<s>
Permutation	B-General_Concept
tests	I-General_Concept
rely	O
on	O
resampling	O
the	O
original	O
data	O
assuming	O
the	O
null	O
hypothesis	O
.	O
</s>
<s>
Bootstrapping	B-Algorithm
is	O
a	O
statistical	O
method	O
for	O
estimating	O
the	O
sampling	B-General_Concept
distribution	I-General_Concept
of	O
an	O
estimator	O
by	O
sampling	O
with	O
replacement	O
from	O
the	O
original	O
sample	O
,	O
most	O
often	O
with	O
the	O
purpose	O
of	O
deriving	O
robust	O
estimates	O
of	O
standard	B-General_Concept
errors	I-General_Concept
and	O
confidence	O
intervals	O
of	O
a	O
population	O
parameter	O
like	O
a	O
mean	O
,	O
median	O
,	O
proportion	O
,	O
odds	O
ratio	O
,	O
correlation	O
coefficient	O
or	O
regression	B-General_Concept
coefficient	I-General_Concept
.	O
</s>
<s>
It	O
has	O
been	O
called	O
the	O
plug-in	B-Algorithm
principle	I-Algorithm
,	O
as	O
it	O
is	O
the	O
method	O
of	O
estimation	O
of	O
functionals	O
of	O
a	O
population	O
distribution	O
by	O
evaluating	O
the	O
same	O
functionals	O
at	O
the	O
empirical	B-General_Concept
distribution	I-General_Concept
based	O
on	O
a	O
sample	O
.	O
</s>
<s>
For	O
example	O
,	O
when	O
estimating	O
the	O
population	O
mean	O
,	O
this	O
method	O
uses	O
the	O
sample	O
mean	O
;	O
to	O
estimate	O
the	O
population	O
median	O
,	O
it	O
uses	O
the	O
sample	O
median	O
;	O
to	O
estimate	O
the	O
population	O
regression	B-General_Concept
line	I-General_Concept
,	O
it	O
uses	O
the	O
sample	O
regression	B-General_Concept
line	I-General_Concept
.	O
</s>
<s>
It	O
is	O
often	O
used	O
as	O
a	O
robust	O
alternative	O
to	O
inference	O
based	O
on	O
parametric	O
assumptions	O
when	O
those	O
assumptions	O
are	O
in	O
doubt	O
,	O
or	O
where	O
parametric	O
inference	O
is	O
impossible	O
or	O
requires	O
very	O
complicated	O
formulas	O
for	O
the	O
calculation	O
of	O
standard	B-General_Concept
errors	I-General_Concept
.	O
</s>
<s>
Bootstrapping	B-Algorithm
techniques	O
are	O
also	O
used	O
in	O
the	O
updating-selection	O
transitions	O
of	O
particle	B-Algorithm
filters	I-Algorithm
,	O
genetic	B-Algorithm
type	I-Algorithm
algorithms	I-Algorithm
and	O
related	O
resample/reconfiguration	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
used	O
in	O
computational	B-Algorithm
physics	I-Algorithm
.	O
</s>
<s>
In	O
this	O
context	O
,	O
the	B-Application
bootstrap	I-Application
is	O
used	O
to	O
replace	O
sequentially	O
empirical	O
weighted	O
probability	O
measures	O
by	O
empirical	O
measures	O
.	O
</s>
<s>
The	B-Application
bootstrap	I-Application
allows	O
to	O
replace	O
the	O
samples	O
with	O
low	O
weights	O
by	O
copies	O
of	O
the	O
samples	O
with	O
high	O
weights	O
.	O
</s>
<s>
Cross-validation	B-Application
is	O
a	O
statistical	O
method	O
for	O
validating	O
a	O
predictive	B-General_Concept
model	I-General_Concept
.	O
</s>
<s>
Cross-validation	B-Application
is	O
employed	O
repeatedly	O
in	O
building	O
decision	O
trees	O
.	O
</s>
<s>
One	O
form	O
of	O
cross-validation	B-Application
leaves	O
out	O
a	O
single	O
observation	O
at	O
a	O
time	O
;	O
this	O
is	O
similar	O
to	O
the	O
jackknife	B-Algorithm
.	O
</s>
<s>
Another	O
,	O
K-fold	O
cross-validation	B-Application
,	O
splits	O
the	O
data	O
into	O
K	O
subsets	O
;	O
each	O
is	O
held	O
out	O
in	O
turn	O
as	O
the	O
validation	O
set	O
.	O
</s>
<s>
For	O
comparison	O
,	O
in	O
regression	O
analysis	O
methods	O
such	O
as	O
linear	B-General_Concept
regression	I-General_Concept
,	O
each	O
y	O
value	O
draws	O
the	O
regression	B-General_Concept
line	I-General_Concept
toward	O
itself	O
,	O
making	O
the	O
prediction	O
of	O
that	O
value	O
appear	O
more	O
accurate	O
than	O
it	O
really	O
is	O
.	O
</s>
<s>
Cross-validation	B-Application
applied	O
to	O
linear	B-General_Concept
regression	I-General_Concept
predicts	O
the	O
y	O
value	O
for	O
each	O
observation	O
without	O
using	O
that	O
observation	O
.	O
</s>
<s>
Without	O
cross-validation	B-Application
,	O
adding	O
predictors	O
always	O
reduces	O
the	O
residual	O
sum	O
of	O
squares	O
(	O
or	O
possibly	O
leaves	O
it	O
unchanged	O
)	O
.	O
</s>
<s>
Subsampling	O
is	O
an	O
alternative	O
method	O
for	O
approximating	O
the	O
sampling	B-General_Concept
distribution	I-General_Concept
of	O
an	O
estimator	O
.	O
</s>
<s>
The	O
two	O
key	O
differences	O
to	O
the	B-Application
bootstrap	I-Application
are	O
:	O
</s>
<s>
The	O
advantage	O
of	O
subsampling	O
is	O
that	O
it	O
is	O
valid	O
under	O
much	O
weaker	O
conditions	O
compared	O
to	O
the	B-Application
bootstrap	I-Application
.	O
</s>
<s>
There	O
are	O
many	O
cases	O
of	O
applied	O
interest	O
where	O
subsampling	O
leads	O
to	O
valid	O
inference	O
whereas	O
bootstrapping	B-Algorithm
does	O
not	O
;	O
for	O
example	O
,	O
such	O
cases	O
include	O
examples	O
where	O
the	O
rate	O
of	O
convergence	O
of	O
the	O
estimator	O
is	O
not	O
the	O
square	O
root	O
of	O
the	O
sample	O
size	O
or	O
when	O
the	O
limiting	O
distribution	O
is	O
non-normal	O
.	O
</s>
<s>
When	O
both	O
subsampling	O
and	O
the	B-Application
bootstrap	I-Application
are	O
consistent	O
,	O
the	B-Application
bootstrap	I-Application
is	O
typically	O
more	O
accurate	O
.	O
</s>
<s>
RANSAC	B-Algorithm
is	O
a	O
popular	O
algorithm	O
using	O
subsampling	O
.	O
</s>
<s>
Jackknifing	O
(	O
jackknife	B-Algorithm
cross-validation	B-Application
)	O
,	O
is	O
used	O
in	O
statistical	O
inference	O
to	O
estimate	O
the	O
bias	O
and	O
standard	B-General_Concept
error	I-General_Concept
(	O
variance	O
)	O
of	O
a	O
statistic	O
,	O
when	O
a	O
random	O
sample	O
of	O
observations	O
is	O
used	O
to	O
calculate	O
it	O
.	O
</s>
<s>
Historically	O
,	O
this	O
method	O
preceded	O
the	O
invention	O
of	O
the	B-Application
bootstrap	I-Application
with	O
Quenouille	O
inventing	O
this	O
method	O
in	O
1949	O
and	O
Tukey	O
extending	O
it	O
in	O
1958	O
.	O
</s>
<s>
The	O
basic	O
idea	O
behind	O
the	O
jackknife	B-Algorithm
variance	O
estimator	O
lies	O
in	O
systematically	O
recomputing	O
the	O
statistic	O
estimate	O
,	O
leaving	O
out	O
one	O
or	O
more	O
observations	O
at	O
a	O
time	O
from	O
the	O
sample	O
set	O
.	O
</s>
<s>
Instead	O
of	O
using	O
the	O
jackknife	B-Algorithm
to	O
estimate	O
the	O
variance	O
,	O
it	O
may	O
instead	O
be	O
applied	O
to	O
the	O
log	O
of	O
the	O
variance	O
.	O
</s>
<s>
For	O
many	O
statistical	O
parameters	O
the	O
jackknife	B-Algorithm
estimate	O
of	O
variance	O
tends	O
asymptotically	O
to	O
the	O
true	O
value	O
almost	O
surely	O
.	O
</s>
<s>
In	O
technical	O
terms	O
one	O
says	O
that	O
the	O
jackknife	B-Algorithm
estimate	O
is	O
consistent	O
.	O
</s>
<s>
The	O
jackknife	B-Algorithm
is	O
consistent	O
for	O
the	O
sample	O
means	O
,	O
sample	O
variances	O
,	O
central	O
and	O
non-central	O
t-statistics	O
(	O
with	O
possibly	O
non-normal	O
populations	O
)	O
,	O
sample	O
coefficient	O
of	O
variation	O
,	O
maximum	O
likelihood	O
estimators	O
,	O
least	O
squares	O
estimators	O
,	O
correlation	O
coefficients	O
and	O
regression	B-General_Concept
coefficients	I-General_Concept
.	O
</s>
<s>
In	O
the	O
case	O
of	O
a	O
unimodal	O
variate	O
the	O
ratio	O
of	O
the	O
jackknife	B-Algorithm
variance	O
to	O
the	O
sample	O
variance	O
tends	O
to	O
be	O
distributed	O
as	O
one	O
half	O
the	O
square	O
of	O
a	O
chi	O
square	O
distribution	O
with	O
two	O
degrees	O
of	O
freedom	O
.	O
</s>
<s>
The	O
jackknife	B-Algorithm
,	O
like	O
the	O
original	O
bootstrap	B-Application
,	O
is	O
dependent	O
on	O
the	O
independence	O
of	O
the	O
data	O
.	O
</s>
<s>
Extensions	O
of	O
the	O
jackknife	B-Algorithm
to	O
allow	O
for	O
dependence	O
in	O
the	O
data	O
have	O
been	O
proposed	O
.	O
</s>
<s>
Jackknife	B-Algorithm
is	O
equivalent	O
to	O
the	O
random	O
(	O
subsampling	O
)	O
leave-one-out	O
cross-validation	B-Application
,	O
it	O
only	O
differs	O
in	O
the	O
goal	O
.	O
</s>
<s>
Both	O
methods	O
,	O
the	B-Application
bootstrap	I-Application
and	O
the	O
jackknife	B-Algorithm
,	O
estimate	O
the	O
variability	O
of	O
a	O
statistic	O
from	O
the	O
variability	O
of	O
that	O
statistic	O
between	O
subsamples	O
,	O
rather	O
than	O
from	O
parametric	O
assumptions	O
.	O
</s>
<s>
For	O
the	O
more	O
general	O
jackknife	B-Algorithm
,	O
the	O
delete-m	O
observations	O
jackknife	B-Algorithm
,	O
the	B-Application
bootstrap	I-Application
can	O
be	O
seen	O
as	O
a	O
random	O
approximation	O
of	O
it	O
.	O
</s>
<s>
Although	O
there	O
are	O
huge	O
theoretical	O
differences	O
in	O
their	O
mathematical	O
insights	O
,	O
the	O
main	O
practical	O
difference	O
for	O
statistics	O
users	O
is	O
that	O
the	B-Application
bootstrap	I-Application
gives	O
different	O
results	O
when	O
repeated	O
on	O
the	O
same	O
data	O
,	O
whereas	O
the	O
jackknife	B-Algorithm
gives	O
exactly	O
the	O
same	O
result	O
each	O
time	O
.	O
</s>
<s>
Because	O
of	O
this	O
,	O
the	O
jackknife	B-Algorithm
is	O
popular	O
when	O
the	O
estimates	O
need	O
to	O
be	O
verified	O
several	O
times	O
before	O
publishing	O
(	O
e.g.	O
,	O
official	O
statistics	O
agencies	O
)	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
when	O
this	O
verification	O
feature	O
is	O
not	O
crucial	O
and	O
it	O
is	O
of	O
interest	O
not	O
to	O
have	O
a	O
number	O
but	O
just	O
an	O
idea	O
of	O
its	O
distribution	O
,	O
the	B-Application
bootstrap	I-Application
is	O
preferred	O
(	O
e.g.	O
,	O
studies	O
in	O
physics	O
,	O
economics	O
,	O
biological	O
sciences	O
)	O
.	O
</s>
<s>
Whether	O
to	O
use	O
the	B-Application
bootstrap	I-Application
or	O
the	O
jackknife	B-Algorithm
may	O
depend	O
more	O
on	O
operational	O
aspects	O
than	O
on	O
statistical	O
concerns	O
of	O
a	O
survey	O
.	O
</s>
<s>
The	O
jackknife	B-Algorithm
,	O
originally	O
used	O
for	O
bias	O
reduction	O
,	O
is	O
more	O
of	O
a	O
specialized	O
method	O
and	O
only	O
estimates	O
the	O
variance	O
of	O
the	O
point	O
estimator	O
.	O
</s>
<s>
The	B-Application
bootstrap	I-Application
,	O
on	O
the	O
other	O
hand	O
,	O
first	O
estimates	O
the	O
whole	O
distribution	O
(	O
of	O
the	O
point	O
estimator	O
)	O
and	O
then	O
computes	O
the	O
variance	O
from	O
that	O
.	O
</s>
<s>
"	O
The	B-Application
bootstrap	I-Application
can	O
be	O
applied	O
to	O
both	O
variance	O
and	O
distribution	O
estimation	O
problems	O
.	O
</s>
<s>
However	O
,	O
the	B-Application
bootstrap	I-Application
variance	O
estimator	O
is	O
not	O
as	O
good	O
as	O
the	O
jackknife	B-Algorithm
or	O
the	O
balanced	O
repeated	O
replication	O
(	O
BRR	O
)	O
variance	O
estimator	O
in	O
terms	O
of	O
the	O
empirical	O
results	O
.	O
</s>
<s>
Furthermore	O
,	O
the	B-Application
bootstrap	I-Application
variance	O
estimator	O
usually	O
requires	O
more	O
computations	O
than	O
the	O
jackknife	B-Algorithm
or	O
the	O
BRR	O
.	O
</s>
<s>
Thus	O
,	O
the	B-Application
bootstrap	I-Application
is	O
mainly	O
recommended	O
for	O
distribution	O
estimation.	O
"	O
</s>
<s>
There	O
is	O
a	O
special	O
consideration	O
with	O
the	O
jackknife	B-Algorithm
,	O
particularly	O
with	O
the	O
delete-1	O
observation	O
jackknife	B-Algorithm
.	O
</s>
<s>
It	O
should	O
only	O
be	O
used	O
with	O
smooth	O
,	O
differentiable	O
statistics	O
(	O
e.g.	O
,	O
totals	O
,	O
means	O
,	O
proportions	O
,	O
ratios	O
,	O
odd	O
ratios	O
,	O
regression	B-General_Concept
coefficients	I-General_Concept
,	O
etc	O
.	O
</s>
<s>
This	O
disadvantage	O
is	O
usually	O
the	O
argument	O
favoring	O
bootstrapping	B-Algorithm
over	O
jackknifing	O
.	O
</s>
<s>
More	O
general	O
jackknifes	O
than	O
the	O
delete-1	O
,	O
such	O
as	O
the	O
delete-m	O
jackknife	B-Algorithm
or	O
the	O
delete-all-but-2	O
Hodges	B-General_Concept
–	I-General_Concept
Lehmann	I-General_Concept
estimator	I-General_Concept
,	O
overcome	O
this	O
problem	O
for	O
the	O
medians	O
and	O
quantiles	O
by	O
relaxing	O
the	O
smoothness	O
requirements	O
for	O
consistent	O
variance	O
estimation	O
.	O
</s>
<s>
Usually	O
the	O
jackknife	B-Algorithm
is	O
easier	O
to	O
apply	O
to	O
complex	O
sampling	O
schemes	O
than	O
the	B-Application
bootstrap	I-Application
.	O
</s>
<s>
Theoretical	O
aspects	O
of	O
both	O
the	B-Application
bootstrap	I-Application
and	O
the	O
jackknife	B-Algorithm
can	O
be	O
found	O
in	O
Shao	O
and	O
Tu	O
(	O
1995	O
)	O
,	O
whereas	O
a	O
basic	O
introduction	O
is	O
accounted	O
in	O
Wolter	O
(	O
2007	O
)	O
.	O
</s>
<s>
The	B-Application
bootstrap	I-Application
estimate	O
of	O
model	O
prediction	O
bias	O
is	O
more	O
precise	O
than	O
jackknife	B-Algorithm
estimates	O
with	O
linear	O
models	O
such	O
as	O
linear	O
discriminant	O
function	O
or	O
multiple	O
regression	O
.	O
</s>
