Hellinger distance estimation for nonregular spectra

For Gaussian stationary processes, a time series Hellinger distance $T(f,g)$ for spectra $f$ and $g$ is derived. Evaluating $T(f_\theta,f_{\theta+h})$ of the form $O(h^\alpha)$, we give $1/\alpha$-consistent asymptotics of the maximum likelihood estimator of $\theta$ for nonregular spectra. For regular spectra, we introduce the minimum Hellinger distance estimator $\widehat{\theta}=\operatorname{arg}\min_\theta T(f_\theta,\widehat{g}_n)$, where $\widehat{g}_n$ is a nonparametric spectral density estimator. We show that $\widehat\theta$ is asymptotically efficient and more robust than the Whittle estimator. Brief numerical studies are provided.