Yan Liu’s research while affiliated with Waseda University and other places

A new family of circular distributions has been proposed as circular distributions generated from general spectral densities of time series models. These circular models have shown promising potential for analyzing and interpreting circular data. However, a significant challenge arises in the statistical inference for those circular data due to the loss of explicit forms associated with the normalizing constants in these circular models. We consider the empirical likelihood for data coming from these circular distributions. The proposed empirical likelihood ratio test has a chi-squared limiting distribution. The theoretical results are illustrated by numerical simulations and real data analysis.

This paper studies the second-order asymptotics of maximum likelihood estimator (MLE) and Whittle estimator under $\varepsilon$-contaminated model for Gaussian stationary processes. We evaluate the robustness of MLE and Whittle estimator based on the second-order Edgeworth expansion with an $\varepsilon$-disturbance spectral density. The measures of second-order robustness of MLE and Whittle estimator are investigated for concrete models with numerical study. The findings show that the MLE of Gaussian autoregressive process is robust in second-order term to a disturbance in spectral density under the middle level of spectral frequency, while it is more sensitive to a contamination under a too low frequency spectral mass. The Whittle estimator is robust to a moving average contamination when the Gaussian autoregressive process is not near unit root case, while it is sensitive to the disturbance under a nonregular situation in the case of near unit root.

We consider the sparse principal component analysis for high‐dimensional stationary processes. The standard principal component analysis performs poorly when the dimension of the process is large. We establish oracle inequalities for penalized principal component estimators for the large class of processes including heavy‐tailed time series. The rate of convergence of the estimators is established. We also elucidate the theoretical rate for choosing the tuning parameter in penalized estimators. The performance of the sparse principal component analysis is demonstrated by numerical simulations. The utility of the sparse principal component analysis for time series data is exemplified by the application to average temperature data.

We consider the problem of testing for the existence of fixed effects and random effects in one-way models, where the groups are correlated and the disturbances are dependent. The classical F-statistic in the analysis of variance is not asymptotically distribution-free in this setting. To overcome this problem, we propose a new test statistic for this problem without any distributional assumptions, so that the test statistic is asymptotically distribution-free. The proposed test statistic takes the form of a natural extension of the classical F-statistic in the sense of distribution-freeness. The new tests are shown to be asymptotically size α and consistent. The nontrivial power under local alternatives is also elucidated. The theoretical results are justified by numerical simulations for the model with disturbances from linear time series with innovations of symmetric random variables, heavy-tailed variables, and skewed variables, and furthermore from GARCH models. The proposed test is applied to log-returns for stock prices and uncovers random effects in sectors. Supplementary information: The online version contains supplementary material available at 10.1007/s11749-022-00828-9.

We consider the minimax estimation of time series in view of higher order asymptotic theory. Under the framework of Bayesian inference, we focus on the Bayes estimator and the Bayesian Whittle estimator for parameter estimation. It is shown that these estimators are minimax with respect to the Bayes risk of higher order bias appeared in their asymptotic expansion. The minimax problem in the boundary issue with parameter on the boundary of parameter space is also discussed. Our theoretical discovery is justified by simulation studies even when the sample size is small.

This paper deals with shrinkage estimators for the mean of p-dimensional Gaussian stationary processes. The shrinkage estimators are expressed by a shrinkage function, including the sample mean and the James–Stein estimator as special cases. We evaluate the mean squared error of such shrinkage estimators from the true mean of a p-dimensional Gaussian vector stationary process with $p \ge 3$. A sufficient condition for shrinkage estimators improving the mean squared error upon the sample mean is given in terms of the shrinkage function and the spectral density matrix. In addition, a shrinkage estimator, providing the most significant improvement to the sample mean, is proposed as a theoretical result. The remarkable performance of the proposed shrinkage estimator, compared with the sample mean and the James–Stein estimator, is illustrated by a thorough numerical simulation. A real data analysis also witnesses the applicability of the proposed estimator for multivariate time series.

We consider the sparse principal component analysis for high-dimensional stationary processes. The standard principal component analysis performs poorly when the dimension of the process is large. We establish the oracle inequalities for penalized principal component estimators for the processes including heavy-tailed time series. The consistency of the estimators is established even when the dimension grows at the exponential rate of the sample size. We also elucidate the theoretical rate for choosing the tuning parameter in penalized estimators. The performance of the sparse principal component analysis is demonstrated by numerical simulations. The utility of the sparse principal component analysis for time series data is exemplified by the application to average temperature data.

The minimax principle is very important for all the fields of statistical science. The minimax approach is to choose an estimator which protects against the largest risk possible. In this paper we show that the Whittle estimator becomes a minimax estimator for the prediction error loss. It is shown that the Whittle estimator is a Bayes estimator for Jeffreys’ prior. Because the minimax approach is very immature in time series analysis, the result shows another advantage of the Whittle estimator.

Granger causality has been employed to investigate causality relations between components of stationary multiple time series. Here, we generalize this concept by developing statistical inference for local Granger causality for multivariate locally stationary processes. Thus, our proposed local Granger causality approach captures time-evolving causality relationships in nonstationary processes. The proposed local Granger causality is well represented in the frequency domain and estimated based on the parametric time-varying spectral density matrix using the local Whittle likelihood. Under regularity conditions, we demonstrate that the estimators converge weakly to a Gaussian process. Additionally, the test statistic for the local Granger causality is shown to be asymptotically distributed as a quadratic form of a multivariate normal distribution. The finite sample performance is confirmed with several simulation studies for multivariate time-varying VAR models. For practical demonstration, the proposed local Granger causality method uncovered new functional connectivity relationships between channels in brain signals. Moreover, the method was able to identify structural changes of Granger causality in financial data.