# Goodness-of-fit test for noisy directional data

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1 LM-Orsay - Laboratoire de Mathématiques d-Orsay

Abstract : We consider spherical data $X i$ noised by a random rotation $\varepsilon i\in$ SO3 so that only the sample $Z i=\varepsilon iX i$, $i=1,\dots, N$ is observed. We define a nonparametric test procedure to distinguish $H 0:$ -the density $f$ of $X i$ is the uniform density $f 0$ on the sphere- and $H 1:$ -$\|f-f 0\| 2^2\geq \C\psi N$ and $f$ is in a Sobolev space with smoothness $s$-. For a noise density $f \varepsilon$ with smoothness index $u$, we show that an adaptive procedure i.e. $s$ is not assumed to be known cannot have a faster rate of separation than $\psi N^{ad}s=N-\sqrt{\log\logN}^{-2s-2s+2 u+1}$ and we provide a procedure which reaches this rate. We also deal with the case of super smooth noise. We illustrate the theory by implementing our test procedure for various kinds of noise on SO3 and by comparing it to other procedures. Applications to real data in astrophysics and paleomagnetism are provided.

Keywords : spherical harmonics nonparametric alternatives minimax hypothesis testing Adaptive testing spherical deconvolution

Autor: Claire Lacour - Thanh Mai Pham Ngoc -

Fuente: https://hal.archives-ouvertes.fr/

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