Bernoulli

  • Bernoulli
  • Volume 20, Number 4 (2014), 2131-2168.

Goodness-of-fit test for noisy directional data

Claire Lacour and Thanh Mai Pham Ngoc

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Abstract

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.

Article information

Source
Bernoulli, Volume 20, Number 4 (2014), 2131-2168.

Dates
First available in Project Euclid: 19 September 2014

Permanent link to this document
https://projecteuclid.org/euclid.bj/1411134456

Digital Object Identifier
doi:10.3150/13-BEJ553

Mathematical Reviews number (MathSciNet)
MR3263101

Zentralblatt MATH identifier
1357.62192

Keywords
adaptive testing minimax hypothesis testing nonparametric alternatives spherical deconvolution spherical harmonics

Citation

Lacour, Claire; Pham Ngoc, Thanh Mai. Goodness-of-fit test for noisy directional data. Bernoulli 20 (2014), no. 4, 2131--2168. doi:10.3150/13-BEJ553. https://projecteuclid.org/euclid.bj/1411134456


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