g01ez returns the probability associated with the upper tail of the Kolmogorov–Smirnov two sample distribution.

Syntax

C#
public static double g01ez(
	int n1,
	int n2,
	double d,
	out int ifail
)
Visual Basic
Public Shared Function g01ez ( _
	n1 As Integer, _
	n2 As Integer, _
	d As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double
Visual C++
public:
static double g01ez(
	int n1, 
	int n2, 
	double d, 
	[OutAttribute] int% ifail
)
F#
static member g01ez : 
        n1 : int * 
        n2 : int * 
        d : float * 
        ifail : int byref -> float 

Parameters

n1
Type: System..::..Int32
On entry: the number of observations in the first sample, n1.
Constraint: n11.
n2
Type: System..::..Int32
On entry: the number of observations in the second sample, n2.
Constraint: n21.
d
Type: System..::..Double
On entry: the test statistic Dn1,n2, for the two sample Kolmogorov–Smirnov goodness-of-fit test, that is the maximum difference between the empirical cumulative distribution functions (CDFs) of the two samples.
Constraint: 0.0d1.0.
ifail
Type: System..::..Int32%
On exit: ifail=0 unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Return Value

g01ez returns the probability associated with the upper tail of the Kolmogorov–Smirnov two sample distribution.

Description

Let Fn1x and Gn2x denote the empirical cumulative distribution functions for the two samples, where n1 and n2 are the sizes of the first and second samples respectively.
The function g01ez computes the upper tail probability for the Kolmogorov–Smirnov two sample two-sided test statistic Dn1,n2, where
Dn1,n2=supxFn1x-Gn2x.
The probability is computed exactly if n1,n210000 and maxn1,n22500 using a method given by Kim and Jenrich (1973). For the case where minn1,n210% of the maxn1,n2 and minn1,n280 the Smirnov approximation is used. For all other cases the Kolmogorov approximation is used. These two approximations are discussed in Kim and Jenrich (1973).

References

Conover W J (1980) Practical Nonparametric Statistics Wiley
Feller W (1948) On the Kolmogorov–Smirnov limit theorems for empirical distributions Ann. Math. Statist. 19 179–181
Kendall M G and Stuart A (1973) The Advanced Theory of Statistics (Volume 2) (3rd Edition) Griffin
Kim P J and Jenrich R I (1973) Tables of exact sampling distribution of the two sample Kolmogorov–Smirnov criterion Dmnm<n Selected Tables in Mathematical Statistics 1 80–129 American Mathematical Society
Siegel S (1956) Non-parametric Statistics for the Behavioral Sciences McGraw–Hill
Smirnov N (1948) Table for estimating the goodness of fit of empirical distributions Ann. Math. Statist. 19 279–281

Error Indicators and Warnings

Errors or warnings detected by the method:
ifail=1
On entry,n1<1,
orn2<1.
ifail=2
On entry,d<0.0,
ord>1.0.
ifail=3
The approximation solution did not converge in 500 iterations. A tail probability of 1.0 is returned by g01ez.
ifail=-9000
An error occured, see message report.

Accuracy

The large sample distributions used as approximations to the exact distribution should have a relative error of less than 5% for most cases.

Parallelism and Performance

None.

Further Comments

The upper tail probability for the one-sided statistics, Dn1,n2+ or Dn1,n2-, can be approximated by halving the two-sided upper tail probability returned by g01ez, that is p/2. This approximation to the upper tail probability for either Dn1,n2+ or Dn1,n2- is good for small probabilities, (e.g., p0.10) but becomes poor for larger probabilities.
The time taken by the method increases with n1 and n2, until n1n2>10000 or maxn1,n22500. At this point one of the approximations is used and the time decreases significantly. The time then increases again modestly with n1 and n2.

Example

The following example reads in 10 different sample sizes and values for the test statistic Dn1,n2. The upper tail probability is computed and printed for each case.

Example program (C#): g01eze.cs

Example program data: g01eze.d

Example program results: g01eze.r

See Also