<s>
Kuiper	B-General_Concept
's	I-General_Concept
test	I-General_Concept
is	O
used	O
in	O
statistics	O
to	O
test	O
that	O
whether	O
a	O
given	O
distribution	O
,	O
or	O
family	O
of	O
distributions	O
,	O
is	O
contradicted	O
by	O
evidence	O
from	O
a	O
sample	O
of	O
data	O
.	O
</s>
<s>
Kuiper	B-General_Concept
's	I-General_Concept
test	I-General_Concept
is	O
closely	O
related	O
to	O
the	O
better-known	O
Kolmogorov	B-General_Concept
–	I-General_Concept
Smirnov	I-General_Concept
test	I-General_Concept
(	O
or	O
K-S	B-General_Concept
test	I-General_Concept
as	O
it	O
is	O
often	O
called	O
)	O
.	O
</s>
<s>
As	O
with	O
the	O
K-S	B-General_Concept
test	I-General_Concept
,	O
the	O
discrepancy	O
statistics	O
D+	O
and	O
D−	O
represent	O
the	O
absolute	O
sizes	O
of	O
the	O
most	O
positive	O
and	O
most	O
negative	O
differences	O
between	O
the	O
two	O
cumulative	O
distribution	O
functions	O
that	O
are	O
being	O
compared	O
.	O
</s>
<s>
The	O
trick	O
with	O
Kuiper	B-General_Concept
's	I-General_Concept
test	I-General_Concept
is	O
to	O
use	O
the	O
quantity	O
D++D−	O
as	O
the	O
test	O
statistic	O
.	O
</s>
<s>
This	O
small	O
change	O
makes	O
Kuiper	B-General_Concept
's	I-General_Concept
test	I-General_Concept
as	O
sensitive	O
in	O
the	O
tails	O
as	O
at	O
the	O
median	O
and	O
also	O
makes	O
it	O
invariant	O
under	O
cyclic	O
transformations	O
of	O
the	O
independent	O
variable	O
.	O
</s>
<s>
The	O
Anderson	B-General_Concept
–	I-General_Concept
Darling	I-General_Concept
test	I-General_Concept
is	O
another	O
test	O
that	O
provides	O
equal	O
sensitivity	O
at	O
the	O
tails	O
as	O
the	O
median	O
,	O
but	O
it	O
does	O
not	O
provide	O
the	O
cyclic	O
invariance	O
.	O
</s>
<s>
This	O
invariance	O
under	O
cyclic	O
transformations	O
makes	O
Kuiper	B-General_Concept
's	I-General_Concept
test	I-General_Concept
invaluable	O
when	O
testing	O
for	O
cyclic	O
variations	O
by	O
time	O
of	O
year	O
or	O
day	O
of	O
the	O
week	O
or	O
time	O
of	O
day	O
,	O
and	O
more	O
generally	O
for	O
testing	O
the	O
fit	O
of	O
,	O
and	O
differences	O
between	O
,	O
circular	O
probability	O
distributions	O
.	O
</s>
<s>
The	O
test	O
statistic	O
,	O
V	O
,	O
for	O
Kuiper	B-General_Concept
's	I-General_Concept
test	I-General_Concept
is	O
defined	O
as	O
follows	O
.	O
</s>
<s>
Let	O
F	O
be	O
the	O
continuous	O
cumulative	O
distribution	O
function	O
which	O
is	O
to	O
be	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
.	O
</s>
<s>
To	O
test	O
this	O
,	O
we	O
would	O
collect	O
the	O
dates	O
on	O
which	O
the	O
test	O
set	O
of	O
computers	O
had	O
failed	O
and	O
build	O
an	O
empirical	B-General_Concept
distribution	I-General_Concept
function	I-General_Concept
.	O
</s>
<s>
The	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
that	O
the	O
failures	O
are	O
uniformly	O
distributed	O
.	O
</s>
<s>
Another	O
test	O
statistic	O
having	O
this	O
property	O
is	O
the	O
Watson	O
statistic	O
,	O
which	O
is	O
related	O
to	O
the	O
Cramér	B-General_Concept
–	I-General_Concept
von	I-General_Concept
Mises	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
This	O
inability	O
to	O
distinguish	O
distributions	O
with	O
a	O
comb-like	O
shape	O
from	O
continuous	O
uniform	O
distributions	O
is	O
a	O
key	O
problem	O
with	O
all	O
statistics	O
based	O
on	O
a	O
variant	O
of	O
the	O
K-S	B-General_Concept
test	I-General_Concept
.	O
</s>
<s>
Kuiper	B-General_Concept
's	I-General_Concept
test	I-General_Concept
,	O
applied	O
to	O
the	O
event	O
times	O
modulo	O
one	O
week	O
,	O
is	O
able	O
to	O
detect	O
such	O
a	O
pattern	O
.	O
</s>
<s>
Using	O
event	O
times	O
that	O
have	O
been	O
modulated	O
with	O
the	O
K-S	B-General_Concept
test	I-General_Concept
can	O
result	O
in	O
different	O
results	O
depending	O
on	O
how	O
the	O
data	O
is	O
phased	O
.	O
</s>
<s>
In	O
this	O
example	O
,	O
the	O
K-S	B-General_Concept
test	I-General_Concept
may	O
detect	O
the	O
non-uniformity	O
if	O
the	O
data	O
is	O
set	O
to	O
start	O
the	O
week	O
on	O
Saturday	O
,	O
but	O
fail	O
to	O
detect	O
the	O
non-uniformity	O
if	O
the	O
week	O
starts	O
on	O
Wednesday	O
.	O
</s>
