Bootstrapping Dependent Data
One of the key issues confronting bootstrap resampling approximations is how
to deal with dependent data. Consider a sequence
f
X
t
g
n
t
=1
of dependent random
variables. Clearly it would be a mistake to resample from the sequence scalar
quantities, as the reshu› ed resamples would break the temporal dependence.
Our goal is most often to learn the variance of a general statistic
T
n
(
X
1
;:::;X
n
)
,
we hereafter refer to the unknown variance as
2
. The quantity
2
may not
be calculable analytically because the dependence structure and the underlying
distribution of the innovations are not assumed to be known.
In 1985, Hall examined the problem of bootstrap estimation for data that
was spatial in character. His proposed methods could be applied to timeseries
1
For
m
nonoverlapping
blocks of equal length, each block has length
n
m
. For the movingblock bootstrap,
he proposes dividing the series into
n
m
+1
overlapping blocks of equal length
n
m
.
f
x
1
;:::;x
4
g
bootstrap is obtained by constructing the statistic of interest for each member of
the set
f
(
x
1
;x
2
)
;
(
x
3
;x
4
)
g
:
The movingblock bootstrap is obtained by constructing the statistic of interest
for each member of the set
f
(
x
1
;x
2
)
;
(
x
2
;x
3
)
;
(
x
3
;x
4
)
g
:
The intuition underpinning the &xedblock bootstrap is as follows. The moving
block bootstrap has many samples that share a large number of observations, in
Further, if
m
grows with
n
, then a statistic constructed from a given subsample
will eventually behave as though it is independent of all but two (the adjacent two)
of the statistics constructed from the other subsamples. In addition,
m
should
grow with
n
to allow for longlived dynamics to be captured. One natural choice
for
m
would be
m
=
cn
, with
0
< c <
1
, as the subsamples would be of the same
order of magnitude as the original data. Unfortunately, such an approach would
1
To see why consider the case of two dimensional spatial data. Rather than a sequence of
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 Fall '08
 Staff
 Statistics, Tn, …xedblock bootstrap

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