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LAN theorem for non-Gaussian locally stationary processes and its applications. (English) Zbl 1077.62070
Summary: For a class of locally stationary processes introduced by R. Dahlhaus [J. Nonparametric Stat. 6, No. 2–3, 171–191 (1996; Zbl 0879.62025); Stochastic Processes Appl. 62, No. 1, 139–168 (1996; Zbl 0849.60032); Ann. Stat. 25, No. 1, 1–37 (1997; Zbl 0871.62080)], we derive the LAN theorem under non-Gaussianity and apply the results to asymptotically optimal estimation and testing problems. For a class \(\mathcal F\) of statistics, which includes important statistics, we derive the asymptotic distributions of statistics in \(\mathcal F\) under contiguous alternatives of the unknown parameters. Because the asymptotics depend on the non-Gaussianity of the process, we discuss non-Gaussian robustness. An interesting feature of effects of non-Gaussianity is elucidated in terms of LAN. Furthermore, the LAN theorem is applied to adaptive estimation when the innovation density is unknown.

MSC:
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G10 Nonparametric hypothesis testing
62F12 Asymptotic properties of parametric estimators
62E20 Asymptotic distribution theory in statistics
62F35 Robustness and adaptive procedures (parametric inference)
62F05 Asymptotic properties of parametric tests
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