equations (2.1.12) and (2.1.16), while ( DistKS3 ) can be found in [39] . Formula ( DistKS2 ) becomes numerically unstable as \(nx\) increases. The approximation ( DistKS3 ) is simpler to compute and excellent when \(nx\) is large. The relative error on \(F_n(x) = P[D_n^+ \le x]\) is always less than \(10^{-5}\).

Definition at line 145 of file KolmogorovSmirnovPlusDist.java.