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Local asymptotic normality property for fractional Gaussian noise under high-frequency observations. (English) Zbl 1411.62045

Local Asymptotic Normality (LAN) is proven for fractional Gaussian noise under high-frequency observations. The idea is the use of nondiagonal rate matrices. The self-similarity property is the key for this analysis. The paper is well-organized with correct mathematical formulation of the definitions and theorems, using very interesting principles.

MSC:

62F05 Asymptotic properties of parametric tests
62F12 Asymptotic properties of parametric estimators
60G22 Fractional processes, including fractional Brownian motion
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References:

[1] Aït-Sahalia, Y. and Jacod, J. (2008). Fisher’s information for discretely sampled Lévy processes. Econometrica76 727–761. · Zbl 1144.62070
[2] Berzin, C. and León, J. R. (2008). Estimation in models driven by fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat.44 191–213.
[3] Choi, S., Hall, W. J. and Schick, A. (1996). Asymptotically uniformly most powerful tests in parametric and semiparametric models. Ann. Statist.24 841–861. · Zbl 0860.62020
[4] Clément, E. and Gloter, A. (2015). Local asymptotic mixed normality property for discretely observed stochastic differential equations driven by stable Lévy processes. Stochastic Process. Appl.125 2316–2352. · Zbl 1312.60075
[5] Coeurjolly, J.-F. and Istas, J. (2001). Cramèr–Rao bounds for fractional Brownian motions. Statist. Probab. Lett.53 435–447. · Zbl 1092.62574
[6] Cohen, S., Gamboa, F., Lacaux, C. and Loubes, J.-M. (2013). LAN property for some fractional type Brownian motion. ALEA Lat. Am. J. Probab. Math. Stat.10 91–106. · Zbl 1281.62061
[7] Dahlhaus, R. (1989). Efficient parameter estimation for self-similar processes. Ann. Statist.17 1749–1766. · Zbl 0703.62091
[8] Dahlhaus, R. (2006). Correction: “Efficient parameter estimation for self-similar processes” [Ann. Statist. 17 (1989), no. 4, 1749–1766; MR1026311]. Ann. Statist.34 1045–1047. · Zbl 0703.62091
[9] Gloter, A. and Jacod, J. (2001). Diffusions with measurement errors. I. Local asymptotic normality. ESAIM Probab. Stat.5 225–242. · Zbl 1008.60089
[10] Gloter, A. and Jacod, J. (2001). Diffusions with measurement errors. II. Optimal estimators. ESAIM Probab. Stat.5 243–260. · Zbl 1009.60065
[11] Gobet, E. (2001). Local asymptotic mixed normality property for elliptic diffusion: A Malliavin calculus approach. Bernoulli7 899–912. · Zbl 1003.60057
[12] Gobet, E. (2002). LAN property for ergodic diffusions with discrete observations. Ann. Inst. Henri Poincaré Probab. Stat.38 711–737. · Zbl 1018.60076
[13] Ibragimov, I. A. and Has’minskiĭ, R. Z. (1981). Statistical Estimation: Asymptotic Theory. Applications of Mathematics16. Springer, New York. Translated from the Russian by Samuel Kotz.
[14] Istas, J. and Lang, G. (1997). Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann. Inst. Henri Poincaré Probab. Stat.33 407–436. · Zbl 0882.60032
[15] Kawai, R. (2013). Fisher information for fractional Brownian motion under high-frequency discrete sampling. Comm. Statist. Theory Methods42 1628–1636. · Zbl 1411.60059
[16] Kawai, R. (2013). Local asymptotic normality property for Ornstein–Uhlenbeck processes with jumps under discrete sampling. J. Theoret. Probab.26 932–967. · Zbl 1281.62175
[17] Kawai, R. and Masuda, H. (2013). Local asymptotic normality for normal inverse Gaussian Lévy processes with high-frequency sampling. ESAIM Probab. Stat.17 13–32. · Zbl 1333.60095
[18] Kutoyants, Y. A. (2004). Statistical Inference for Ergodic Diffusion Processes. Springer, London. · Zbl 1038.62073
[19] Le Cam, L. (1972). Limits of experiments. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. I: Theory of Statistics 245–261. Univ. California Press, Berkeley, CA.
[20] Lieberman, O., Rosemarin, R. and Rousseau, J. (2012). Asymptotic theory for maximum likelihood estimation of the memory parameter in stationary Gaussian processes. Econometric Theory28 457–470. · Zbl 1298.62044
[21] Masuda, H. (2009). Joint estimation of discretely observed stable Lévy processes with symmetric Lévy density. J. Japan Statist. Soc.39 49–75.
[22] Reiß, M. (2011). Asymptotic equivalence for inference on the volatility from noisy observations. Ann. Statist.39 772–802. · Zbl 1215.62113
[23] Roussas, G. G. (1972). Contiguity of Probability Measures: Some Applications in Statistics. Cambridge Tracts in Mathematics and Mathematical Physics63. Cambridge Univ. Press, London. · Zbl 0265.60003
[24] Taniguchi, M. and Kakizawa, Y. (2000). Asymptotic Theory of Statistical Inference for Time Series. Springer, New York. · Zbl 0955.62088
[25] van der Vaart, A.
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