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AN EMPIRICAL LIKELIHOOD APPROACH FOR NON-GAUSSIAN VECTOR STATIONARY PROCESSES AND ITS APPLICATION TO MINIMUM CONTRAST ESTIMATION

Authors

  • Hiroaki Ogata,

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    • School of Business Administration, Faculty of Urban Liberal Arts, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji-shi, Tokyo, 192-0397, Japan.

  • Masanobu Taniguchi

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    • Department of Applied Mathematics, School of Fundamental Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo, 169-8555, Japan.
      e-mail: taniguchi@waseda.jp


  • Acknowledgments. The authors are grateful to an editor and a referee for their helpful comments, which improved the original version greatly. This work was supported by Grant-in-Aid for Scientific Research (A) (19204009, 21241040) and Grant-in-Aid for Young Scientists (B) (22700291).

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Summary

We develop the empirical likelihood approach for a class of vector-valued, not necessarily Gaussian, stationary processes with unknown parameters. In time series analysis, it is known that the Whittle likelihood is one of the most fundamental tools with which to obtain a good estimator of unknown parameters, and that the score functions are asymptotically normal. Motivated by the Whittle likelihood, we apply the empirical likelihood approach to its derivative with respect to unknown parameters. We also consider the empirical likelihood approach to minimum contrast estimation based on a spectral disparity measure, and apply the approach to the derivative of the spectral disparity.

This paper provides rigorous proofs on the convergence of our two empirical likelihood ratio statistics to sums of gamma distributions. Because the fitted spectral model may be different from the true spectral structure, the results enable us to construct confidence regions for various important time series parameters without assuming specified spectral structures and the Gaussianity of the process.

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