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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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##### References:
 [1] Bickel, P.J., On adaptive estimation, Ann. statist., 10, 647-671, (1982) · Zbl 0489.62033 [2] Brillinger, D.R., Time seriesdata analysis and theory, (1981), Holden-Day San Francisco [3] Dahlhaus, R., On the kullback – leibler information divergence of locally stationary processes, Stochatic process. appl., 62, 139-168, (1996) · Zbl 0849.60032 [4] Dahlhaus, R., Maximum likelihood estimation and model selection for locally stationary processes, J. nonparametr. statist., 6, 171-191, (1996) · Zbl 0879.62025 [5] Dahlhaus, R., Fitting time series models to nonstationary process, Ann. statist., 25, 1-37, (1997) · Zbl 0871.62080 [6] Dahlhaus, R., A likelihood approximation for locally stationary processes, Ann. statist., 28, 1762-1794, (2000) · Zbl 1010.62078 [7] Garel, B.; Hallin, M., Local asymptotic normality of multivariate ARMA processes with a linear trend, Ann. inst. statist. math., 47, 551-579, (1995) · Zbl 0841.62076 [8] Hallin, M.; Taniguchi, M.; Serroukh, A.; Choy, K., Local asymptotic normality for regression models with long-memory disturbance, Ann. statist., 27, 2054-2080, (1999) · Zbl 0957.62077 [9] Kreiss, J.P., On adaptive estimation in stationary ARMA processes, Ann. statist., 15, 112-133, (1987) · Zbl 0616.62042 [10] Kreiss, J.P., Local asymptotic normality for autoregression with infinite order, J. statist. plan. inf., 26, 185-219, (1990) · Zbl 0778.62083 [11] LeCam, L., Asymptotic methods in statistical decision theory, (1986), Springer New York [12] Linton, O., Adaptive estimation in ARCH models, Econom. theory, 9, 539-569, (1993) [13] Sakiyama, K.; Taniguchi, M., Testing composite hypotheses for locally stationary processes, J. time series anal., 24, 483-504, (2003) · Zbl 1036.62067 [14] Sakiyama, K.; Taniguchi, M., Discriminant analysis for locally stationary processes, J. multivariate anal., 90, 282-300, (2004) · Zbl 1050.62066 [15] Stone, C., Adaptive maximum likelihood estimation of a location parameter, Ann. statist., 3, 267-284, (1975) · Zbl 0303.62026 [16] Strasser, H., Mathematical theory of statistics, (1985), W. de Gruyter Berlin [17] Swensen, A.R., The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend, J. multivariate anal., 16, 54-70, (1985) · Zbl 0563.62065 [18] Taniguchi, M.; Kakizawa, Y., Asymptotic theory of statistical inference for time series, (2000), Springer New York · Zbl 0955.62088
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