aidedboot {boodd}R Documentation

Aided Frequency Bootstrap

Description

The Aided Frequency Bootstrap (AFB) is a variation of the Frequency Domain Bootstrap (FDB). The idea is to fit a sieve AR(p) model and to generate the corresponding bootstrapped time series (by resampling centered residuals) with periodogram I^{\ast}_{AR}. Then the we estimate the quotient of the two spectral densities q(\omega)=\frac{f(\omega)}{f_{AR}(\omega)} and generate bootstrap periodogram by multiplying I^{\ast}_{AR} by this quantity q(\omega)=\frac{f(\omega)}{f_{AR}(\omega)}.

Usage

aidedboot(x, XI, g, B, order = NULL, kernel = "normal", bandwidth)

Arguments

x

A numeric vector representing a time series.

XI

A list of functions defined on the interval [0, \pi].

g

A numeric function taking length(XI) as arguments.

B

A positive integer; the number of bootstrap samples.

order

The order of the autoregressive sieve process (integer). If not specified, it is set by default as

\left \lfloor 4 * (length(x) \log(length(x)))^{1/4}\right \rfloor.

kernel

A character string specifying the smoothing kernel. The possible choices are: "normal", "box" and "epanechnikov".

bandwidth

The kernel bandwidth smoothing parameter. If missing, the bandwidth is automaticly computed by bandw1.

Details

The idea underlying the Aided Frequency Bootstrap is importance sampling. It was introduced by Kreiss and Paparoditis (2003) and allows to better mimic the asymptotic covariance structure of the periodogram in the bootstrap world. Kreiss and Paparoditis (2003) considered a spectral density which is easy to estimate (typically based on a sieve AR representation of the time series), say f_{AR}(\omega). The argument x is supposed to be a sample of a real valued zero-mean stationary time series.

The autoregressive sieve process of order l=l(n) is modelled as

X_{t}=\sum_{k=1}^{l}\psi_{k}X_{t-k}+\epsilon_{t}

with E(\epsilon_{t})=0, Var(\epsilon_{t})=\sigma^{2}(l).

We estimate functionals of the spectral density T(f) of the form

T(f)=g(A(\xi,f))

where g is a third order differentiable function,

A(\xi,f)=\left( \int_{0}^{\pi}\xi_{1}(\omega)f(\omega)d\omega,\int_{0}^{\pi }\xi_{2}(\omega)f(\omega)d\omega,\dots,\int_{0}^{\pi}\xi_{p}(\omega )f(\omega)d\omega\right)

and

\xi=(\xi_{1},\dots,\xi_{p}): [0,\pi] \rightarrow R^p.

If the order argument is not specified, its default value is l=\left \lfloor (4*(n\log(n))^{1/4})\right \rfloor, where n is the length of x.

The kernel argument has the same meaning as in the freqboot function.

Value

aidedboot returns an object of class boodd (see class.boodd).

References

Bertail, P. and Dudek, A. (2025). Bootstrap for Dependent Data, with an R package (by Bernard Desgraupes and Karolina Marek)- submitted.

Kreiss, J.-P. and Paparoditis, E. (2003). Autoregressive aided periodogram bootstrap for time series. Ann. Stat. 31 1923–1955.

See Also

freqboot.

Examples

n <- 200
x <- arima.sim(list(order=c(4,0,0),ar=c(0.7,0.4,-0.3,-0.1)),n=n)
B <- 299
one <- function(x) {1}
XI <- list(cos,one)
g <- function(x,y) {return(x/y)}
ord <- 2*floor(n^(1/3))
boo <- aidedboot(x,XI,g,B,order=ord) 
plot(boo)

[Package boodd version 0.1 Index]