tseries (version 0.8-3)

bootstrap: Bootstrap for General Stationary Data

Description

bootstrap generates bootstrap samples for general stationary data and computes the bootstrap estimate of standard error and bias if a statistic is given.

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" (blockwise bootstrap with block length

...

additional arguments for statistic which are passed unchanged each time statistic is called.

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 bootstrap estimate of the bias of statistic.
• se
  the bootstrap estimate of the standard error of statistic.
• call
  the original call of bootstrap.

Details

If type is "stationary", then the stationary bootstrap scheme with mean block length b according to Politis and Romano (1994) is used to generate the simulated series. For type equals "block", the blockwise bootstrap with block length b according to Kuensch (1989) is used.

Note, that this bootstrap procedure does not implement the block of blocks bootstrap. Hence, it may only be used for symmetric statistics, i.e., for statistics which are invariant under permutations of x. For consistency, the (mean) block length b should grow with n at an appropriate rate. If b is not given, then a default growth rate of const * n^(1/3) is used (n is the number of observations in x). 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 process which has generated x. Therefore, a default value for const has been determined by a Monte Carlo simulation using a Gaussian AR(1) process (AR(1)-parameter of 0.5, 500 observations). const has been 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.

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

sample, surrogate

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)])

bootstrap(x, nb=500, statistic=mean)  # ok 
bootstrap(x, nb=500, statistic=mean, type="block", b=1)  # wrong here

# std. error for mean should approximately be
sqrt(1/(n*(1-a)^2))

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