Geodesic
Hellinger distance estimation for nonregular spectra
Teoriâ veroâtnostej i ee primeneniâ, Tome 69 (2024) no. 1, pp. 188-200 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: Gaussian stationary process, Hellinger distance estimator, nonregular spectra, asymptotically efficient, robust. 
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M. Taniguchi; Y. Xue. Hellinger distance estimation for nonregular spectra. Teoriâ veroâtnostej i ee primeneniâ, Tome 69 (2024) no. 1, pp. 188-200. http://geodesic.mathdoc.fr/item/TVP_2024_69_1_a9/

[1] A. Amimour, K. Belaide, O. Hili, “Minimum Hellinger distance estimates for a periodically time-varying long memory parameter”, C. R. Math. Acad. Sci. Paris, 360 (2022), 1153–1162 ; (2020), 14 pp., arXiv: 2011.10880 | DOI | MR | Zbl

[2] T. Anderson, Statisticheskii analiz vremennykh ryadov, Mir, M., 1976, 756 pp.; T. W. Anderson, Jr., The statistical analysis of time series, Wiley Ser. Probab. Stat., John Wiley Sons, Inc., New York–London–Sydney, 1971, xiv+704 pp. | DOI | MR | Zbl

[3] R. Beran, “Minimum Hellinger distance estimates for parametric models”, Ann. Statist., 5:3 (1977), 445–463 | DOI | MR | Zbl

[4] P. Billingsli, Skhodimost veroyatnostnykh mer, Nauka, M., 1977, 351 pp. ; P. Billingsley, Convergence of probability measures, John Wiley Sons, Inc., New York–London–Sydney, 1968, xii+253 pp. | MR | MR | Zbl

[5] K. O. Dzhaparidze, Otsenka parametrov i proverka gipotez v spektralnom analize statsionarnykh vremennykh ryadov, Tbilisskii gos. un-t, Tbilisi, 1981, 264 pp. ; K. Dzhaparidze, Parameter estimation and hypothesis testing in spectral analysis of stationary time series, Springer Ser. Statist., Springer-Verlag, New York, 1986, vi+324 pp. | MR | Zbl | DOI | MR | Zbl

[6] M. S. Ermakov, “Asimptoticheski effektivnye statisticheskie vyvody dlya veroyatnostei umerennykh uklonenii”, Teoriya veroyatn. i ee primen., 48:4 (2003), 676–700 ; M. S. Ermakov, “On asymptotically efficient statistical inference for moderate deviation probabilities”, Theory Probab. Appl., 48:4 (2004), 622–641 | DOI | MR | Zbl | DOI

[7] F. R. Hampel, “The influence curve and its role in robust estimation”, J. Amer. Statist. Assoc., 69:346 (1974), 383–393 | DOI | MR | Zbl

[8] E. Khennan, Mnogomernye vremennye ryady, Mir, M., 1974, 576 pp. ; E. J. Hannan, Multiple time series, John Wiley Sons, Inc., New York–London–Sydney, 1970, xi+536 pp. | Zbl | DOI | MR | Zbl

[9] I. A. Ibragimov, R. Z. Khasminskii, Asimptoticheskaya teoriya otsenivaniya, Nauka, M., 1979, 527 pp. ; I. A. Ibragimov, R. Z. Has'minskii, Statistical estimation. Asymptotic theory, Appl. Math., 16, Springer-Verlag, New York–Berlin, 1981, vii+403 pp. | MR | Zbl | DOI | MR | Zbl

[10] L. H. Koopmans, The spectral analysis of time series, Probab. Math. Statist., 22, Academic Press, New York–London, 1974, xiv+366 pp. | MR | Zbl

[11] B. G. Lindsay, “Efficiency versus robustness: the case for minimum Hellinger distance and related methods”, Ann. Statist., 22:2 (1994), 1081–1114 | DOI | MR | Zbl

[12] D. G. Simpson, “Minimum Hellinger distance estimation for the analysis of count data”, J. Amer. Statist. Assoc., 82:399 (1987), 802–807 | DOI | MR | Zbl

[13] M. Taniguchi, “On estimation of parameters of Gaussian stationary processes”, J. Appl. Probab., 16:3 (1979), 575–591 | DOI | MR | Zbl

[14] M. Taniguchi, “An estimation procedure of parameters of a certain spectral density model”, J. Roy. Statist. Soc. Ser. B, 43:1 (1981), 34–40 | MR | Zbl

[15] M. Taniguchi, “Minimum contrast estimation for spectral densities of stationary processes”, J. Roy. Statist. Soc. Ser. B, 49:3 (1987), 315–325 | MR | Zbl

[16] M. Taniguchi, “Non-regular estimation theory for piecewise continuous spectral densities”, Stochastic Process. Appl., 118:2 (2008), 153–170 | DOI | MR | Zbl

[17] Yujie Xue, M. Taniguchi, “Local Whittle likelihood approach for generalized divergence”, Scand. J. Stat., 47:1 (2020), 182–195 | DOI | MR | Zbl