Histograms for stationary linear random fields |
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Authors: | Michel Carbon |
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Institution: | 1. Université Laval de Québec, 2450, rue de Bilbao. Apartment 201, Québec?, QC?, G2C 0E4, Canada
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Abstract: | Denote the integer lattice points in the \(N\) -dimensional Euclidean space by \(\mathbb {Z}^N\) and assume that \(X_\mathbf{n}\) , \(\mathbf{n} \in \mathbb {Z}^N\) is a linear random field. Sharp rates of convergence of histogram estimates of the marginal density of \(X_\mathbf{n}\) are obtained. Histograms can achieve optimal rates of convergence \(({\hat{\mathbf{n}}}^{-1} \log {\hat{\mathbf{n}}})^{1/3}\) where \({\hat{\mathbf{n}}}=n_1 \times \cdots \times n_N\) . The assumptions involved can easily be checked. Histograms appear to be very simple and good estimators from the point of view of uniform convergence. |
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