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\name{tsbootstrap}
\alias{tsbootstrap}
\alias{print.resample.statistic}
\title{Bootstrap for General Stationary Data}
\description{
\code{tsbootstrap} generates bootstrap samples for general stationary
data and computes the bootstrap estimate of standard error and bias
if a statistic is given.
}
\usage{
tsbootstrap(x, nb = 1, statistic = NULL, m = 1, b = NULL,
type = c("stationary","block"), \dots)
}
\arguments{
\item{x}{a numeric vector or time series giving the original data.}
\item{nb}{the number of bootstrap series to compute.}
\item{statistic}{a function which when applied to a time series
returns a vector containing the statistic(s) of interest.}
\item{m}{the length of the basic blocks in the block of blocks
bootstrap.}
\item{b}{if \code{type} is \code{"stationary"}, then \code{b} is the
mean block length. If \code{type} is \code{"block"}, then \code{b}
is the fixed block length.}
\item{type}{the type of bootstrap to generate the simulated time
series. The possible input values are \code{"stationary"}
(stationary bootstrap with mean block length \code{b}) and
\code{"block"} (blockwise bootstrap with block length
\code{b}). Default to \code{"stationary"}.}
\item{\dots}{additional arguments for \code{statistic} which are
passed unchanged each time \code{statistic} is called.}
}
\details{
If \code{type} is \code{"stationary"}, then the stationary
bootstrap scheme with mean block length \code{b} according to Politis
and Romano (1994) is computed. For \code{type} equals \code{"block"},
the blockwise bootstrap with block length \code{b} according to
Kuensch (1989) is used.
If \code{m > 1}, then the block of blocks bootstrap is computed
(see Kuensch, 1989). The basic sampling scheme is the same as for
the case \code{m = 1}, except that the bootstrap is applied to a series
\code{y} containing blocks of length \code{m}, where each block of \code{y} is
defined as \eqn{y[t] = (x[t], \dots, x[t-m+1])}. Therefore, for the block
of blocks bootstrap the first argument of \code{statistic} is given by
a \code{n x m} matrix \code{yb}, where each row of \code{yb} contains one
bootstrapped basic block observation \eqn{y[t]} (\code{n} is the number of
observations in \code{x}).
Note, that for statistics which are functions of the empirical
\code{m}-dimensional marginal (\code{m > 1}) only this procedure
yields asymptotically valid bootstrap estimates. The
case \code{m = 1} may only be used for symmetric statistics (i.e., for
statistics which are invariant under permutations of \code{x}).
\code{\link[boot]{tsboot}} does not implement the block of blocks
bootstrap, and, therefore, the first example in \code{\link[boot]{tsboot}}
yields inconsistent estimates.
For consistency, the (mean) block length \code{b} should grow with
\code{n} at an appropriate rate. If \code{b} is not given, then a
default growth rate of \code{const * n^(1/3)} is used. This rate is
"optimal" under certain conditions (see the references for more
details). However, in general the growth rate depends on the specific
properties of the data generation process. A default value for
\code{const} has been determined by a Monte Carlo simulation using a
Gaussian AR(1) process (AR(1)-parameter of 0.5, 500
observations). \code{const} has been chosen such that the mean square
error for the bootstrap estimate of the variance of the empirical mean
is minimized.
Note, that the computationally intensive parts are fully implemented
in \code{C} which makes \code{tsbootstrap} about 10 to 30 times faster
than \code{\link[boot]{tsboot}}.
Missing values are not allowed.
There is a special print method for objects of class
\code{"resample.statistic"} which by default uses
\code{max(3, getOption("digits") - 3)} digits to format real numbers.
}
\value{
If \code{statistic} is \code{NULL}, then it returns a matrix or time
series with \code{nb} columns and \code{length(x)} rows containing the
bootstrap data. Each column contains one bootstrap sample.
If \code{statistic} is given, then a list of class
\code{"resample.statistic"} with the following elements is returned:
\item{statistic}{the results of applying \code{statistic} to each of
the simulated time series.}
\item{orig.statistic}{the results of applying \code{statistic} to the
original series.}
\item{bias}{the bootstrap estimate of the bias of \code{statistic}.}
\item{se}{the bootstrap estimate of the standard error of \code{statistic}.}
\item{call}{the original call of \code{tsbootstrap}.}
}
\references{
H. R. Kuensch (1989):
The Jackknife and the Bootstrap for General Stationary Observations.
\emph{The Annals of Statistics} \bold{17}, 1217--1241.
\doi{10.1214/aos/1176347265}.
D. N. Politis and J. P. Romano (1994):
The Stationary Bootstrap.
\emph{Journal of the American Statistical Association} \bold{89},
1303--1313.
\doi{10.1080/01621459.1994.10476870}.
}
\author{A. Trapletti}
\seealso{
\code{\link{sample}},
\code{\link{surrogate}},
\code{\link[boot]{tsboot}}
}
\examples{
n <- 500 # Generate AR(1) process
a <- 0.6
e <- rnorm(n+100)
x <- double(n+100)
x[1] <- rnorm(1)
for(i in 2:(n+100)) {
x[i] <- a * x[i-1] + e[i]
}
x <- ts(x[-(1:100)])
tsbootstrap(x, nb=500, statistic=mean)
# Asymptotic formula for the std. error of the mean
sqrt(1/(n*(1-a)^2))
acflag1 <- function(x)
{
xo <- c(x[,1], x[1,2])
xm <- mean(xo)
return(mean((x[,1]-xm)*(x[,2]-xm))/mean((xo-xm)^2))
}
tsbootstrap(x, nb=500, statistic=acflag1, m=2)
# Asymptotic formula for the std. error of the acf at lag one
sqrt(((1+a^2)-2*a^2)/n)
}
\keyword{ts}
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