h = kstest2(x1,x2) returns
a test decision for the null hypothesis that the data in vectors x1 and x2 are
from the same continuous distribution, using the two-sample Kolmogorov-Smirnov
test. The alternative hypothesis is that x1 and x2 are
from different continuous distributions. The result h is 1 if
the test rejects the null hypothesis at the 5% significance level,
and 0 otherwise.

h = kstest2(x1,x2,Name,Value) returns
a test decision for a two-sample Kolmogorov-Smirnov test with additional
options specified by one or more name-value pair arguments. For example,
you can change the significance level or conduct a one-sided test.

Generate sample data from two different Weibull distributions.

rng(1); % For reproducibility
x1 = wblrnd(1,1,1,50);
x2 = wblrnd(1.2,2,1,50);

Test the null hypothesis that data in vectors x1 and x2 comes
from populations with the same distribution, against the alternative
hypothesis that the cdf of the distribution of x1 is
larger than the cdf of the distribution of x2.

[h,p,k] = kstest2(x1,x2,'Tail','larger')

h =
1
p =
0.0158
k =
0.2800

The returned value of h = 1 indicates that kstest rejects
the null hypothesis, in favor of the alternative hypothesis that the
cdf of the distribution of x1 is larger than the
cdf of the distribution of x2, at the default 5%
significance level. The returned value of k is
the test statistic for the two-sample Kolmogorov-Smirnov test.

Sample data from the second sample, specified as a vector. Data
vectors x1 and x2 do not
need to be the same size.

Data Types: single | double

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments.
Name is the argument
name and Value is the corresponding
value. Name must appear
inside single quotes (' ').
You can specify several name and value pair
arguments in any order as Name1,Value1,...,NameN,ValueN.

Type of alternative hypothesis to evaluate, specified as the
comma-separated pair consisting of 'Tail' and one
of the following.

'unequal'

Test the alternative hypothesis that the empirical cdf of x1 is
unequal to the empirical cdf of x2.

'larger'

Test the alternative hypothesis that the empirical cdf of x1 is
larger than the empirical cdf of x2.

'smaller'

Test the alternative hypothesis that the empirical cdf of x1 is
smaller than the empirical cdf of x2.

If the data values in x1 tend to be larger
than those in x2, the empirical distribution
function of x1 tends to be smaller than that
of x2, and vice versa.

Asymptotic p-value of the test, returned
as a scalar value in the range (0,1). p is the
probability of observing a test statistic as extreme as, or more extreme
than, the observed value under the null hypothesis. The asymptotic p-value
becomes very accurate for large sample sizes, and is believed to be
reasonably accurate for sample sizes n1 and n2,
such that (n1*n2)/(n1 + n2) ≥ 4.

The two-sample Kolmogorov-Smirnov test is a
nonparametric hypothesis test that evaluates the difference between
the cdfs of the distributions of the two sample data vectors over
the range of x in each data set.

The two-sided test uses the maximum absolute difference between
the cdfs of the distributions of the two data vectors. The test statistic
is

where $${\widehat{F}}_{1}\left(x\right)$$ is the proportion of x1 values
less than or equal to x and $${\widehat{F}}_{2}\left(x\right)$$ is the proportion of x2 values
less than or equal to x.

The one-sided test uses the actual value of the difference between
the cdfs of the distributions of the two data vectors rather than
the absolute value. The test statistic is

In kstest2, the decision to reject the
null hypothesis is based on comparing the p-value p with
the significance level Alpha, not by comparing
the test statistic ks2stat with a critical value.

References

[1] Massey, F. J. "The Kolmogorov-Smirnov
Test for Goodness of Fit." Journal of the American
Statistical Association. Vol. 46, No. 253, 1951, pp. 68–78.

[2] Miller, L. H. "Table of Percentage
Points of Kolmogorov Statistics." Journal of the
American Statistical Association. Vol. 51, No. 273, 1956,
pp. 111–121.

[3] Marsaglia, G., W. Tsang, and J. Wang.
"Evaluating Kolmogorov's Distribution." Journal
of Statistical Software. Vol. 8, Issue 18, 2003.