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

Teamah, A. A. M.; Bakouch, H. S.

Vladikavkaz. Mat. Zh. 9, No. 1, 56-61 (2007).

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

Keywords:

stationary time series; non-overlapped segments; overlapped segments; multitapering; kernel; bandwidth; spectrum estimate