public static double g01ey(
	int n,
	double d,
	out int ifail
)

Public Shared Function g01ey ( _
	n As Integer, _
	d As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double

public:
static double g01ey(
	int n, 
	double d, 
	[OutAttribute] int% ifail
)

static member g01ey : 
        n : int * 
        d : float * 
        ifail : int byref -> float 

g01ey returns the upper tail probability associated with the one sample Kolmogorov–Smirnov distribution.

Let $S_n\left(x\right)$ be the sample cumulative distribution function and $F_0\left(x\right)$ the hypothesised theoretical distribution function.

g01ey returns the upper tail probability, $p$, associated with the one-sided Kolmogorov–Smirnov test statistic $D_n^{+}$ or $D_n^{-}$, where these one-sided statistics are defined as follows; If $n\le 100$ an exact method is used; for the details see Conover (1980). Otherwise a large sample approximation derived by Smirnov is used; see Feller (1948), Kendall and Stuart (1973) or Smirnov (1948).

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

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

Errors or warnings detected by the method:

The large sample distribution used as an approximation to the exact distribution should have a relative error of less than $2.5$% for most cases.

None.

The upper tail probability for the two-sided statistic, $D_n=\max \left(D_n^{+},D_n^{-}\right)$, can be approximated by twice the probability returned via g01ey, that is $2p$. (Note that if the probability from g01ey is greater than $0.5$ then the two-sided probability should be truncated to $1.0$). This approximation to the tail probability for $D_n$ is good for small probabilities, (e.g., $p\le 0.10$) but becomes very poor for larger probabilities.

The time taken by the method increases with $n$, until $n>100$. At this point the approximation is used and the time decreases significantly. The time then increases again modestly with $n$.

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