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NAG Toolbox

NAG Toolbox: nag_stat_prob_kolmogorov2 (g01ez)

Purpose

nag_stat_prob_kolmogorov2 (g01ez) returns the probability associated with the upper tail of the Kolmogorov–Smirnov two sample distribution.

Syntax

[result, ifail] = g01ez(n1, n2, d)
[result, ifail] = nag_stat_prob_kolmogorov2(n1, n2, d)

Description

Let Fn1(x)Fn1(x) and Gn2(x)Gn2(x) denote the empirical cumulative distribution functions for the two samples, where n1n1 and n2n2 are the sizes of the first and second samples respectively.
The function nag_stat_prob_kolmogorov2 (g01ez) computes the upper tail probability for the Kolmogorov–Smirnov two sample two-sided test statistic Dn1,n2Dn1,n2, where
Dn1,n2 = supx|Fn1(x)Gn2(x)|.
Dn1,n2=supx|Fn1(x)-Gn2(x)|.
The probability is computed exactly if n1,n210000n1,n210000 and max (n1,n2)2500max(n1,n2)2500 using a method given by Kim and Jenrich (1973). For the case where min (n1,n2) 10 % min(n1,n2) 10 %  of the max (n1,n2)max(n1,n2) and min (n1,n2) 80 min(n1,n2) 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 Dmn(m < n)Dmn(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

Parameters

Compulsory Input Parameters

1:     n1 – int64int32nag_int scalar
The number of observations in the first sample, n1n1.
Constraint: n11n11.
2:     n2 – int64int32nag_int scalar
The number of observations in the second sample, n2n2.
Constraint: n21n21.
3:     d – double scalar
The test statistic Dn1,n2Dn1,n2, 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: 0.0d1.00.0d1.0.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     result – double scalar
The result of the function.
2:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
On entry,n1 < 1n1<1,
orn2 < 1n2<1.
  ifail = 2ifail=2
On entry,d < 0.0d<0.0,
ord > 1.0d>1.0.
  ifail = 3ifail=3
The approximation solution did not converge in 500500 iterations. A tail probability of 1.01.0 is returned by nag_stat_prob_kolmogorov2 (g01ez).

Accuracy

The large sample distributions used as approximations to the exact distribution should have a relative error of less than 5% for most cases.

Further Comments

The upper tail probability for the one-sided statistics, Dn1,n2 + Dn1,n2+ or Dn1,n2Dn1,n2-, can be approximated by halving the two-sided upper tail probability returned by nag_stat_prob_kolmogorov2 (g01ez), that is p / 2p/2. This approximation to the upper tail probability for either Dn1,n2 + Dn1,n2+ or Dn1,n2Dn1,n2- is good for small probabilities, (e.g., p0.10p0.10) but becomes poor for larger probabilities.
The time taken by the function increases with n1n1 and n2n2, until n1n2 > 10000n1n2>10000 or max (n1,n2)2500max(n1,n2)2500. At this point one of the approximations is used and the time decreases significantly. The time then increases again modestly with n1n1 and n2n2.

Example

function nag_stat_prob_kolmogorov2_example
n1 = int64(5);
n2 = int64(10);
d = 0.5;
[result, ifail] = nag_stat_prob_kolmogorov2(n1, n2, d)
 

result =

    0.3506


ifail =

                    0


function g01ez_example
n1 = int64(5);
n2 = int64(10);
d = 0.5;
[result, ifail] = g01ez(n1, n2, d)
 

result =

    0.3506


ifail =

                    0



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Chapter Contents
Chapter Introduction
NAG Toolbox

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