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Histograms for stationary linear random fields

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  • Michel Carbon

Abstract

Denote the integer lattice points in the $$N$$ N -dimensional Euclidean space by $$\mathbb {Z}^N$$ Z N and assume that $$X_\mathbf{n}$$ X n , $$\mathbf{n} \in \mathbb {Z}^N$$ n ∈ Z N is a linear random field. Sharp rates of convergence of histogram estimates of the marginal density of $$X_\mathbf{n}$$ X n are obtained. Histograms can achieve optimal rates of convergence $$({\hat{\mathbf{n}}}^{-1} \log {\hat{\mathbf{n}}})^{1/3}$$ ( n ^ - 1 log n ^ ) 1 / 3 where $${\hat{\mathbf{n}}}=n_1 \times \cdots \times n_N$$ n ^ = n 1 × ⋯ × 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. Copyright Springer Science+Business Media Dordrecht 2014

Suggested Citation

  • Michel Carbon, 2014. "Histograms for stationary linear random fields," Statistical Inference for Stochastic Processes, Springer, vol. 17(3), pages 245-266, October.
  • Handle: RePEc:spr:sistpr:v:17:y:2014:i:3:p:245-266
    DOI: 10.1007/s11203-014-9099-0
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    References listed on IDEAS

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    1. Gérard Biau & Benoît Cadre, 2004. "Nonparametric Spatial Prediction," Statistical Inference for Stochastic Processes, Springer, vol. 7(3), pages 327-349, October.
    2. Carbon, Michel & Tran, Lanh Tat & Wu, Berlin, 1997. "Kernel density estimation for random fields (density estimation for random fields)," Statistics & Probability Letters, Elsevier, vol. 36(2), pages 115-125, December.
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    6. Tran, L. T. & Yakowitz, S., 1993. "Nearest Neighbor Estimators for Random Fields," Journal of Multivariate Analysis, Elsevier, vol. 44(1), pages 23-46, January.
    7. Marc Hallin & Zudi Lu & Lanh T. Tran, 2001. "Density estimation for spatial linear processes," ULB Institutional Repository 2013/2109, ULB -- Universite Libre de Bruxelles.
    8. Tran, Lanh Tat, 1990. "Kernel density estimation on random fields," Journal of Multivariate Analysis, Elsevier, vol. 34(1), pages 37-53, July.
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