NAG Library Function Document
nag_prob_1_sample_ks (g01eyc)
1
Purpose
nag_prob_1_sample_ks (g01eyc) returns the upper tail probability associated with the one sample Kolmogorov–Smirnov distribution.
2
Specification
#include <nag.h> |
#include <nagg01.h> |
double |
nag_prob_1_sample_ks (Integer n,
double d,
NagError *fail) |
|
3
Description
Let be the sample cumulative distribution function and the hypothesised theoretical distribution function.
nag_prob_1_sample_ks (g01eyc) returns the upper tail probability,
, associated with the one-sided Kolmogorov–Smirnov test statistic
or
, where these one-sided statistics are defined as follows;
If
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) or
Smirnov (1948).
4
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
5
Arguments
- 1:
– IntegerInput
-
On entry: , the number of observations in the sample.
Constraint:
.
- 2:
– doubleInput
-
On entry: contains the test statistic, or .
Constraint:
.
- 3:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_INT
-
On entry, .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- NE_REAL
-
On entry, or : .
7
Accuracy
The large sample distribution used as an approximation to the exact distribution should have a relative error of less than % for most cases.
8
Parallelism and Performance
nag_prob_1_sample_ks (g01eyc) is not threaded in any implementation.
The upper tail probability for the two-sided statistic, , can be approximated by twice the probability returned via nag_prob_1_sample_ks (g01eyc), that is . (Note that if the probability from nag_prob_1_sample_ks (g01eyc) is greater than then the two-sided probability should be truncated to ). This approximation to the tail probability for is good for small probabilities, (e.g., ) but becomes very poor for larger probabilities.
The time taken by the function increases with , until . At this point the approximation is used and the time decreases significantly. The time then increases again modestly with .
10
Example
The following example reads in different sample sizes and values for the test statistic . The upper tail probability is computed and printed for each case.
10.1
Program Text
Program Text (g01eyce.c)
10.2
Program Data
Program Data (g01eyce.d)
10.3
Program Results
Program Results (g01eyce.r)