<s>
In	O
statistics	O
,	O
the	O
kth	O
order	B-General_Concept
statistic	I-General_Concept
of	O
a	O
statistical	O
sample	O
is	O
equal	O
to	O
its	O
kth-smallest	O
value	O
.	O
</s>
<s>
Together	O
with	O
rank	O
statistics	O
,	O
order	B-General_Concept
statistics	I-General_Concept
are	O
among	O
the	O
most	O
fundamental	O
tools	O
in	O
non-parametric	B-General_Concept
statistics	I-General_Concept
and	O
inference	O
.	O
</s>
<s>
Important	O
special	O
cases	O
of	O
the	O
order	B-General_Concept
statistics	I-General_Concept
are	O
the	O
minimum	O
and	O
maximum	O
value	O
of	O
a	O
sample	O
,	O
and	O
(	O
with	O
some	O
qualifications	O
discussed	O
below	O
)	O
the	O
sample	O
median	O
and	O
other	O
sample	O
quantiles	O
.	O
</s>
<s>
When	O
using	O
probability	O
theory	O
to	O
analyze	O
order	B-General_Concept
statistics	I-General_Concept
of	O
random	O
samples	O
from	O
a	O
continuous	O
distribution	O
,	O
the	O
cumulative	O
distribution	O
function	O
is	O
used	O
to	O
reduce	O
the	O
analysis	O
to	O
the	O
case	O
of	O
order	B-General_Concept
statistics	I-General_Concept
of	O
the	O
uniform	O
distribution	O
.	O
</s>
<s>
where	O
the	O
subscript	O
enclosed	O
in	O
parentheses	O
indicates	O
the	O
th	O
order	B-General_Concept
statistic	I-General_Concept
of	O
the	O
sample	O
.	O
</s>
<s>
The	O
first	O
order	B-General_Concept
statistic	I-General_Concept
(	O
or	O
smallest	O
order	B-General_Concept
statistic	I-General_Concept
)	O
is	O
always	O
the	O
minimum	O
of	O
the	O
sample	O
,	O
that	O
is	O
,	O
</s>
<s>
Similarly	O
,	O
for	O
a	O
sample	O
of	O
size	O
,	O
the	O
th	O
order	B-General_Concept
statistic	I-General_Concept
(	O
or	O
largest	O
order	B-General_Concept
statistic	I-General_Concept
)	O
is	O
the	O
maximum	O
,	O
that	O
is	O
,	O
</s>
<s>
The	O
sample	B-General_Concept
range	I-General_Concept
is	O
the	O
difference	O
between	O
the	O
maximum	O
and	O
minimum	O
.	O
</s>
<s>
It	O
is	O
a	O
function	O
of	O
the	O
order	B-General_Concept
statistics	I-General_Concept
:	O
</s>
<s>
A	O
similar	O
important	O
statistic	O
in	O
exploratory	B-General_Concept
data	I-General_Concept
analysis	I-General_Concept
that	O
is	O
simply	O
related	O
to	O
the	O
order	B-General_Concept
statistics	I-General_Concept
is	O
the	O
sample	O
interquartile	B-General_Concept
range	I-General_Concept
.	O
</s>
<s>
The	O
sample	O
median	O
may	O
or	O
may	O
not	O
be	O
an	O
order	B-General_Concept
statistic	I-General_Concept
,	O
since	O
there	O
is	O
a	O
single	O
middle	O
value	O
only	O
when	O
the	O
number	O
of	O
observations	O
is	O
odd	O
.	O
</s>
<s>
More	O
precisely	O
,	O
if	O
for	O
some	O
integer	O
,	O
then	O
the	O
sample	O
median	O
is	O
and	O
so	O
is	O
an	O
order	B-General_Concept
statistic	I-General_Concept
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
when	O
is	O
even	O
,	O
and	O
there	O
are	O
two	O
middle	O
values	O
,	O
and	O
,	O
and	O
the	O
sample	O
median	O
is	O
some	O
function	O
of	O
the	O
two	O
(	O
usually	O
the	O
average	O
)	O
and	O
hence	O
not	O
an	O
order	B-General_Concept
statistic	I-General_Concept
.	O
</s>
<s>
Given	O
any	O
random	O
variables	O
X1	O
,	O
X2	O
...	O
,	O
Xn	O
,	O
the	O
order	B-General_Concept
statistics	I-General_Concept
X(1 )	O
,	O
X(2 )	O
,	O
...	O
,	O
X(n )	O
are	O
also	O
random	O
variables	O
,	O
defined	O
by	O
sorting	B-Algorithm
the	O
values	O
(	O
realizations	O
)	O
of	O
X1	O
,	O
...	O
,	O
Xn	O
in	O
increasing	O
order	O
.	O
</s>
<s>
(	O
where	O
r	O
specifies	O
which	O
order	B-General_Concept
statistic	I-General_Concept
)	O
:	O
</s>
<s>
In	O
this	O
section	O
we	O
show	O
that	O
the	O
order	B-General_Concept
statistics	I-General_Concept
of	O
the	O
uniform	O
distribution	O
on	O
the	O
unit	O
interval	O
have	O
marginal	O
distributions	O
belonging	O
to	O
the	O
beta	O
distribution	O
family	O
.	O
</s>
<s>
We	O
also	O
give	O
a	O
simple	O
method	O
to	O
derive	O
the	O
joint	O
distribution	O
of	O
any	O
number	O
of	O
order	B-General_Concept
statistics	I-General_Concept
,	O
and	O
finally	O
translate	O
these	O
results	O
to	O
arbitrary	O
continuous	O
distributions	O
using	O
the	O
cdf	O
.	O
</s>
<s>
Note	O
that	O
the	O
order	B-General_Concept
statistics	I-General_Concept
also	O
satisfy	O
.	O
</s>
<s>
that	O
is	O
,	O
the	O
kth	O
order	B-General_Concept
statistic	I-General_Concept
of	O
the	O
uniform	O
distribution	O
is	O
a	O
beta-distributed	O
random	O
variable	O
.	O
</s>
<s>
Perhaps	O
surprisingly	O
,	O
the	O
joint	O
density	O
of	O
the	O
n	O
order	B-General_Concept
statistics	I-General_Concept
turns	O
out	O
to	O
be	O
constant	O
:	O
</s>
<s>
different	O
permutations	O
of	O
the	O
sample	O
corresponding	O
to	O
the	O
same	O
sequence	O
of	O
order	B-General_Concept
statistics	I-General_Concept
.	O
</s>
<s>
Using	O
the	O
above	O
formulas	O
,	O
one	O
can	O
derive	O
the	O
distribution	O
of	O
the	O
range	O
of	O
the	O
order	B-General_Concept
statistics	I-General_Concept
,	O
that	O
is	O
the	O
distribution	O
of	O
,	O
i.e.	O
</s>
<s>
The	O
Laplace	O
transform	O
of	O
order	B-General_Concept
statistics	I-General_Concept
may	O
be	O
sampled	O
from	O
an	O
Erlang	O
distribution	O
via	O
a	O
path	O
counting	O
method	O
.	O
</s>
<s>
to	O
derive	O
the	O
following	O
probability	O
density	O
functions	O
for	O
the	O
order	B-General_Concept
statistics	I-General_Concept
of	O
a	O
sample	O
of	O
size	O
n	O
drawn	O
from	O
the	O
distribution	O
of	O
X	O
:	O
</s>
<s>
An	O
interesting	O
question	O
is	O
how	O
well	O
the	O
order	B-General_Concept
statistics	I-General_Concept
perform	O
as	O
estimators	O
of	O
the	O
quantiles	O
of	O
the	O
underlying	O
distribution	O
.	O
</s>
<s>
In	O
that	O
case	O
,	O
the	O
sample	O
median	O
is	O
usually	O
defined	O
as	O
the	O
midpoint	O
of	O
the	O
interval	O
delimited	O
by	O
the	O
3rd	O
and	O
4th	O
order	B-General_Concept
statistics	I-General_Concept
.	O
</s>
<s>
This	O
asymptotic	O
analysis	O
suggests	O
that	O
the	O
mean	O
outperforms	O
the	O
median	O
in	O
cases	O
of	O
low	O
kurtosis	B-Error_Name
,	O
and	O
vice	O
versa	O
.	O
</s>
<s>
Moments	O
of	O
the	O
distribution	O
for	O
the	O
first	O
order	B-General_Concept
statistic	I-General_Concept
can	O
be	O
used	O
to	O
develop	O
a	O
non-parametric	B-General_Concept
density	O
estimator	O
.	O
</s>
<s>
The	O
expected	O
value	O
of	O
the	O
first	O
order	B-General_Concept
statistic	I-General_Concept
given	O
a	O
sample	O
of	O
total	O
observations	O
yields	O
,	O
</s>
<s>
This	O
equation	O
in	O
combination	O
with	O
a	O
jackknifing	B-Algorithm
technique	O
becomes	O
the	O
basis	O
for	O
the	O
following	O
density	O
estimation	O
algorithm	O
,	O
</s>
<s>
In	O
contrast	O
to	O
the	O
bandwidth/length	O
based	O
tuning	O
parameters	O
for	O
histogram	B-Algorithm
and	O
kernel	B-General_Concept
based	O
approaches	O
,	O
the	O
tuning	O
parameter	O
for	O
the	O
order	B-General_Concept
statistic	I-General_Concept
based	O
density	O
estimator	O
is	O
the	O
size	O
of	O
sample	O
subsets	O
.	O
</s>
<s>
Such	O
an	O
estimator	O
is	O
more	O
robust	O
than	O
histogram	B-Algorithm
and	O
kernel	B-General_Concept
based	O
approaches	O
,	O
for	O
example	O
densities	O
like	O
the	O
Cauchy	O
distribution	O
(	O
which	O
lack	O
finite	O
moments	O
)	O
can	O
be	O
inferred	O
without	O
the	O
need	O
for	O
specialized	O
modifications	O
such	O
as	O
IQR	B-Application
based	I-Application
bandwidths	I-Application
.	O
</s>
<s>
This	O
is	O
because	O
the	O
first	O
moment	O
of	O
the	O
order	B-General_Concept
statistic	I-General_Concept
always	O
exists	O
if	O
the	O
expected	O
value	O
of	O
the	O
underlying	O
distribution	O
does	O
,	O
but	O
the	O
converse	O
is	O
not	O
necessarily	O
true	O
.	O
</s>
<s>
The	O
problem	O
of	O
computing	O
the	O
kth	O
smallest	O
(	O
or	O
largest	O
)	O
element	O
of	O
a	O
list	O
is	O
called	O
the	O
selection	B-Algorithm
problem	I-Algorithm
and	O
is	O
solved	O
by	O
a	O
selection	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Although	O
this	O
problem	O
is	O
difficult	O
for	O
very	O
large	O
lists	O
,	O
sophisticated	O
selection	B-Algorithm
algorithms	I-Algorithm
have	O
been	O
created	O
that	O
can	O
solve	O
this	O
problem	O
in	O
time	O
proportional	O
to	O
the	O
number	O
of	O
elements	O
in	O
the	O
list	O
,	O
even	O
if	O
the	O
list	O
is	O
totally	O
unordered	O
.	O
</s>
<s>
In	O
many	O
applications	O
all	O
order	B-General_Concept
statistics	I-General_Concept
are	O
required	O
,	O
in	O
which	O
case	O
a	O
sorting	B-Algorithm
algorithm	I-Algorithm
can	O
be	O
used	O
and	O
the	O
time	O
taken	O
is	O
O(n log n )	O
.	O
</s>
