Description Usage Arguments Details Value References Examples
This routine computes nonparametric confidence intervals for bootstrap estimates. For reproducibility, save or set the random number state before calling this routine.
1 2 3 4 5 6 7 8 9 10 11 12 |
x |
an n \times p data matrix, rows are observed
p-vectors, assumed to be independently sampled from
target population. If p is 1 then |
B |
number of bootstrap replications. It can also be a vector
of |
func |
function \hat{θ}=func(x) computing estimate of the parameter of interest; func(x) should return a real value for any n^\prime \times p matrix x^\prime, n^\prime not necessarily equal to n |
... |
additional arguments for |
m |
an integer less than or equal to n; the routine
collects the n rows of |
mr |
if m < n then |
K |
a non-negative integer. If |
J |
the number of groups into which the bootstrap replications are split |
alpha |
percentiles desired for the bca confidence limits. One
only needs to provide |
verbose |
logical for verbose progress messages |
Bootstrap confidence intervals depend on three elements:
the cdf of the B bootstrap replications t_i^*, i=1… B
the bias-correction number z_0=Φ(∑_i^B I(t_i^* < t_0) / B ) where t_0=f(x) is the original estimate
the acceleration number a that measures the rate of change in σ_{t_0} as x, the data changes.
The first two of these depend only on the bootstrap distribution,
and not how it is generated: parametrically or
non-parametrically. Program bcajack can be used in a hybrid fashion
in which the vector tt
of B bootstrap replications is first
generated from a parametric model.
So, in the diabetes example below, we might first draw bootstrap
samples y^* \sim N(X\hat{β}, \hat{σ}^2 I) where
\hat{β} and \hat{σ} were obtained from
lm(y~X)
; each y^* would then provide a bootstrap
replication tstar = rfun(cbind(X, ystar))
. Then we could get bca
intervals from bcajack(Xy, tt, rfun ....)
with tt
,
the vector of B tstar
values. The only difference from a full
parametric bca analysis would lie in the nonparametric estimation
of a, often a negligible error.
a named list of several items
lims : first column shows the estimated bca confidence limits
at the requested alpha percentiles. These can be compared with
the standard limits \hat{θ} +
\hat{σ}z_{α}, third column. The second column
jacksd
gives the internal standard errors for the bca limits,
quite small in the example. Column 4, pct
, gives the
percentiles of the ordered B bootstrap replications
corresponding to the bca limits, eg the 897th largest
replication equalling the .975 bca limit .557.
stats : top line of stats shows 5 estimates: theta is
f(x), original point estimate of the parameter of
interest; sdboot
is its bootstrap estimate of standard error;
z0
is the bca bias correction value, in this case quite
negative; a
is the acceleration, a component of the bca
limits (nearly zero here); sdjack
is the jackknife estimate
of standard error for theta. Bottom line gives the internal
standard errors for the five quantities above. This is
substantial for z0
above.
B.mean : bootstrap sample size B, and the mean of the B bootstrap replications \hat{θ^*}
ustats : The bias-corrected estimator 2 * t0 - mean(tt)
,
and an estimate sdu
of its sampling error
seed : The random number state for reproducibility
DiCiccio T and Efron B (1996). Bootstrap confidence intervals. Statistical Science 11, 189-228
Efron B (1987). Better bootstrap confidence intervals. JASA 82 171-200
B. Efron and B. Narasimhan. Automatic Construction of Bootstrap Confidence Intervals, 2018.
1 2 3 4 5 6 7 8 9 10 |
$call
bcajack(x = Xy, B = 1000, func = rfun, m = 34, verbose = FALSE)
$lims
bca jacksd std pct
0.025 0.4349782 0.003100685 0.4421256 0.006
0.05 0.4425891 0.007211787 0.4524850 0.016
0.1 0.4545379 0.003161749 0.4644287 0.037
0.16 0.4643673 0.003564197 0.4738671 0.068
0.5 0.4989099 0.002384237 0.5065603 0.311
0.84 0.5312118 0.002510736 0.5392536 0.692
0.9 0.5408992 0.003948006 0.5486919 0.784
0.95 0.5527242 0.002534187 0.5606356 0.875
0.975 0.5615867 0.001996201 0.5709950 0.928
$stats
theta sdboot z0 a sdjack
est 0.5065603 0.0328754548 -0.2455895 -0.001290911 0.01864547
jsd 0.0000000 0.0007288154 0.0483231 0.000000000 0.00000000
$B.mean
[1] 1000.0000000 0.5148997
$ustats
ustat sdu
0.49822094 0.04073378
attr(,"class")
[1] "bcaboot"
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