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

# Syntax

C#
```public static double g01ey(
int n,
double d,
out int ifail
)```
Visual Basic
```Public Shared Function g01ey ( _
n As Integer, _
d As Double, _
<OutAttribute> ByRef ifail As Integer _
) As Double```
Visual C++
```public:
static double g01ey(
int n,
double d,
[OutAttribute] int% ifail
)```
F#
```static member g01ey :
n : int *
d : float *
ifail : int byref -> float
```

#### Parameters

n
Type: System..::..Int32
On entry: $n$, the number of observations in the sample.
Constraint: ${\mathbf{n}}\ge 1$.
d
Type: System..::..Double
On entry: contains the test statistic, ${D}_{n}^{+}$ or ${D}_{n}^{-}$.
Constraint: $0.0\le {\mathbf{d}}\le 1.0$.
ifail
Type: System..::..Int32%
On exit: ${\mathbf{ifail}}={0}$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

#### Return Value

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

# Description

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;
 $Dn+=supxSnx-F0x,Dn-=supxF0x-Snx.$
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).

# 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
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:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{n}}<1$.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{d}}<0.0$, or ${\mathbf{d}}>1.0$.
${\mathbf{ifail}}=-9000$
An error occured, see message report.

# Accuracy

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.

# Parallelism and Performance

None.

The upper tail probability for the two-sided statistic, ${D}_{n}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({D}_{n}^{+},{D}_{n}^{-}\right)$, 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 $n$.

# Example

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.

Example program (C#): g01eye.cs

Example program data: g01eye.d

Example program results: g01eye.r