naginterfaces.library.stat.prob_​kolmogorov1

naginterfaces.library.stat.prob_kolmogorov1(n, d)

prob_kolmogorov1 returns the upper tail probability associated with the one sample Kolmogorov–Smirnov distribution.

For full information please refer to the NAG Library document for g01ey

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/g01/g01eyf.html

Parameters

nint

$n$, the number of observations in the sample.

dfloat

Contains the test statistic, $D_n^{+}$ or $D_n^{-}$.

Returns

pfloat

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

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $2$)

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

Notes

Let $S_n\left(x\right)$ be the sample cumulative distribution function and $F_0\left(x\right)$ the hypothesised theoretical distribution function.

prob_kolmogorov1 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;

$$\begin{array}{c}\begin{array}{ccc}{D}_n^{+}& =& {sup}_x[S_n\left(x\right)-F_0\left(x\right)]\text{,}\\ & & \\ D_n^{-}& =& {sup}_x[F_0\left(x\right)-S_n\left(x\right)[\text{.}\end{array}\end{array}$$

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) and Smirnov (1948).

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

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