## Some asymptotic properties of a kernel spectrum estimate with different multitapers.(English)Zbl 1324.62055

Summary: Let $$X(t)$$, $$t=0,\pm 1,\dots$$, be a zero mean real-valued stationary time series with spectrum $$f_{XX}(\lambda)$$, $$-\pi\leq\lambda\leq\pi$$. Given the realization $$X(1),X(2),\dots,X(N)$$, we construct $$L$$ different multitapered periodograms $$I_{XX}^{(mt)_{j}}(\lambda)$$, $$j=1,2,\dots,L$$, on non-overlapped and overlapped segments $$X^{(j)}(t)$$, $$1\leq t<N$$. Also, we give asymptotic expressions of the mean and variance of the average of these different multitapered periodograms. We obtain an estimate of $$f_{XX}(\lambda)$$ via $$I_{XX}^{(mt)_{j}}(\lambda)$$ and different kernels $$W_{\beta}^{(j)}(\alpha)$$, $$j=1,2,\dots,L$$; $$-\pi<\alpha\leq\pi$$; $$\beta$$ is a bandwidth. We find asymptotic expressions of the first and second-order moments of this estimate. Moreover, we propose a choice of the considered bandwidth. An asymptotic expression of the integrated relative mean squared error (IMSE) of the estimate is formulated.

### MSC:

 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62M15 Inference from stochastic processes and spectral analysis 62G07 Density estimation 62G20 Asymptotic properties of nonparametric inference
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