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 |
g |
A numeric function taking |
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
|
kernel |
A character string specifying the smoothing kernel. The possible choices are:
|
bandwidth |
The kernel bandwidth smoothing parameter. If missing, the bandwidth
is automaticly computed by |
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
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)