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\iteman{io-port 05875162}
\itemau{Ahn, V.V.; Leonenko, N.N.; Sakhno, L.M.}
\itemti{Evaluation of bias in higher-order spectral estimation.}
\itemso{Teor. Jmovirn. Mat. Stat. 80, 1-14 (2009) and Theory Probab. Math. Stat. 80, 1-14 (2010).}
\itemab
Let $X(t),\ t\in I,$ be a real-valued measurable, strictly stationary zero-mean random field with spectral densities $f_k(\lambda_1,\dots,\lambda_{k-1})\in L_1(\mathbb S^{k-1})$ of order $k=2,3,\dots$, where $\mathbb S=\mathbb R^d$ or $\mathbb S= (-\pi,\pi]^d$ for the continuous-parameter or discrete-parameter cases, respectively. $I\subset\mathbb R^d$ with measure $\nu(\cdot)$. The authors deal with estimation of integrals of spectral functionals $$ J_k(\varphi_k)=\int_{\mathbb S^{k-1}}\varphi_k(\lambda)f_k(\lambda)d\lambda $$ for appropriate functions $\varphi_k(\lambda)$ based on the observations of the field $X(t)$ over the domain $D_T = [-T,T]^d\subset I$. To estimate the value of the functional $J_k(\varphi_k)$ they use the empirical spectral functional of $k$-th order $$ J_{k,T}(\varphi_k)=\int_{\mathbb S^{k-1}}\varphi_k(\lambda) I_{k,T}^{h}(\lambda)d\lambda, $$ where $I_{k,T}^{h}(\lambda)$ is the tapered periodograms of $k$-th order $$I_{k,T}^{h}(\lambda_1\dots,\lambda_{k-1})=\left( (2\pi)^{(k-1)d}H_{k,T}(0)\right)^{-1} \prod_{i=1}^kd^h_T(\lambda_i),$$ where $d^h_T(\lambda)$ and $H_{k,T}(\lambda)$ are the finite Fourier transforms $$d^h_T(\lambda)=\int h_T(t)X(t)e^{-i(\lambda,t)}\nu(dt),\quad H_{k,T}(\lambda)=\int (h_T(t))^kX(t)e^{-i(\lambda,t)}\nu(dt),$$ related to the tapered data $\{h_T (t)X(t), t\in D_T\}$. The main results of the paper are the evaluation of the bias of the proposed estimators of the spectral functionals based on tapered periodograms and conditions which guarantee an appropriate rate of convergence of the bias to zero, especially for spatial data ($d\geq2$) when the bias can be subject to edge effects.
\itemrv{Rostyslav E. Yamnenko (Ky\"{\i}v)}
\itemcc{}
\itemut{higher-order spectral density; minimum contrast estimation}
\itemli{doi:10.1090/S0094-9000-2010-00790-3}
\end