Open Access
November 2014 Goodness-of-fit test for noisy directional data
Claire Lacour, Thanh Mai Pham Ngoc
Bernoulli 20(4): 2131-2168 (November 2014). DOI: 10.3150/13-BEJ553


We consider spherical data $X_{i}$ noised by a random rotation $\varepsilon_{i}\in\operatorname{SO} (3)$ so that only the sample $Z_{i}=\varepsilon_{i}X_{i}$, $i=1,\dots,N$ 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 $H_{1}$: “$\|f-f_{0}\|_{2}^{2}\geq\mathcal{C} \psi_{N}$ and $f$ is in a Sobolev space with smoothness $s$”. For a noise density $f_{\varepsilon}$ with smoothness index $\nu$, we show that an adaptive procedure (i.e., $s$ is not assumed to be known) cannot have a faster rate of separation than $\psi_{N}^{\mathrm{ad}}(s)=(N/\sqrt{\log\log(N)})^{-2s/(2s+2\nu+1)}$ 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 $\operatorname{SO}(3)$ and by comparing it to other procedures. Applications to real data in astrophysics and paleomagnetism are provided.


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Claire Lacour. Thanh Mai Pham Ngoc. "Goodness-of-fit test for noisy directional data." Bernoulli 20 (4) 2131 - 2168, November 2014.


Published: November 2014
First available in Project Euclid: 19 September 2014

zbMATH: 1357.62192
MathSciNet: MR3263101
Digital Object Identifier: 10.3150/13-BEJ553

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

Rights: Copyright © 2014 Bernoulli Society for Mathematical Statistics and Probability

Vol.20 • No. 4 • November 2014
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