g01 Chapter Contents
g01 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_prob_1_sample_ks (g01eyc)

## 1  Purpose

nag_prob_1_sample_ks (g01eyc) returns the upper tail probability associated with the one sample Kolmogorov–Smirnov distribution.

## 2  Specification

 #include #include
 double nag_prob_1_sample_ks (Integer n, double d, NagError *fail)

## 3  Description

Let ${S}_{n}\left(x\right)$ be the sample cumulative distribution function and ${F}_{0}\left(x\right)$ the hypothesised theoretical distribution function.
nag_prob_1_sample_ks (g01eyc) 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).
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

## 5  Arguments

1:     nIntegerInput
On entry: $n$, the number of observations in the sample.
Constraint: ${\mathbf{n}}\ge 1$.
2:     ddoubleInput
On entry: contains the test statistic, ${D}_{n}^{+}$ or ${D}_{n}^{-}$.
Constraint: $0.0\le {\mathbf{d}}\le 1.0$.
3:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL
On entry, ${\mathbf{d}}<0.0$ or ${\mathbf{d}}>1.0$: ${\mathbf{d}}=⟨\mathit{\text{value}}⟩$.

## 7  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.

## 8  Parallelism and Performance

Not applicable.

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 nag_prob_1_sample_ks (g01eyc), that is $2p$. (Note that if the probability from nag_prob_1_sample_ks (g01eyc) 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 function 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 $n$.

## 10  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.

### 10.1  Program Text

Program Text (g01eyce.c)

### 10.2  Program Data

Program Data (g01eyce.d)

### 10.3  Program Results

Program Results (g01eyce.r)