<s>
In	O
statistics	O
,	O
the	O
jackknife	B-Algorithm
(	O
jackknife	B-Algorithm
cross-validation	B-Application
)	O
is	O
a	O
cross-validation	B-Application
technique	O
and	O
,	O
therefore	O
,	O
a	O
form	O
of	O
resampling	B-General_Concept
.	O
</s>
<s>
The	O
jackknife	B-Algorithm
pre-dates	O
other	O
common	O
resampling	B-General_Concept
methods	O
such	O
as	O
the	B-Application
bootstrap	I-Application
.	O
</s>
<s>
Given	O
a	O
sample	O
of	O
size	O
,	O
a	O
jackknife	B-Algorithm
estimator	O
can	O
be	O
built	O
by	O
aggregating	O
the	O
parameter	O
estimates	O
from	O
each	O
subsample	O
of	O
size	O
obtained	O
by	O
omitting	O
one	O
observation	O
.	O
</s>
<s>
The	O
jackknife	B-Algorithm
technique	O
was	O
developed	O
by	O
Maurice	O
Quenouille	O
(	O
1924	O
–	O
1973	O
)	O
from	O
1949	O
and	O
refined	O
in	O
1956	O
.	O
</s>
<s>
John	O
Tukey	O
expanded	O
on	O
the	O
technique	O
in	O
1958	O
and	O
proposed	O
the	O
name	O
"	O
jackknife	B-Algorithm
"	O
because	O
,	O
like	O
a	O
physical	O
jack-knife	O
(	O
a	O
compact	O
folding	O
knife	O
)	O
,	O
it	O
is	O
a	O
rough-and-ready	O
tool	O
that	O
can	O
improvise	O
a	O
solution	O
for	O
a	O
variety	O
of	O
problems	O
even	O
though	O
specific	O
problems	O
may	O
be	O
more	O
efficiently	O
solved	O
with	O
a	O
purpose-designed	O
tool	O
.	O
</s>
<s>
The	O
jackknife	B-Algorithm
is	O
a	O
linear	O
approximation	O
of	O
the	B-Application
bootstrap	I-Application
.	O
</s>
<s>
The	O
jackknife	B-Algorithm
estimator	O
of	O
a	O
parameter	O
is	O
found	O
by	O
systematically	O
leaving	O
out	O
each	O
observation	O
from	O
a	O
dataset	O
and	O
calculating	O
the	O
parameter	O
estimate	O
over	O
the	O
remaining	O
observations	O
and	O
then	O
aggregating	O
these	O
calculations	O
.	O
</s>
<s>
Then	O
we	O
proceed	O
as	O
follows	O
:	O
For	O
each	O
we	O
compute	O
the	O
mean	O
of	O
the	O
jackknife	B-Algorithm
subsample	O
consisting	O
of	O
all	O
but	O
the	O
-th	O
data	O
point	O
,	O
and	O
this	O
is	O
called	O
the	O
-th	O
jackknife	B-Algorithm
replicate	O
:	O
</s>
<s>
It	O
could	O
help	O
to	O
think	O
that	O
these	O
jackknife	B-Algorithm
replicates	O
give	O
us	O
an	O
approximation	O
of	O
the	O
distribution	O
of	O
the	O
sample	O
mean	O
and	O
the	O
larger	O
the	O
the	O
better	O
this	O
approximation	O
will	O
be	O
.	O
</s>
<s>
Then	O
finally	O
to	O
get	O
the	O
jackknife	B-Algorithm
estimator	O
we	O
take	O
the	O
average	O
of	O
these	O
jackknife	B-Algorithm
replicates	O
:	O
</s>
<s>
From	O
the	O
definition	O
of	O
as	O
the	O
average	O
of	O
the	O
jackknife	B-Algorithm
replicates	O
one	O
could	O
try	O
to	O
calculate	O
explicitly	O
,	O
and	O
the	O
bias	O
is	O
a	O
trivial	O
calculation	O
but	O
the	O
variance	O
of	O
is	O
more	O
involved	O
since	O
the	O
jackknife	B-Algorithm
replicates	O
are	O
not	O
independent	O
.	O
</s>
<s>
For	O
the	O
special	O
case	O
of	O
the	O
mean	O
,	O
one	O
can	O
show	O
explicitly	O
that	O
the	O
jackknife	B-Algorithm
estimate	O
equals	O
the	O
usual	O
estimate	O
:	O
</s>
<s>
This	O
simple	O
example	O
for	O
the	O
case	O
of	O
mean	O
estimation	O
is	O
just	O
to	O
illustrate	O
the	O
construction	O
of	O
a	O
jackknife	B-Algorithm
estimator	O
,	O
while	O
the	O
real	O
subtleties	O
(	O
and	O
the	O
usefulness	O
)	O
emerge	O
for	O
the	O
case	O
of	O
estimating	O
other	O
parameters	O
,	O
such	O
as	O
higher	O
moments	O
than	O
the	O
mean	O
or	O
other	O
functionals	O
of	O
the	O
distribution	O
.	O
</s>
<s>
A	O
jackknife	B-Algorithm
estimate	O
of	O
the	O
variance	O
of	O
can	O
be	O
calculated	O
from	O
the	O
variance	O
of	O
the	O
jackknife	B-Algorithm
replicates	O
:	O
</s>
<s>
The	O
jackknife	B-Algorithm
technique	O
can	O
be	O
used	O
to	O
estimate	O
(	O
and	O
correct	O
)	O
the	O
bias	O
of	O
an	O
estimator	O
calculated	O
over	O
the	O
entire	O
sample	O
.	O
</s>
<s>
In	O
this	O
kind	O
of	O
situation	O
the	O
jackknife	B-Algorithm
resampling	I-Algorithm
technique	O
may	O
be	O
of	O
help	O
.	O
</s>
<s>
We	O
construct	O
the	O
jackknife	B-Algorithm
replicates	O
:	O
</s>
<s>
where	O
each	O
replicate	O
is	O
a	O
"	O
leave-one-out	O
"	O
estimate	O
based	O
on	O
the	O
jackknife	B-Algorithm
subsample	O
consisting	O
of	O
all	O
but	O
one	O
of	O
the	O
data	O
points	O
:	O
</s>
<s>
The	O
jackknife	B-Algorithm
estimate	O
of	O
the	O
bias	O
of	O
is	O
given	O
by	O
:	O
</s>
<s>
and	O
the	O
resulting	O
bias-corrected	O
jackknife	B-Algorithm
estimate	O
of	O
is	O
given	O
by	O
:	O
</s>
<s>
The	O
jackknife	B-Algorithm
technique	O
can	O
be	O
also	O
used	O
to	O
estimate	O
the	O
variance	O
of	O
an	O
estimator	O
calculated	O
over	O
the	O
entire	O
sample	O
.	O
</s>
