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Inference of weighted \(V\)-statistics for nonstationary time series and its applications. (English) Zbl 1306.62205

The article deals with statistical inference in piece-wise locally stationary (PLS), potentially nonlinear, time series models. Central and non-central limit theorems are proved for weighted \(V\)-statistics of PLS time series data, both in the nondegenerate and in the degenerate case. To this end, under regularity assumptions, a Fourier integral representation of the \(V\)-statistic kernel is employed. This leads to a mathematically tractable structure of the degenerate and the nondegenerate part appearing in the Hoeffding-based decomposition of the \(V\)-statistic.
Applications of the main results comprise asymptotic distributional theory for quadratic forms of PLS processes, point-wise central limit theorems for certain nonparametric estimators of time series parameter functions, and asymptotic distributional results in the context of a spectral analysis based on the (empirical) periodogram of the PLS time series.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G05 Nonparametric estimation
62E20 Asymptotic distribution theory in statistics
60F05 Central limit and other weak theorems
62M15 Inference from stochastic processes and spectral analysis
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References:

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