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双曲平面上的莫比乌斯反卷积及阻抗密度估计AI 解读

In this paper we consider a novel statistical inverse problem on the Poincaré, or Lobachevsky, upper (complex) half plane. Here the Riemannian structure is hyperbolic and a transitive group action comes from the space of 2 × 2 real matrices of determinant one via Möbius transformations. Our approach is based on a deconvolution technique which relies on the Helgason–Fourier calculus adapted to this hyperbolic space. This gives a minimax nonparametric density estimator of a…

长周期彗星轨道的本征分布与选择偏差AI 解读

A question that arises in the study of cometary orbits is whether or not the directed normals to the orbits are uniformly distributed on the celestial sphere. Previous studies by statisticians have not taken selection effects into account and have tended to reject uniformity. Here a plausible selection mechanism is proposed that gives rise to a one-parameter family of distributions on the sphere. Data on long-period comets are analyzed using this one-parameter family. A…

线性逆问题中的对数密度估计AI 解读

We estimate a probability density function p which is related by a linear operator K to a density function q in sequences of regular exponential families based on a random sample from q. In this paper deconvolution and positron emission tomography are considered. The logarithm of the density function is approximated by basis functions consisting of singular functions of K. While direct maximum likelihood (or minimum Kullback-Leibler) density estimation in exponential…

非参数统计逆问题的最优收敛速率AI 解读

Consider an unknown regression function $f$ of the response $Y$ on a $d$-dimensional measurement variable $X$. It is assumed that $f$ belongs to a class of functions having a smoothness measure $p$. Let $T$ denote a known linear operator of order $q$ which maps $f$ to another function $T(f)$ in a space $G$. Let $\hat{T}_n$ denote an estimator of $T(f)$ based on a random sample of size $n$ from the distribution of $(X, Y)$, and let $\|\hat{T}_n - T(f)\|_G$ be a norm of $\hat{T}_n - T(f)$…