In linear regression of Y on X(∈ R^p) with parameters 𝛽(∈ R^{p+1}), statistical inference is unreliable when observations are obtained from gross-error model, F_{ϵ,G} = (1 − ϵ)F + ϵG, instead of the assumed probability F;G is gross-error probability, 0 < ϵ < 1. Residual’s influence index (RINFIN) at (x, y) is introduced, with components measuring also the local influence of x in the residual and large value flagging a bad leverage case (from G), thus causing unmasking. Large sample properties of RINFIN are presented to confirm significance of the findings, but often the large difference in the RINFIN scores of the data is indicative. RINFIN is successful with microarray data, simulated, high dimensional data and classic regression data sets. RINFIN’s performance improves as p increases and can be used in multiple response linear regression.

Centroidal Voronoi tessellation (CVT) is a special type of Voronoi diagram such that the generating point of each Voronoi cell is also its center of mass. The CVT has broad applications in computer graphics, such as meshing, stippling, sampling, etc. The existing methods for computing CVTs on meshes either require a global parameterization or compute it in the restricted sense (that is, intersecting a 3D CVT with the surface). Therefore, these approaches often fail on models with complicated geometry and/or topology. This paper presents two intrinsic algorithms for computing CVT on triangle meshes. The first algorithm adopts the Lloyd framework, which iteratively moves the generator of each geodesic Voronoi diagram to its mass center. Based on the discrete exponential map, our method can efficiently compute the Riemannian center and the center of mass for any geodesic Voronoi diagram. The second algorithm uses the L-BFGS method to accelerate the intrinsic CVT computation. Thanks to the intrinsic feature, our methods are independent of the embedding space, and work well for models with arbitrary topology and complicated geometry, where the existing extrinsic approaches often fail. The promising experimental results show the advantages of our method.

This paper mainly focuses on the CR analogue of the three-circle theorem in a complete noncompact pseudohermitian manifold of vanishing torsion being odd dimensional counterpart of Kaehler geometry. In this paper, we show that the CR three-circle theorem holds if its pseudohermitian sectional curvature is nonnegative. As an application, we confirm the first CR Yau uniformization conjecture and obtain the CR analogue of the sharp dimension estimate for CR holomorphic functions of polynomial growth and its rigidity when the pseudohermitian sectional curvature is nonnegative. This is also the first step toward second and third CR Yauís uniformization conjecture. Moreover, in the course of the proof of the CR three-circle theorem, we derive CR sub-Laplacian comparison theorem. Then Liouville theorem holds for positive pseudoharmonic functions in a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion and nonnegative pseudohermitian Ricci curvature.

Divergence functions are the non-symmetric “distance” on the manifold, Μθ, of parametric probability density functions over a measure space, ( Χ,μ). Classical information geometry prescribes, on Μθ: (i) a Riemannian metric given by the Fisher information; (ii) a pair of dual connections (giving rise to the family of α-connections) that preserve the metric under parallel transport by their joint actions; and (iii) a family of divergence functions ( α-divergence) defined on Μθ x？ Μθ, which induce the metric and the dual connections. Here, we construct an extension of this differential geometric structure from Μθ (that of parametric probability density functions) to the manifold, Μ, of non-parametric functions on X, removing the positivity and normalization constraints. The generalized Fisher information and α-connections on M are induced by an α-parameterized family of divergence functions, reflecting the fundamental convex inequality associated with any smooth and strictly convex function. The infinite-dimensional manifold, M, has zero curvature for all these α-connections; hence, the generally non-zero curvature of M can be interpreted as arising from an embedding of Μθ into Μ. Furthermore, when a parametric model (after a monotonic scaling) forms an affine submanifold, its natural and expectation parameters form biorthogonal coordinates, and such a submanifold is dually flat for α？= ± 1, generalizing the results of Amari’s α-embedding. The present analysis illuminates two different types of duality in information geometry, one concerning the referential status of a point (measurable function) expressed in the divergence function (“referential duality”) and the other concerning its representation under an arbitrary monotone scaling (“representational duality”).

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