# bootstrap

##### Generate Bootstrap Data and Statistics

Generates `nb`

bootstrap samples from the original data
`x`

and computes the bootstrap estimate of standard error and
bias for `statistic`

, if `statistic`

is given.

- Keywords
- ts

##### Usage

```
bootstrap (x, nb = 1, statistic = NULL, b = NULL, type =
c("stationary","block"), ...)
print (obj, digits = max(3,.Options$digits-3), ...)
```

##### Arguments

- x
- a numeric vector or time series.
- 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 leng - object
- a list with class
`"resample.statistic"`

. - digits
- the number of digits to format real numbers.
- ...
- either additional arguments for
`statistic`

which are passed unchanged each time`statistic`

is called (`bootstrap`

), or additional arguments for`print`

(`print.resample.statistic`

).

##### 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.

##### 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. *J. Amer. Statist. Assoc.* **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 0.7-4, License: GPL (see file COPYING), except for ./src/muin2ser.f and ./misc which are free for non-commercial purposes. See file README for details.*