IMSL_KOLMOGOROV2

Syntax | Return Value | Arguments | Keywords | Discussion | Example | Version History

The IMSL_KOLMOGOROV2 function performs a Kolmogorov-Smirnov two-sample test.

Note
This routine requires an IDL Advanced Math and Stats license. For more information, contact your ITT Visual Information Solutions sales or technical support representative.

Syntax

Result = KOLMORGOROV2(x, y [, DIFFERENCES=variable] [, /DOUBLE] [, NMISSINGX=variable] [, NMISSINGY=variable] )

Return Value

One-dimensional array of length 3 containing Z, p₁, and p2 .

Arguments

x

One-dimensional array containing the observations from sample one.

y

One-dimensional array containing the observations from sample two.

Keywords

DIFFERENCES

Named variable into which a one-dimensional array containing Dn , Dn⁺, Dn^- is stored.

DOUBLE

If present and nonzero, double precision is used.

NMISSINGX

Named variable into which the number of missing values in the x sample is stored.

NMISSINGY

Named variable into which the number of missing values in the y sample is stored.

Discussion

The IMSL_KOLMOGOROV2 function computes Kolmogorov-Smirnov two-sample test statistics for testing that two continuous cumulative distribution functions (CDF's) are identical based upon two random samples. One- or two-sided alternatives are allowed. If n_observations_x = N_ELEMENTS(x) and n_observations_y = N_ELEMENTS(y), then the exact p-values are computed for the two-sided test when n_observations_x * n_observations_y is less than 104.

Let F_n(x) denote the empirical CDF in the X sample, let G_m(y) denote the empirical CDF in the Y sample, where n = n_observations_x - Nmissingx and m = n_observations_y - Nmissingy, and let the corresponding population distribution functions be denoted by F(x) and G(y), respectively. Then, the hypotheses tested by IMSL_KOLMOGOROV2 are as follows:

The test statistics are given as follows:

Asymptotically, the distribution of the statistic

(returned in Result (0)) converges to a distribution given by Smirnov (1939).

Exact probabilities for the two-sided test are computed when m * n is less than or equal to 10⁴, according to an algorithm given by Kim and Jennrich (1973;). When m * n is greater than 10⁴, the very good approximations given by Kim and Jennrich are used to obtain the two-sided p-values. The one-sided probability is taken as one half the two-sided probability. This is a very good approximation when the p-value is small (say, less than 0.10) and not very good for large p-values.

Example

The following example illustrates the IMSL_KOLMOGOROV2 routine with two randomly generated samples from a uniform(0,1) distribution. Since the two theoretical distributions are identical, we would not expect to reject the null hypothesis.

IMSL_RANDOMOPT, set  =  123457 
x  =  IMSL_RANDOM(100, /Uniform) 
y  =  IMSL_RANDOM(60, /Uniform) 
stats  =  IMSL_KOLMOGOROV2(x, y, DIFFERENCES = d, $ 
   NMISSINGX = nmx, NMISSINGY = nmy) 
PRINT, 'D  =', d(0) 
PRINT, 'D+ =', d(1) 
PRINT, 'D- =', d(2) 
PRINT, 'Z  =', stats(0) 
PRINT, 'Prob greater D one sided =', stats(1) 
PRINT, 'Prob greater D two sided =', stats(2) 
PRINT, 'Missing X =', nmx 
PRINT, 'Missing Y =', nmy 
 
D  =     0.180000 
D+ =     0.180000 
D- =    0.0100001 
Z  =      1.10227 
Prob greater D one sided =    0.0720105 
Prob greater D two sided =     0.144021 
Missing X =           0 
Missing Y =           0 

Version History