HELLINGER DISTANCE ESTIMATION FOR NONREGULAR SPECTRA M. Taniguchi, Y. Xue Theory of Probability and Its Applications, 2024 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.
INTRODUCTION OF GENERAL DISTRIBUTIONS ON SPHERE AND TORUS IN VIEW OF TIME SERIES SPECTRA Taniguchi, Masanobu, Xue, Yujie Statistica, 2023 There are various fields where observations are taken on directions in three dimensions, e.g., sphere and torus. Herewe will introduce a very general family of distributions on sphere and torus by use of time series spectra, which includes a lot of proposed classical one as special cases. Because time series spectra can be described by a lot of famous parametric models, e.g., AR, ARMA etc., we can develop the systematic model selection in this field by use of AIC, BIC, etc. Applications are very wide.
Modified LASSO estimators for time series regression models with dependent disturbances Yujie Xue, Masanobu Taniguchi Statistical Methods and Applications, 2020 This paper applies the modified least absolute shrinkage and selection operator (LASSO) to the regression model with dependent disturbances, especially, long-memory disturbances. Assuming the norm of different column in the regression matrix may have different order of observation length n , we introduce a modified LASSO estimator where the tuning parameter $$\\lambda$$ λ is not a scalar but vector. When the dimension of parameters is fixed, we derive the asymptotic distribution of the modified LASSO estimators under certain regularity condition. When the dimension of parameters increases with respect to n , the consistency on the probability of the correct selection of penalty parameters is shown under certain regularity conditions. Some simulation studies are examined.
Local Whittle likelihood approach for generalized divergence Yujie Xue, Masanobu Taniguchi Scandinavian Journal of Statistics, 2020 There are many approaches in the estimation of spectral density. With regard to parametric approaches, different divergences are proposed in fitting a certain parametric family of spectral densities. Moreover, nonparametric approaches are also quite common considering the situation when we cannot specify the model of process. In this paper, we develop a local Whittle likelihood approach based on a general score function, with some special cases of which, the approach applies to more applications. This paper highlights the effective asymptotics of our general local Whittle estimator, and presents a comparison with other estimators. Additionally, for a special case, we construct the one‐step ahead predictor based on the form of the score function. Subsequently, we show that it has a smaller prediction error than the classical exponentially weighted linear predictor. The provided numerical studies show some interesting features of our local Whittle estimator.
Robust Linear Interpolation and Extrapolation of Stationary Time Series in Lp Yan Liu, Yujie Xue, Masanobu Taniguchi Journal of Time Series Analysis, 2020 To deal with uncertainty of the spectral distribution, we consider minimax interpolation and extrapolation problems in Lp for stationary processes. The interpolation and extrapolation problems can be regarded as a linear approximation problem on the unit disk in the complex plane. Although the robust one‐step‐ahead predictor and interpolator has already been considered separately in the previous literature, we give two conditions for the uncertainty class to find the minimax interpolator and extrapolator in the general framework from both the point of view of the observation set and the point of view of evaluation on the interpolation and extrapolation error under the Lp‐norm. We show that there exists a minimax interpolator and extrapolator for the class of spectral densities ε‐contaminated by unknown spectral densities under our conditions. When the uncertainty class contains spectral distribution functions which are not absolutely continuous to the Lebesgue measure, we show that there exists an approximate interpolator and extrapolator in Lp such that its maximal interpolation and extrapolation error is arbitrarily close to the minimax error when the spectral distributions have densities. Our results are applicable to the stationary harmonizable stable processes.
