Hellinger Distance Estimation for Non-regular Spectra

For Gaussian stationary process, 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 non-regular spectra. For regular spectra, we introduce the minimum Hellinger distance estimator \(\hat{\theta }=\arg \min _\theta T(f_\theta ,\hat{g}_n)\) , where \(\hat{g}_n\) is a nonparametric spectral density estimator. We show that \(\hat{\theta }\) is asymptotically efficient and more robust than the Whittle estimator. Brief numerical studies are provided. This chapter is mainly based on Taniguchi and Xue (2024).