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A Bayesian approach to the estimation of maps between Riemannian manifolds. (English) Zbl 1229.62006
Summary: Let \(\Theta \) be a smooth compact oriented manifold without boundary, imbedded in a Euclidean space \(E^{s}\), and let \(\gamma \) be a smooth map of \(\Theta \) into a Riemannian manifold \(\Lambda \). An unknown state \(\theta \in \Theta \) is observed via \(X = \theta + \varepsilon \xi \), where \(\varepsilon > 0\) is a small parameter and \(\xi \) is a white Gaussian noise. For a given smooth prior \(\lambda \) on \(\Theta \) and smooth estimators \(g(X)\) of the map \(\gamma \) we derive a second-order asymptotic expansion for the related Bayesian risk. The calculation involves the geometry of the underlying spaces \(\Theta \) and \(\Lambda \), in particular, the integration-by-parts formula. Using this result, a second-order minimax estimator of \(\gamma \) is found based on the modern theory of harmonic maps and hypoelliptic differential operators.

MSC:
62C10 Bayesian problems; characterization of Bayes procedures
62C20 Minimax procedures in statistical decision theory
62F12 Asymptotic properties of parametric estimators
53B20 Local Riemannian geometry
53C17 Sub-Riemannian geometry
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References:
[1] S. Amari, Differential-Geometrical Methods in Statistics, in Lecture Notes in Statist. (Springer, Berlin, 1985). · Zbl 0559.62001
[2] P. Berkhin and B. Levit, ”Second Order Asymptotically Minimax Estimates of the Mean of a Normal Population”, Problems Inform. Transmission 16, 212–229 (1980). · Zbl 0466.62033
[3] J. Eells and L. Lemaire, Selected Topics in Harmonic Maps (C.B.M.S. Regional Conference Series, AMS, 1983). · Zbl 0515.58011
[4] J. Jost, Riemannian Geometry and Geometric Analysis, 4th ed. (Springer, Berlin, 2005). · Zbl 1083.53001
[5] Z. Landsman and B. Levit, ”Second Order Asymptotically Minimax Estimation in the Presence of Nuisance Parameters”, Problems Inform. Transmission 26, 50–66 (1990). · Zbl 0711.62009
[6] B. Levit, ”On Second Order Asymptotically Minimax Estimates”, Theor. Probab. Appl. 25, 552–568 (1980). · Zbl 0494.62035 · doi:10.1137/1125066
[7] B. Levit, ”Minimax Estimation and the Positive Solutions of Elliptic Equations”, Theor. Probab. Appl. 27, 525–546 (1982). · Zbl 0501.62019
[8] R. Montgomery, A Tour of Sub-Riemannian Geometries, Their Geodesics and Applications, in Math. Surveys and Monographs (AMS, Providence, RI, 2002), Vol. 91. · Zbl 1044.53022
[9] A. Bloch, Nonholonomic Mechanics and Control, with the collaboration of J. Baillieul, P. Crouch and J. Marsden, with scientific input from P. S. Krishnaprasad, R. M. Murray, and D. Zenkov, in Interdisciplinary Applied Math., Vol. 24: Systems and Control (Springer, New York, 2003).
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