kolmogorov_one
Performs a Kolmogorov-Smirnov one-sample test for continuous distributions.
Synopsis
#include <imsls.h>
float *imsls_f_kolmogorov_one (float cdf(), int n_observations, float x[], ..., 0)
The type double function is imsls_d_kolmogorov_one.
Required Arguments
float cdf (float x) (Input)
User-supplied function to compute the cumulative distribution function (CDF) at a given value. The form is CDF(x), where x is the value at which cdf is to be evaluated (Input) and cdf is the value of CDF at x. (Output)
int n_observations (Input)
Number of observations.
float x[] (Input)
Array of size n_observations containing the observations.
Return Value
Pointer to an array of length 3 containing Z, p1, and p2.
Synopsis with Optional Arguments
#include <imsls.h>
float *imsls_f_kolmogorov_one (float cdf(), int n_observations, float x[],
IMSLS_DIFFERENCES, float **differences,
IMSLS_DIFFERENCES_USER, float differences[],
IMSLS_N_MISSING, int *n_missing,
IMSLS_RETURN_USER, float test_statistic[],
IMSLS_FCN_W_DATA, float cdf(), void *data,
0)
Optional Arguments
IMSLS_DIFFERENCES, float **differences (Output)
Address of a pointer to the internally allocated array containing Dn , Dn+, Dn-.
IMSLS_DIFFERENCES_USER, float differences[] (Output)
Storage for the array differences is provided by the user.
See IMSLS_DIFFERENCES.
IMSLS_N_MISSING, int *n_missing (Ouput)
Number of missing values is returned in *n_missing.
IMSLS_RETURN_USER, float test_statistics[] (Output)
If specified, the Z-score and the p-values for hypothesis test against both one-sided and two-sided alternatives is stored in array test_statistics provided by the user.
IMSLS_FCN_W_DATA, float cdf (float x) , void *data, (Input)
User-supplied function to compute the cumulative distribution function, which also accepts a pointer to data that is supplied by the user. data is a pointer to the data to be passed to the user-supplied function. See the Passing Data to User-Supplied Functions section at the beginning of this manual for more details.
Description
The routine imsls_f_kolmogorov_one performs a Kolmogorov-Smirnov goodness-of-fit test in one sample. The hypotheses tested follow:
where F is the cumulative distribution function (CDF) of the random variable, and the theoretical cdf, F*, is specified via the user-supplied function cdf. Let n = n_observations - n_missing. The test statistics for both one-sided alternatives
and
and the two-sided (Dn = differences[0]) alternative are computed as well as an asymptotic z-score (test_statistics[0]) and p-values associated with the one-sided (test_statistics[1]) and two-sided (test_statistics[2]) hypotheses. For n > 80, asymptotic p-values are used (see Gibbons 1971). For n ≤ 80, exact one-sided p-values are computed according to a method given by Conover (1980, page 350). An approximate two-sided test p-value is obtained as twice the one-sided p-value. The approximation is very close for one-sided p-values less than 0.10 and becomes very bad as the one-sided p-values get larger.
Programming Notes
| 1. | The theoretical CDF is assumed to be continuous. If the CDF is not continuous, the statistics |
will not be computed correctly.
| 2. | Estimation of parameters in the theoretical CDF from the sample data will tend to make the p-values associated with the test statistics too liberal. The empirical CDF will tend to be closer to the theoretical CDF than it should be. |
| 3. | No attempt is made to check that all points in the sample are in the support of the theoretical CDF. If all sample points are not in the support of the CDF, the null hypothesis must be rejected. |
Example
In this example, a random sample of size 100 is generated via routine imsls_f_random_uniform (Chapter 12, Random Number Generation) for the uniform (0, 1) distribution. We want to test the null hypothesis that the cdf is the standard normal distribution with a mean of 0.5 and a variance equal to the uniform (0, 1) variance (1/12).
#include <imsls.h>
#include <stdio.h>
float cdf(float);
int main()
{
float *statistics=NULL, *diffs = NULL, *x=NULL;
int nobs = 100, nmiss;
imsls_random_seed_set(123457);
x = imsls_f_random_uniform(nobs, 0);
statistics = imsls_f_kolmogorov_one(cdf, nobs, x,
IMSLS_N_MISSING, &nmiss,
IMSLS_DIFFERENCES, &diffs,
0);
printf("D = %8.4f\n", diffs[0]);
printf("D+ = %8.4f\n", diffs[1]);
printf("D- = %8.4f\n", diffs[2]);
printf("Z = %8.4f\n", statistics[0]);
printf("Prob greater D one sided = %8.4f\n", statistics[1]);
printf("Prob greater D two sided = %8.4f\n", statistics[2]);
printf("N missing = %d\n", nmiss);
}
float cdf(float x)
{
float mean = .5, std = .2886751, z;
z = (x-mean)/std;
return(imsls_f_normal_cdf(z));
}
Output
D = 0.1471
D+ = 0.0810
D- = 0.1471
Z = 1.4708
Prob greater D one-sided = 0.0132
Prob greater D two-sided = 0.0264
N missing = 0
Warning Errors
|
IMSLS_TIE_DETECTED |
# ties were detected in the sample. |
Fatal Errors
|
IMSLS_STOP_USER_FCN |
Request from user supplied function to stop algorithm. |
