# bootstrap

##### Generate Bootstrap Data and Statistics

Generates bootstrap samples from the original data and computes the bootstrap estimate of standard error and bias for a given statistic (if this is given).

- Keywords
- ts

##### Usage

```
bootstrap(x, nb = 1, statistic = NULL, b = NULL,
type = c("stationary","block"), ...)
```

##### Arguments

- x
- a numeric vector or time series giving the original data.
- nb
- the number of bootstrap series to compute.
- statistic
- a function which when applied to a time series returns a vector containing the statistic(s) of interest.
- b
- if
`type`

is`"stationary"`

, then`b`

is the mean block length. If`type`

is`"block"`

, then`b`

is the fixed block length. - type
- the type of bootstrap to generate the simulated time
series. The possible input values are
`"stationary"`

(stationary bootstrap with mean block length`b`

) and`"block"`

(moving blocks bootstrap with block len - ...
- additional arguments for
`statistic`

which are passed unchanged each time`statistic`

is called.

##### Details

If `type`

is `"stationary"`

, then the stationary
bootstrap scheme with mean block length `b`

generates the
simulated series. If `type`

is `"block"`

, then the moving
blocks bootstrap with block length `b`

generates the
simulated series.

For consistency, the (mean) block length `b`

should grow with
`n`

as `const * n^(1/3)`

, where `n`

is the number of
observations in `x`

. Note, that in general `const`

depends
on intricate properties of the process `x`

. The default value for
`const`

has been determined by a Monte Carlo simulation using a
Gaussian AR(1) (AR(1)-parameter of 0.5, 500 observations) process for
`x`

. It is chosen such that the mean square error for
the bootstrap estimate of the variance of the empirical mean is
minimized.

Missing values are not allowed.

There is a special print method for objects of class
`"resample.statistic"`

which by default uses
`max(3, getOption("digits") - 3)`

digits to format real numbers.

##### Value

- If
`statistic`

is`NULL`

, then it returns a matrix or time series with`nb`

columns and`length(x)`

rows containing the bootstrap data. Each column contains one bootstrap sample.If

`statistic`

is given, then a list of class`"resample.statistic"`

with the following elements is returned: statistic the results of applying `statistic`

to each of the simulated time series.orig.statistic the results of applying `statistic`

to the original series.bias the bias of the statistics computed as in a bootstrap setup. se the standard error of the statistics computed as in a bootstrap setup. call the original call of `bootstrap`

.

##### References

H. R. Kuensch (1989):
The Jackknife and the Bootstrap for General Stationary Observations.
*The Annals of Statistics* **17**, 1217--1241.

D. N. Politis and J. P. Romano (1994):
The Stationary Bootstrap.
*Journal of the American Statistical Association* **89**,
1303--1313.

##### See Also

##### Examples

```
n <- 500 # Generate AR(1) process
e <- rnorm(n)
x <- double(n)
x[1] <- rnorm(1)
for(i in 2:n) {
x[i] <- 0.5 * x[i-1] + e[i]
}
x <- ts(x)
theta <- function(x) # Autocorrelations up to lag 10
return(acf(x, plot=FALSE)$acf[2:11])
bootstrap(x, nb=50, statistic=theta)
```

*Documentation reproduced from package tseries, version 0.8-0, License: GPL (see file COPYING), except for ./src/muin2ser.f and ./misc which are free for non-commercial purposes. See file README for details.*