naginterfaces.library.stat.prob_​kolmogorov2

naginterfaces.library.stat.prob_kolmogorov2(n1, n2, d)

prob_kolmogorov2 returns the probability associated with the upper tail of the Kolmogorov–Smirnov two sample distribution.

For full information please refer to the NAG Library document for g01ez

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g01/g01ezf.html

Parameters

n1int

The number of observations in the first sample, $n_1$.

n2int

The number of observations in the second sample, $n_2$.

dfloat

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.

Returns

pfloat

The probability associated with the upper tail of the Kolmogorov–Smirnov two sample distribution.

Raises

NagValueError

(errno $1$)

On entry, $n1=⟨value⟩$ and $n2=⟨value⟩$.

Constraint: $n1\ge 1$ and $n2\ge 1$.

(errno $2$)

On entry, $d<0.0$ or $d>1.0$: $d=⟨value⟩$.

(errno $3$)

The Smirnov approximation used for large samples did not converge in $200$ iterations. The probability is set to $1.0$.

Notes

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 prob_kolmogorov2 computes the upper tail probability for the Kolmogorov–Smirnov two sample two-sided test statistic $D_{n_1,n_2}$, where

$$D_{n_1,n_2}={sup}_x|F_{n_1}\left(x\right)-G_{n_2}\left(x\right)|\text{.}$$

The probability is computed exactly if $n_1,n_2\le 10000$ and $max(n_1,n_2)\le 2500$ using a method given by Kim and Jenrich (1973). For the case where $min(n_1,n_2)\le 10\%$ of the $max(n_1,n_2)$ and $min(n_1,n_2)\le 80$ the Smirnov approximation is used. For all other cases the Kolmogorov approximation is used. These two approximations are discussed in Kim and Jenrich (1973).

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

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 Statistics (1), 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