<s>
In	O
statistics	O
,	O
the	O
Hodges	B-General_Concept
–	I-General_Concept
Lehmann	I-General_Concept
estimator	I-General_Concept
is	O
a	O
robust	O
and	O
nonparametric	B-General_Concept
estimator	I-General_Concept
of	O
a	O
population	O
's	O
location	O
parameter	O
.	O
</s>
<s>
For	O
populations	O
that	O
are	O
symmetric	O
about	O
one	O
median	O
,	O
such	O
as	O
the	O
(	O
Gaussian	O
)	O
normal	O
distribution	O
or	O
the	O
Student	O
t-distribution	O
,	O
the	O
Hodges	B-General_Concept
–	I-General_Concept
Lehmann	I-General_Concept
estimator	I-General_Concept
is	O
a	O
consistent	O
and	O
median-unbiased	O
estimate	O
of	O
the	O
population	O
median	O
.	O
</s>
<s>
For	O
non-symmetric	O
populations	O
,	O
the	O
Hodges	B-General_Concept
–	I-General_Concept
Lehmann	I-General_Concept
estimator	I-General_Concept
estimates	O
the	O
"	O
pseudo	O
–	O
median	O
"	O
,	O
which	O
is	O
closely	O
related	O
to	O
the	O
population	O
median	O
.	O
</s>
<s>
The	O
Hodges	B-General_Concept
–	I-General_Concept
Lehmann	I-General_Concept
estimator	I-General_Concept
was	O
proposed	O
originally	O
for	O
estimating	O
the	O
location	O
parameter	O
of	O
one-dimensional	O
populations	O
,	O
but	O
it	O
has	O
been	O
used	O
for	O
many	O
more	O
purposes	O
.	O
</s>
<s>
It	O
has	O
been	O
generalized	O
from	O
univariate	O
populations	O
to	O
multivariate	B-General_Concept
populations	I-General_Concept
,	O
which	O
produce	O
samples	O
of	O
vectors	O
.	O
</s>
<s>
It	O
is	O
based	O
on	O
the	O
Wilcoxon	B-General_Concept
signed-rank	I-General_Concept
statistic	I-General_Concept
.	O
</s>
<s>
In	O
statistical	O
theory	O
,	O
it	O
was	O
an	O
early	O
example	O
of	O
a	O
rank-based	O
estimator	O
,	O
an	O
important	O
class	O
of	O
estimators	O
both	O
in	O
nonparametric	B-General_Concept
statistics	I-General_Concept
and	O
in	O
robust	O
statistics	O
.	O
</s>
<s>
The	O
Hodges	B-General_Concept
–	I-General_Concept
Lehmann	I-General_Concept
estimator	I-General_Concept
was	O
proposed	O
in	O
1963	O
independently	O
by	O
Pranab	O
Kumar	O
Sen	O
and	O
by	O
Joseph	O
Hodges	O
and	O
Erich	O
Lehmann	O
,	O
and	O
so	O
it	O
is	O
also	O
called	O
the	O
"	O
Hodges	B-General_Concept
–	I-General_Concept
Lehmann	I-General_Concept
–	O
Sen	O
estimator	O
"	O
.	O
</s>
<s>
In	O
the	O
simplest	O
case	O
,	O
the	O
"	O
Hodges	B-General_Concept
–	I-General_Concept
Lehmann	I-General_Concept
"	O
statistic	O
estimates	O
the	O
location	O
parameter	O
for	O
a	O
univariate	O
population	O
.	O
</s>
<s>
For	O
each	O
such	O
subset	O
,	O
the	O
mean	O
is	O
computed	O
;	O
finally	O
,	O
the	O
median	O
of	O
these	O
n(n-1 )	O
/2	O
averages	O
is	O
defined	O
to	O
be	O
the	O
Hodges	B-General_Concept
–	I-General_Concept
Lehmann	I-General_Concept
estimator	I-General_Concept
of	O
location	O
.	O
</s>
<s>
The	O
Hodges	B-General_Concept
–	I-General_Concept
Lehmann	I-General_Concept
statistic	O
also	O
estimates	O
the	O
difference	O
between	O
two	O
populations	O
.	O
</s>
<s>
The	O
Hodges	B-General_Concept
–	I-General_Concept
Lehmann	I-General_Concept
statistic	O
is	O
the	O
median	O
of	O
the	O
m×n	O
differences	O
.	O
</s>
<s>
For	O
a	O
population	O
that	O
is	O
symmetric	O
,	O
the	O
Hodges	B-General_Concept
–	I-General_Concept
Lehmann	I-General_Concept
statistic	O
estimates	O
the	O
population	O
's	O
median	O
.	O
</s>
<s>
The	O
Hodges	B-General_Concept
–	I-General_Concept
Lehmann	I-General_Concept
estimator	I-General_Concept
is	O
much	O
better	O
than	O
the	O
sample	O
mean	O
when	O
estimating	O
mixtures	O
of	O
normal	O
distributions	O
,	O
also	O
.	O
</s>
<s>
For	O
symmetric	O
distributions	O
,	O
the	O
Hodges	B-General_Concept
–	I-General_Concept
Lehmann	I-General_Concept
statistic	O
has	O
greater	O
efficiency	O
than	O
does	O
the	O
sample	O
median	O
.	O
</s>
<s>
For	O
the	O
normal	O
distribution	O
,	O
the	O
Hodges-Lehmann	B-General_Concept
statistic	O
is	O
nearly	O
as	O
efficient	O
as	O
the	O
sample	O
mean	O
.	O
</s>
<s>
For	O
the	O
Cauchy	O
distribution	O
(	O
Student	O
t-distribution	O
with	O
one	O
degree	O
of	O
freedom	O
)	O
,	O
the	O
Hodges-Lehmann	B-General_Concept
is	O
infinitely	O
more	O
efficient	O
than	O
the	O
sample	O
mean	O
,	O
which	O
is	O
not	O
a	O
consistent	O
estimator	O
of	O
the	O
median	O
.	O
</s>
<s>
For	O
non-symmetric	O
populations	O
,	O
the	O
Hodges-Lehmann	B-General_Concept
statistic	O
estimates	O
the	O
population	O
's	O
"	O
pseudo-median	O
"	O
,	O
a	O
location	O
parameter	O
that	O
is	O
closely	O
related	O
to	O
the	O
median	O
.	O
</s>
<s>
The	O
one-sample	O
Hodges	B-General_Concept
–	I-General_Concept
Lehmann	I-General_Concept
statistic	O
need	O
not	O
estimate	O
any	O
populationmean	O
,	O
which	O
for	O
many	O
distributions	O
does	O
not	O
exist	O
.	O
</s>
<s>
The	O
two-sample	O
Hodges	B-General_Concept
–	I-General_Concept
Lehmann	I-General_Concept
estimator	I-General_Concept
need	O
not	O
estimate	O
the	O
difference	O
of	O
two	O
means	O
or	O
the	O
difference	O
of	O
two	O
(	O
pseudo	O
-	O
)	O
medians	O
;	O
rather	O
,	O
it	O
estimates	O
the	O
differences	O
between	O
the	O
population	O
of	O
the	O
paired	O
random	O
–	O
variables	O
drawn	O
respectively	O
from	O
the	O
populations	O
.	O
</s>
<s>
The	O
Hodges	B-General_Concept
–	I-General_Concept
Lehmann	I-General_Concept
univariate	O
statistics	O
have	O
several	O
generalizations	O
in	O
multivariate	B-General_Concept
statistics	I-General_Concept
:	O
</s>
