This function will find control variate estimates from a bootstrap output object. It can either find the adjusted bias estimate using post-simulation balancing or it can estimate the bias, variance, third cumulant and quantiles, using the linear approximation as a control variate.
control(boot.out, L = NULL, distn = NULL, index = 1, t0 = NULL,
t = NULL, bias.adj = FALSE, alpha = NULL, …)A bootstrap output object returned from boot. The bootstrap
replicates must have been generated using the usual nonparametric
bootstrap.
The empirical influence values for the statistic of interest. If
L is not supplied then empinf is called to calculate
them from boot.out.
If present this must be the output from smooth.spline giving
the distribution function of the linear approximation. This is used
only if bias.adj is FALSE. Normally this would be
found using a saddlepoint approximation. If it is not supplied in
that case then it is calculated by saddle.distn.
The index of the variable of interest in the output of
boot.out$statistic.
The observed value of the statistic of interest on the original data
set boot.out$data. This argument is used only if
bias.adj is FALSE. The input value is ignored if
t is not also supplied. The default value is is
boot.out$t0[index].
The bootstrap replicate values of the statistic of interest. This
argument is used only if bias.adj is FALSE. The input
is ignored if t0 is not supplied also. The default value is
boot.out$t[,index].
A logical variable which if TRUE specifies that the adjusted
bias estimate using post-simulation balance is all that is required.
If bias.adj is FALSE (default) then the linear
approximation to the statistic is calculated and used as a control
variate in estimates of the bias, variance and third cumulant as
well as quantiles.
The alpha levels for the required quantiles if bias.adj is
FALSE.
Any additional arguments that boot.out$statistic requires.
These are passed unchanged every time boot.out$statistic is
called. boot.out$statistic is called once if bias.adj
is TRUE, otherwise it may be called by empinf for
empirical influence calculations if L is not supplied.
If bias.adj is TRUE then the returned value is the
adjusted bias estimate.
If bias.adj is FALSE then the returned value is a list
with the following components
The empirical influence values used. These are the input values if
supplied, and otherwise they are the values calculated by
empinf.
The linear approximations to the bootstrap replicates t of
the statistic of interest.
The control estimate of bias using the linear approximation to
t as a control variate.
The control estimate of variance using the linear approximation to
t as a control variate.
The control estimate of the third cumulant using the linear
approximation to t as a control variate.
A matrix with two columns; the first column are the alpha levels
used for the quantiles and the second column gives the corresponding
control estimates of the quantiles using the linear approximation to
t as a control variate.
An output object from smooth.spline describing the
saddlepoint approximation to the bootstrap distribution of the
linear approximation to t. If distn was supplied on
input then this is the same as the input otherwise it is calculated
by a call to saddle.distn.
If bias.adj is FALSE then the linear approximation to
the statistic is found and evaluated at each bootstrap replicate.
Then using the equation T* = Tl*+(T*-Tl*), moment estimates can
be found. For quantile estimation the distribution of the linear
approximation to t is approximated very accurately by
saddlepoint methods, this is then combined with the bootstrap
replicates to approximate the bootstrap distribution of t and
hence to estimate the bootstrap quantiles of t.
Davison, A.C. and Hinkley, D.V. (1997) Bootstrap Methods and Their Application. Cambridge University Press.
Davison, A.C., Hinkley, D.V. and Schechtman, E. (1986) Efficient bootstrap simulation. Biometrika, 73, 555--566.
Efron, B. (1990) More efficient bootstrap computations. Journal of the American Statistical Association, 55, 79--89.
boot, empinf, k3.linear, linear.approx, saddle.distn, smooth.spline, var.linear
# NOT RUN {
# Use of control variates for the variance of the air-conditioning data
mean.fun <- function(d, i)
{ m <- mean(d$hours[i])
n <- nrow(d)
v <- (n-1)*var(d$hours[i])/n^2
c(m, v)
}
air.boot <- boot(aircondit, mean.fun, R = 999)
control(air.boot, index = 2, bias.adj = TRUE)
air.cont <- control(air.boot, index = 2)
# Now let us try the variance on the log scale.
air.cont1 <- control(air.boot, t0 = log(air.boot$t0[2]),
t = log(air.boot$t[, 2]))
# }
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