  Fitting parametric models by optimizing frequency domain objective functions
is an attractive approach of parameter estimation in time series analysis.
Whittle estimators are a prominent example in this context. Under weak
conditions and the (realistic) assumption that the true spectral density of the
underlying process does not necessarily belong to the parametric class of
spectral densities fitted, the distribution of Whittle estimators typically
depends on difficult to estimate characteristics of the underlying process.
This makes the implementation of asymptotic results for the construction of
confidence intervals or for assessing the variability of estimators, difficult
in practice. This paper proposes a frequency domain bootstrap method to
estimate the distribution of Whittle estimators which is asymptotically valid
under assumptions that not only allow for (possible) model misspecification but
also for weak dependence conditions which are satisfied by a wide range of
stationary stochastic processes. Adaptions of the bootstrap procedure developed
to incorporate different modifications of Whittle estimators proposed in the
literature, like for instance, tapered, de-biased or boundary extended Whittle
estimators, are also considered. Simulations demonstrate the capabilities of
the bootstrap method proposed and its good finite sample performance. A
real-life data analysis also is presented.
