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\name{control}
\alias{control}
\title{
Control Variate Calculations
}
\description{
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.
}
\usage{
control(boot.out, L = NULL, distn = NULL, index = 1, t0 = NULL,
t = NULL, bias.adj = FALSE, alpha = NULL, \dots)
}
\arguments{
\item{boot.out}{
A bootstrap output object returned from \code{boot}. The bootstrap
replicates must have been generated using the usual nonparametric
bootstrap.
}
\item{L}{
The empirical influence values for the statistic of interest. If
\code{L} is not supplied then \code{empinf} is called to calculate
them from \code{boot.out}.
}
\item{distn}{
If present this must be the output from \code{smooth.spline} giving
the distribution function of the linear approximation. This is used
only if \code{bias.adj} is \code{FALSE}. Normally this would be
found using a saddlepoint approximation. If it is not supplied in
that case then it is calculated by \code{saddle.distn}.
}
\item{index}{
The index of the variable of interest in the output of
\code{boot.out$statistic}.
}
\item{t0}{
The observed value of the statistic of interest on the original data
set \code{boot.out$data}. This argument is used only if
\code{bias.adj} is \code{FALSE}. The input value is ignored if
\code{t} is not also supplied. The default value is is
\code{boot.out$t0[index]}.
}
\item{t}{
The bootstrap replicate values of the statistic of interest. This
argument is used only if \code{bias.adj} is \code{FALSE}. The input
is ignored if \code{t0} is not supplied also. The default value is
\code{boot.out$t[,index]}.
}
\item{bias.adj}{
A logical variable which if \code{TRUE} specifies that the adjusted
bias estimate using post-simulation balance is all that is required.
If \code{bias.adj} is \code{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.
}
\item{alpha}{
The alpha levels for the required quantiles if \code{bias.adj} is
\code{FALSE}.
}
\item{\dots}{
Any additional arguments that \code{boot.out$statistic} requires.
These are passed unchanged every time \code{boot.out$statistic} is
called. \code{boot.out$statistic} is called once if \code{bias.adj}
is \code{TRUE}, otherwise it may be called by \code{empinf} for
empirical influence calculations if \code{L} is not supplied.
}
}
\value{
If \code{bias.adj} is \code{TRUE} then the returned value is the
adjusted bias estimate.
If \code{bias.adj} is \code{FALSE} then the returned value is a list
with the following components
\item{L}{
The empirical influence values used. These are the input values if
supplied, and otherwise they are the values calculated by
\code{empinf}.
}
\item{tL}{
The linear approximations to the bootstrap replicates \code{t} of
the statistic of interest.
}
\item{bias}{
The control estimate of bias using the linear approximation to
\code{t} as a control variate.
}
\item{var}{
The control estimate of variance using the linear approximation to
\code{t} as a control variate.
}
\item{k3}{
The control estimate of the third cumulant using the linear
approximation to \code{t} as a control variate.
}
\item{quantiles}{
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
\code{t} as a control variate.
}
\item{distn}{
An output object from \code{smooth.spline} describing the
saddlepoint approximation to the bootstrap distribution of the
linear approximation to \code{t}. If \code{distn} was supplied on
input then this is the same as the input otherwise it is calculated
by a call to \code{saddle.distn}.
}
}
\details{
If \code{bias.adj} is \code{FALSE} then the linear approximation to
the statistic is found and evaluated at each bootstrap replicate.
Then using the equation \emph{T* = Tl*+(T*-Tl*)}, moment estimates can
be found. For quantile estimation the distribution of the linear
approximation to \code{t} is approximated very accurately by
saddlepoint methods, this is then combined with the bootstrap
replicates to approximate the bootstrap distribution of \code{t} and
hence to estimate the bootstrap quantiles of \code{t}.
}
\references{
Davison, A.C. and Hinkley, D.V. (1997)
\emph{Bootstrap Methods and Their Application}. Cambridge University Press.
Davison, A.C., Hinkley, D.V. and Schechtman, E. (1986) Efficient bootstrap
simulation. \emph{Biometrika}, \bold{73}, 555--566.
Efron, B. (1990) More efficient bootstrap computations.
\emph{Journal of the American Statistical Association}, \bold{55}, 79--89.
}
\seealso{
\code{\link{boot}}, \code{\link{empinf}}, \code{\link{k3.linear}}, \code{\link{linear.approx}}, \code{\link{saddle.distn}}, \code{\link{smooth.spline}}, \code{\link{var.linear}}
}
\examples{
# 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]))
}
\keyword{nonparametric}
|
