The routine may be called by the names g01ezf or nagf_stat_prob_kolmogorov2.
3Description
Let ${F}_{{n}_{1}}\left(x\right)$ and ${G}_{{n}_{2}}\left(x\right)$ denote the empirical cumulative distribution functions for the two samples, where ${n}_{1}$ and ${n}_{2}$ are the sizes of the first and second samples respectively.
The function g01ezf computes the upper tail probability for the Kolmogorov–Smirnov two sample two-sided test statistic ${D}_{{n}_{1},{n}_{2}}$, where
The probability is computed exactly if ${n}_{1},{n}_{2}\le 10000$ and $\mathrm{max}\phantom{\rule{0.125em}{0ex}}({n}_{1},{n}_{2})\le 2500$ using a method given by Kim and Jenrich (1973). For the case where $\mathrm{min}\phantom{\rule{0.125em}{0ex}}({n}_{1},{n}_{2})\le 10\%$ of the $\mathrm{max}\phantom{\rule{0.125em}{0ex}}({n}_{1},{n}_{2})$ and $\mathrm{min}\phantom{\rule{0.125em}{0ex}}({n}_{1},{n}_{2})\le 80$ the Smirnov approximation is used. For all other cases the Kolmogorov approximation is used. These two approximations are discussed in Kim and Jenrich (1973).
4References
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
Kim P J and Jenrich R I (1973) Tables of exact sampling distribution of the two sample Kolmogorov–Smirnov criterion ${D}_{mn}(m<n)$Selected Tables in Mathematical Statistics1 80–129 American Mathematical Society
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
5Arguments
1: $\mathbf{n1}$ – IntegerInput
On entry: the number of observations in the first sample, ${n}_{1}$.
Constraint:
${\mathbf{n1}}\ge 1$.
2: $\mathbf{n2}$ – IntegerInput
On entry: the number of observations in the second sample, ${n}_{2}$.
Constraint:
${\mathbf{n2}}\ge 1$.
3: $\mathbf{d}$ – Real (Kind=nag_wp)Input
On entry: the test statistic ${D}_{{n}_{1},{n}_{2}}$, for the two sample Kolmogorov–Smirnov goodness-of-fit test, that is the maximum difference between the empirical cumulative distribution functions (CDFs) of the two samples.
Constraint:
$0.0\le {\mathbf{d}}\le 1.0$.
4: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, 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, ${\mathbf{n1}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n2}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n1}}\ge 1$ and ${\mathbf{n2}}\ge 1$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{d}}<0.0$ or ${\mathbf{d}}>1.0$: ${\mathbf{d}}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=3$
The Smirnov approximation used for large samples did not converge in $200$ iterations. The probability is set to $1.0$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
The large sample distributions used as approximations to the exact distribution should have a relative error of less than 5% for most cases.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g01ezf is not threaded in any implementation.
9Further Comments
The upper tail probability for the one-sided statistics, ${D}_{{n}_{1},{n}_{2}}^{+}$ or ${D}_{{n}_{1},{n}_{2}}^{-}$, can be approximated by halving the two-sided upper tail probability returned by g01ezf, that is $p/2$. This approximation to the upper tail probability for either ${D}_{{n}_{1},{n}_{2}}^{+}$ or ${D}_{{n}_{1},{n}_{2}}^{-}$ is good for small probabilities, (e.g., $p\le 0.10$) but becomes poor for larger probabilities.
The time taken by the routine increases with ${n}_{1}$ and ${n}_{2}$, until ${n}_{1}{n}_{2}>10000$ or $\mathrm{max}\phantom{\rule{0.125em}{0ex}}({n}_{1},{n}_{2})\ge 2500$. At this point one of the approximations is used and the time decreases significantly. The time then increases again modestly with ${n}_{1}$ and ${n}_{2}$.
10Example
The following example reads in $10$ different sample sizes and values for the test statistic ${D}_{{n}_{1},{n}_{2}}$. The upper tail probability is computed and printed for each case.