NAG Library Routine Document
G08CCF
1 Purpose
G08CCF performs the one sample Kolmogorov–Smirnov distribution test, using a userspecified distribution.
2 Specification
INTEGER 
N, NTYPE, IFAIL 
REAL (KIND=nag_wp) 
X(N), CDF, D, Z, P, SX(N) 
EXTERNAL 
CDF 

3 Description
The data consists of a single sample of $n$ observations, denoted by ${x}_{1},{x}_{2},\dots ,{x}_{n}$. Let ${S}_{n}\left({x}_{\left(i\right)}\right)$ and ${F}_{0}\left({x}_{\left(i\right)}\right)$ represent the sample cumulative distribution function and the theoretical (null) cumulative distribution function respectively at the point ${x}_{\left(i\right)}$, where ${x}_{\left(i\right)}$ is the $i$th smallest sample observation.
The Kolmogorov–Smirnov test provides a test of the null hypothesis
${H}_{0}$: the data are a random sample of observations from a theoretical distribution specified by you (in
CDF) against one of the following alternative hypotheses.
(i) 
${H}_{1}$: the data cannot be considered to be a random sample from the specified null distribution. 
(ii) 
${H}_{2}$: the data arise from a distribution which dominates the specified null distribution. In practical terms, this would be demonstrated if the values of the sample cumulative distribution function ${S}_{n}\left(x\right)$ tended to exceed the corresponding values of the theoretical cumulative distribution function ${F}_{0\left(x\right)}$. 
(iii) 
${H}_{3}$: the data arise from a distribution which is dominated by the specified null distribution. In practical terms, this would be demonstrated if the values of the theoretical cumulative distribution function ${F}_{0}\left(x\right)$ tended to exceed the corresponding values of the sample cumulative distribution function ${S}_{n}\left(x\right)$. 
One of the following test statistics is computed depending on the particular alternative hypothesis specified (see the description of the parameter
NTYPE in
Section 5).
For the alternative hypothesis
${H}_{1}$:
 ${D}_{n}$ – the largest absolute deviation between the sample cumulative distribution function and the theoretical cumulative distribution function. Formally ${D}_{n}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left\{{D}_{n}^{+},{D}_{n}^{}\right\}$.
For the alternative hypothesis
${H}_{2}$:
 ${D}_{n}^{+}$ – the largest positive deviation between the sample cumulative distribution function and the theoretical cumulative distribution function. Formally ${D}_{n}^{+}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left\{{S}_{n}\left({x}_{\left(i\right)}\right){F}_{0}\left({x}_{\left(i\right)}\right),0\right\}$.
For the alternative hypothesis
${H}_{3}$:
 ${D}_{n}^{}$ – the largest positive deviation between the theoretical cumulative distribution function and the sample cumulative distribution function. Formally ${D}_{n}^{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left\{{F}_{0}\left({x}_{\left(i\right)}\right){S}_{n}\left({x}_{\left(i1\right)}\right),0\right\}$. This is only true for continuous distributions. See Section 8 for comments on discrete distributions.
The standardized statistic,
$Z=D\times \sqrt{n}$, is also computed, where
$D$ may be
${D}_{n},{D}_{n}^{+}$ or
${D}_{n}^{}$ depending on the choice of the alternative hypothesis. This is the standardized value of
$D$ with no continuity correction applied and the distribution of
$Z$ converges asymptotically to a limiting distribution, first derived by
Kolmogorov (1933), and then tabulated by
Smirnov (1948). The asymptotic distributions for the onesided statistics were obtained by
Smirnov (1933).
The probability, under the null hypothesis, of obtaining a value of the test statistic as extreme as that observed, is computed. If
$n\le 100$, an exact method given by
Conover (1980) is used. Note that the method used is only exact for continuous theoretical distributions and does not include Conover's modification for discrete distributions. This method computes the onesided probabilities. The twosided probabilities are estimated by doubling the onesided probability. This is a good estimate for small
$p$, that is
$p\le 0.10$, but it becomes very poor for larger
$p$. If
$n>100$ then
$p$ is computed using the Kolmogorov–Smirnov limiting distributions; see
Feller (1948),
Kendall and Stuart (1973),
Kolmogorov (1933),
Smirnov (1933) and
Smirnov (1948).
4 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
Kolmogorov A N (1933) Sulla determinazione empirica di una legge di distribuzione Giornale dell' Istituto Italiano degli Attuari 4 83–91
Siegel S (1956) Nonparametric Statistics for the Behavioral Sciences McGraw–Hill
Smirnov N (1933) Estimate of deviation between empirical distribution functions in two independent samples Bull. Moscow Univ. 2(2) 3–16
Smirnov N (1948) Table for estimating the goodness of fit of empirical distributions Ann. Math. Statist. 19 279–281
5 Parameters
 1: N – INTEGERInput
On entry: $n$, the number of observations in the sample.
Constraint:
${\mathbf{N}}\ge 1$.
 2: X(N) – REAL (KIND=nag_wp) arrayInput
On entry: the sample observations, ${x}_{1},{x}_{2},\dots ,{x}_{n}$.
 3: CDF – REAL (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure
CDF must return the value of the theoretical (null) cumulative distribution function for a given value of its argument.
The specification of
CDF is:
 1: X – REAL (KIND=nag_wp)Input
On entry: the argument for which
CDF must be evaluated.
CDF must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which G08CCF is called. Parameters denoted as
Input must
not be changed by this procedure.
Constraint:
${\mathbf{CDF}}$ must always return a value in the range
$\left[0.0,1.0\right]$ and
CDF must always satify the condition that
${\mathbf{CDF}}\left({x}_{1}\right)\le {\mathbf{CDF}}\left({x}_{2}\right)$ for any
${x}_{1}\le {x}_{2}$.
 4: NTYPE – INTEGERInput
On entry: the statistic to be calculated, i.e., the choice of alternative hypothesis.
 ${\mathbf{NTYPE}}=1$
 Computes ${D}_{n}$, to test ${H}_{0}$ against ${H}_{1}$.
 ${\mathbf{NTYPE}}=2$
 Computes ${D}_{n}^{+}$, to test ${H}_{0}$ against ${H}_{2}$.
 ${\mathbf{NTYPE}}=3$
 Computes ${D}_{n}^{}$, to test ${H}_{0}$ against ${H}_{3}$.
Constraint:
${\mathbf{NTYPE}}=1$, $2$ or $3$.
 5: D – REAL (KIND=nag_wp)Output
On exit: the Kolmogorov–Smirnov test statistic (
${D}_{n}$,
${D}_{n}^{+}$ or
${D}_{n}^{}$ according to the value of
NTYPE).
 6: Z – REAL (KIND=nag_wp)Output
On exit: a standardized value, $Z$, of the test statistic, $D$, without the continuity correction applied.
 7: P – REAL (KIND=nag_wp)Output
On exit: the probability,
$p$, associated with the observed value of
$D$, where
$D$ may
${D}_{n}$,
${D}_{n}^{+}$ or
${D}_{n}^{}$ depending on the value of
NTYPE (see
Section 3).
 8: SX(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the sample observations, ${x}_{1},{x}_{2},\dots ,{x}_{n}$, sorted in ascending order.
 9: IFAIL – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$
On entry,  ${\mathbf{N}}<1$. 
 ${\mathbf{IFAIL}}=2$

On entry,  ${\mathbf{NTYPE}}\ne 1$, $2$ or $3$. 
 ${\mathbf{IFAIL}}=3$
The supplied theoretical cumulative distribution function returns a value less than $0.0$ or greater than $1.0$, thereby violating the definition of the cumulative distribution function.
 ${\mathbf{IFAIL}}=4$
The supplied theoretical cumulative distribution function is not a nondecreasing function thereby violating the definition of a cumulative distribution function, that is ${F}_{0}\left(x\right)>{F}_{0}\left(y\right)$ for some $x<y$.
7 Accuracy
For most cases the approximation for $p$ given when $n>100$ has a relative error of less than $0.01$. The twosided probability is approximated by doubling the onesided probability. This is only good for small $p$, that is $p<0.10$, but very poor for large $p$. The error is always on the conservative side.
The time taken by G08CCF increases with $n$ until $n>100$ at which point it drops and then increases slowly.
For a discrete theoretical cumulative distribution function
${F}_{0}\left(x\right)$,
${D}_{n}^{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left\{{F}_{0}\left({x}_{\left(i\right)}\right){S}_{n}\left({x}_{\left(i\right)}\right),0\right\}$. Thus if you wish to provide a discrete distribution function the following adjustment needs to be made,
 for ${D}_{n}^{+}$, return $F\left(x\right)$ as $x$ as usual;
 for ${D}_{n}^{}$, return $F\left(xd\right)$ at $x$ where $d$ is the discrete jump in the distribution. For example $d=1$ for the Poisson or binomial distributions.
9 Example
The following example performs the one sample Kolmogorov–Smirnov test to test whether a sample of $30$ observations arise firstly from a uniform distribution $U\left(0,1\right)$ or secondly from a Normal distribution with mean $0.75$ and standard deviation $0.5$. The twosided test statistic, ${D}_{n}$, the standardized test statistic, $Z$, and the upper tail probability, $p$, are computed and then printed for each test.
9.1 Program Text
Program Text (g08ccfe.f90)
9.2 Program Data
Program Data (g08ccfe.d)
9.3 Program Results
Program Results (g08ccfe.r)