<s>
In	O
statistics	O
,	O
an	O
empirical	B-General_Concept
distribution	I-General_Concept
function	I-General_Concept
(	O
commonly	O
also	O
called	O
an	O
empirical	B-General_Concept
Cumulative	I-General_Concept
Distribution	I-General_Concept
Function	I-General_Concept
,	O
eCDF	B-General_Concept
)	O
is	O
the	O
distribution	O
function	O
associated	O
with	O
the	O
empirical	O
measure	O
of	O
a	O
sample	O
.	O
</s>
<s>
The	O
empirical	B-General_Concept
distribution	I-General_Concept
function	I-General_Concept
is	O
an	O
estimate	O
of	O
the	O
cumulative	O
distribution	O
function	O
that	O
generated	O
the	O
points	O
in	O
the	O
sample	O
.	O
</s>
<s>
A	O
number	O
of	O
results	O
exist	O
to	O
quantify	O
the	O
rate	O
of	O
convergence	O
of	O
the	O
empirical	B-General_Concept
distribution	I-General_Concept
function	I-General_Concept
to	O
the	O
underlying	O
cumulative	O
distribution	O
function	O
.	O
</s>
<s>
The	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
for	O
the	O
empirical	O
distribution	O
is	O
as	O
follows	O
.	O
</s>
<s>
This	O
expression	O
asserts	O
the	O
pointwise	O
convergence	O
of	O
the	O
empirical	B-General_Concept
distribution	I-General_Concept
function	I-General_Concept
to	O
the	O
true	O
cumulative	O
distribution	O
function	O
.	O
</s>
<s>
The	O
sup-norm	O
in	O
this	O
expression	O
is	O
called	O
the	O
Kolmogorov	B-General_Concept
–	I-General_Concept
Smirnov	I-General_Concept
statistic	I-General_Concept
for	O
testing	O
the	O
goodness-of-fit	O
between	O
the	O
empirical	O
distribution	O
and	O
the	O
assumed	O
true	O
cumulative	O
distribution	O
function	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
L2-norm	O
gives	O
rise	O
to	O
the	O
Cramér	B-General_Concept
–	I-General_Concept
von	I-General_Concept
Mises	I-General_Concept
statistic	I-General_Concept
.	O
</s>
<s>
This	O
result	O
is	O
extended	O
by	O
the	O
Donsker	O
’s	O
theorem	O
,	O
which	O
asserts	O
that	O
the	O
empirical	B-General_Concept
process	I-General_Concept
,	O
viewed	O
as	O
a	O
function	O
indexed	O
by	O
,	O
converges	O
in	O
distribution	O
in	O
the	O
Skorokhod	O
space	O
to	O
the	O
mean-zero	O
Gaussian	B-General_Concept
process	I-General_Concept
,	O
where	O
is	O
the	O
standard	O
Brownian	O
bridge	O
.	O
</s>
<s>
In	O
fact	O
,	O
Kolmogorov	O
has	O
shown	O
that	O
if	O
the	O
cumulative	O
distribution	O
function	O
is	O
continuous	O
,	O
then	O
the	O
expression	O
converges	O
in	O
distribution	O
to	O
,	O
which	O
has	O
the	O
Kolmogorov	B-General_Concept
distribution	I-General_Concept
that	O
does	O
not	O
depend	O
on	O
the	O
form	O
of	O
.	O
</s>
<s>
As	O
per	O
the	O
above	O
bounds	O
,	O
we	O
can	O
plot	O
the	O
Empirical	B-General_Concept
CDF	I-General_Concept
,	O
CDF	O
and	O
Confidence	O
intervals	O
for	O
different	O
distributions	O
by	O
using	O
any	O
one	O
of	O
the	O
Statistical	O
implementations	O
.	O
</s>
<s>
A	O
non-exhaustive	O
list	O
of	O
software	O
implementations	O
of	O
Empirical	B-General_Concept
Distribution	I-General_Concept
function	I-General_Concept
includes	O
:	O
</s>
<s>
In	O
,	O
we	O
compute	O
an	O
empirical	B-General_Concept
cumulative	I-General_Concept
distribution	I-General_Concept
function	I-General_Concept
,	O
with	O
several	O
methods	O
for	O
plotting	O
,	O
printing	O
and	O
computing	O
with	O
such	O
an	O
“	O
ecdf	B-General_Concept
”	O
object	O
.	O
</s>
<s>
,	O
the	O
CDF	O
plot	O
creates	O
a	O
plot	O
of	O
the	O
empirical	B-General_Concept
cumulative	I-General_Concept
distribution	I-General_Concept
function	I-General_Concept
.	O
</s>
