STA 4502 Chapter 5, Page 1
5.4 KOLMOGOROV - SMIRNOV (KS) TEST
A DISTRIBUTION-FREE TEST FOR GENERAL DIFFERENCES
In Chapter 4,
we made inferences on the
difference in location parameters of two populations
,
using two independent samples and the Wilcoxon Rank Sum Test.
In this Chapter
we are interested in testing the hypothesis of
any
difference in distributions of two
populations, using two independent sample of sizes n and m, with
m ≤ n
(
1
).
The hypotheses of interest are:
Ho: The two populations have identical distribution vs
.
Ha
:
The two populations have different distributions.
The null hypothesis is rejected when there is
any
difference (in location and/or dispersion and/or
shape and or anything else) between the two populations. These hypotheses are tested using the
Kolmogorov-Smirnoff (KS) test.
Data:
A total of N = n + m observations obtained by
independent
random samples from two
populations, a population of X’s with dfF
X
(
x
) = P(X ≤
x
) and a population of Y’s with df
F
Y
(
y
) = P(Y ≤
y
).
The sample data are denoted by X
1
, X
2
, X
3
, …, X
m
and
Y
1
, Y
2
, Y
3
, .
.., Y
n
.
Assumptions:
A1.
X
1
, X
2
, X
3
, .
.., X
m
is a
random sample
(i.e., X
j
are independent of each other) from a
continuous
population, say Population 1, that has
df
(
2
), F
X
(t), where F
X
(t) = P(X ≤ t), for all t,
- ∞ < t <∞, [i.e., X, are i.i.d. with df F
X
(t).]
and
Y
1
, Y
2
, Y
3
, .
.., Y
n
is a
random sample
(i.e., Y
j
are independent of each other) from a
continuous population, say Population 2, that has df G
Y
(t), where G
Y
(t) = P(Y ≤ t), for all t, -
∞ < t <∞, [i.e., Y
j
are i.i.d. with df G
Y
(t).]
A2:
X
1
, X
2
, X
3
, .
.., X
m
and Y
1
, Y
2
, Y
3
, .
.., Y
n
are
independent samples.
Hypotheses:
Ho:
[F
x
(t) = G
Y
(t) for all t],
that is, the two populations have identical distributions (
3
) vs.