On the asymptotic theory of new bootstrap confidence bounds
Abstract
We propose a new method, based on sample splitting, for constructing
bootstrap confidence bounds for a parameter appearing in the regular smooth
function model. It has been demonstrated in the literature, for example, by
Hall [
Ann. Statist.
16
(1988) 927–985;
The Bootstrap and Edgeworth Expan-
sion
(1992) Springer], that the well-known
percentile-
t
method for construct-
ing bootstrap confidence bounds typically incurs a coverage error of order
O(n
−
1
)
, with
n
being the sample size. Our version of the percentile-
t
bound
reduces this coverage error to order
O(n
−
3
/
2
)
and in some cases to
O(n
−
2
)
.
Furthermore, whereas the standard
percentile
bounds typically incur coverage
error of
O(n
−
1
/
2
)
, the new bounds have reduced error of
O(n
−
1
)
. In the case
where the parameter of interest is the population mean, we derive for each
confidence bound the exact coefficient of the leading term in an asymptotic
expansion of the coverage error, although similar results may be obtained for
other parameters such as the variance, the correlation coefficient, and the ratio
of two means. We show that equal-tailed confidence intervals with coverage
error at most
O(n
−
2
)
may be obtained from the newly proposed bounds, as
opposed to the typical error
O(n
−
1
)
of the standard intervals. It is also shown
that the good properties of the new percentile-
t
method carry over to regres-
sion problems. Results of independent interest are derived, such as a gener-
alisation of a delta method by Cramér [
Mathematical Methods of Statistics
(1946) Princeton Univ. Press] and Hurt [
Apl. Mat.
21
(1976) 444–456], and
an expression for a polynomial appearing in an Edgeworth expansion of the
distribution of a Studentised statistic for the slope parameter in a regression
model. A small simulation study illustrates the behavior of the confidence
bounds for small to moderate sample sizes
URI
http://hdl.handle.net/10394/26574https://doi.org/10.1214/17-AOS1557
https://projecteuclid.org/download/pdfview_1/euclid.aos/1519268436