cshalizi + density_estimation + statistics_on_manifolds   2

[1411.4040] Kernel Density Estimation on Symmetric Spaces
"We investigate a natural variant of kernel density estimation on a large class of symmetric spaces and prove a minimax rate of convergence as fast as the minimax rate on Euclidean space. We make neither compactness assumptions on the space nor Holder-class assumptions on the densities. A main tool used in proving the convergence rate is the Helgason-Fourier transform, a generalization of the Fourier transform for semisimple Lie groups modulo maximal compact subgroups. This paper obtains a simplified formula in the special case when the symmetric space is the 2-dimensional hyperboloid."
in_NB  density_estimation  statistics  nonparametrics  kith_and_kin  asta.dena  statistics_on_manifolds 
november 2014 by cshalizi
[1102.2450] Concentration Inequalities and Confidence Bands for Needlet Density Estimators on Compact Homogeneous Manifolds
"Let $X_1,...,X_n$ be a random sample from some unknown probability density $f$ defined on a compact homogeneous manifold $mathbf M$ of dimension $d ge 1$. Consider a 'needlet frame' ${phi_{j eta}}$ describing a localised projection onto the space of eigenfunctions of the Laplace operator on $mathbf M$ with corresponding eigenvalues less than $2^{2j}$, as constructed in cite{GP10}. We prove non-asymptotic concentration inequalities for the uniform deviations of the linear needlet density estimator $f_n(j)$ obtained from an empirical estimate of the needlet projection $sum_eta phi_{j eta} int f phi_{j eta}$ of $f$. We apply these results to construct risk-adaptive estimators and nonasymptotic confidence bands for the unknown density $f$. The confidence bands are adaptive over classes of differentiable and H"{older}-continuous functions on $mathbf M$ that attain their H"{o}lder exponents."
to:NB  density_estimation  statistics  nickl.richard  statistics_on_manifolds 
february 2013 by cshalizi

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