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

# Syntax

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

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

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

F# |
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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;

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).

$$\begin{array}{lcl}{D}_{n}^{+}& =& {\mathrm{sup}}_{x}\left[{S}_{n}\left(x\right)-{F}_{0}\left(x\right)\right]\text{,}\\ & & \\ {D}_{n}^{-}& =& {\mathrm{sup}}_{x}\left[{F}_{0}\left(x\right)-{S}_{n}\left(x\right)\right[\text{.}\end{array}$$ |

# References

Conover W J (1980)

*Practical Nonparametric Statistics*WileyFeller W (1948) On the Kolmogorov–Smirnov limit theorems for empirical distributions

*Ann. Math. Statist.***19**179–181Kendall M G and Stuart A (1973)

*The Advanced Theory of Statistics (Volume 2)*(3rd Edition) GriffinSiegel S (1956)

*Non-parametric Statistics for the Behavioral Sciences*McGraw–HillSmirnov 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 ${\mathbf{d}}>1.0$.

# 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.

# Further Comments

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