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NAG Toolbox

NAG Toolbox: nag_nonpar_test_ks_1sample_user (g08cc)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_nonpar_test_ks_1sample_user (g08cc) performs the one sample Kolmogorov–Smirnov distribution test, using a user-specified distribution.

Syntax

[d, z, p, sx, ifail] = g08cc(x, cdf, ntype, 'n', n)
[d, z, p, sx, ifail] = nag_nonpar_test_ks_1sample_user(x, cdf, ntype, 'n', n)

Description

The data consists of a single sample of n observations, denoted by x1,x2,,xn. Let Snxi and F0xi represent the sample cumulative distribution function and the theoretical (null) cumulative distribution function respectively at the point xi, where xi is the ith smallest sample observation.
The Kolmogorov–Smirnov test provides a test of the null hypothesis H0: 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) H1: the data cannot be considered to be a random sample from the specified null distribution.
(ii) H2: 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 Snx tended to exceed the corresponding values of the theoretical cumulative distribution function F0x.
(iii) H3: 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 F0x tended to exceed the corresponding values of the sample cumulative distribution function Snx.
One of the following test statistics is computed depending on the particular alternative hypothesis specified (see the description of the argument ntype in Arguments).
For the alternative hypothesis H1:
For the alternative hypothesis H2:
For the alternative hypothesis H3:
The standardized statistic, Z=D×n, is also computed, where D may be Dn,Dn+ or Dn- 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 one-sided 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 n100, 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 one-sided probabilities. The two-sided probabilities are estimated by doubling the one-sided probability. This is a good estimate for small p, that is p0.10, 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).

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) Non-parametric 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

Parameters

Compulsory Input Parameters

1:     xn – double array
The sample observations, x1,x2,,xn.
2:     cdf – function handle or string containing name of m-file
cdf must return the value of the theoretical (null) cumulative distribution function for a given value of its argument.
[result] = cdf(x)

Input Parameters

1:     x – double scalar
The argument for which cdf must be evaluated.

Output Parameters

1:     result – double scalar
The value of the theoretical (null) cumulative distribution function evaluated at x.
Constraint: cdf must always return a value in the range 0.0,1.0 and cdf must always satify the condition that cdfx1cdfx2 for any x1x2.
3:     ntype int64int32nag_int scalar
The statistic to be calculated, i.e., the choice of alternative hypothesis.
ntype=1
Computes Dn, to test H0 against H1.
ntype=2
Computes Dn+, to test H0 against H2.
ntype=3
Computes Dn-, to test H0 against H3.
Constraint: ntype=1, 2 or 3.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the dimension of the array x.
n, the number of observations in the sample.
Constraint: n1.

Output Parameters

1:     d – double scalar
The Kolmogorov–Smirnov test statistic ( D n , D n + or D n - according to the value of ntype).
2:     z – double scalar
A standardized value, Z, of the test statistic, D, without the continuity correction applied.
3:     p – double scalar
The probability, p, associated with the observed value of D, where D may Dn, Dn+ or Dn- depending on the value of ntype (see Description).
4:     sxn – double array
The sample observations, x1,x2,,xn, sorted in ascending order.
5:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
   ifail=1
On entry,n<1.
   ifail=2
On entry,ntype1, 2 or 3.
   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.
   ifail=4
The supplied theoretical cumulative distribution function is not a nondecreasing function thereby violating the definition of a cumulative distribution function, that is F0x>F0y for some x<y.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

Accuracy

For most cases the approximation for p given when n>100 has a relative error of less than 0.01. The two-sided probability is approximated by doubling the one-sided 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.

Further Comments

The time taken by nag_nonpar_test_ks_1sample_user (g08cc) increases with n until n>100 at which point it drops and then increases slowly.
For a discrete theoretical cumulative distribution function F0x, Dn-=maxF0xi-Snxi,0. Thus if you wish to provide a discrete distribution function the following adjustment needs to be made,

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 U0,1 or secondly from a Normal distribution with mean 0.75 and standard deviation 0.5. The two-sided test statistic, Dn, the standardized test statistic, Z, and the upper tail probability, p, are computed and then printed for each test.
function g08cc_example


fprintf('g08cc example results\n\n');

global xmean std;
xmean = 0.75;
std   = 0.5;

x = [0.01; 0.30; 0.20; 0.90; 1.20; 0.09; 1.30; 0.18; 0.90; 0.48;
     1.98; 0.03; 0.50; 0.07; 0.70; 0.60; 0.95; 1.00; 0.31; 1.45;
     1.04; 1.25; 0.15; 0.75; 0.85; 0.22; 1.56; 0.81; 0.57; 0.55];

% Parameters
ntype  = int64(1);

[d, z, p, sx, ifail] = g08cc( ...
                              x, @cdf, ntype);

fprintf('Test against normal distribution:\n');
fprintf('                     mean = %7.2f\n', xmean);
fprintf('       standard deviation = %7.2f\n', std);

fprintf('\n\nTest statistic D = %8.4f\n', d);
fprintf('Z statistic      = %8.4f\n', z);
fprintf('Tail probability = %8.4f\n', p);



function [result] = cdf(x)
  % Cumulative distribution function.
  % Here: for normal distribution, with mean = 0.75 and s.d. = 0.5
  global xmean std;
  
  z = (x-xmean)/std;
  [result,ifail] = s15ab(z);
g08cc example results

Test against normal distribution:
                     mean =    0.75
       standard deviation =    0.50


Test statistic D =   0.1439
Z statistic      =   0.7882
Tail probability =   0.5262

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Chapter Contents
Chapter Introduction
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