Whittle estimators are important and fundamental in time series estimation.
We apply Whittle estimation to the square transformed ARCH($\infty$) models,
which can be expressed as linear processes.
Whittle estimators for linear processes are known 
to be asymptotically normal with asymptotic variance $V_W=V_2+V_4$, 
where $V_2$ is written in terms of the second-order spectra only,
and $V_4$ includes the fourth-order cumulant spectra.
This note gives a useful and explicit expression of $V_4$, 
and shows that there exists a case of $V_4<0$.
Since $V_2$ can be regarded as the inverse of Fisher information $F^{-1}$ 
in terms of the second-order spectra,
the result implies that there is a case when $V_W<F^{-1}$.
For ARCH models with various innovation distributions, 
we evaluate $V_W$, $V_2$ and $V_4$ numerically.
The numerical studies elucidate some interesting features of the Whittle estimators.