Goodness-of-fit test for noisy directional data

Abstract

We consider spherical data X i noised by a random rotation ε i ∈ SO(3) so that only the sample Z i = ε i X i , i = 1, . . . , 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 ≥ Cψ N and f is in a Sobolev space with smoothness s". For a noise density f ε with smoothness index ν, we show that an adaptive procedure (i.e. s is not assumed to be known) cannot have a faster rate of separation than ψ ad N (s) = (N/ log log(N )) −2s/(2s+2ν+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 SO(3) and by comparing it to other procedures. Applications to real data in astrophysics and paleomagnetism are provided.

Supersmooth testing on the sphere over analytic classes

“…In the same framework, Lacour and Pham Ngoc (2014) showed that considering supersmooth noise and Sobolev class alternatives leads to only logarithmic separation rates. Let us see how introducing analytic classes change notably that conclusion.…”

“…Proof The proof of this lemma is identical to the proof of Lemma A.3 in Lacour and Pham Ngoc (2014). Note that we deal with real valued quantities and do not need complex conjugation.…”

“…Hence contrary to Lacour and Pham Ngoc (2014), we will use a real basis which will lighten the proofs. The supremum norm for functions on S 2 is denoted as…”

“…Note that we deal with real valued quantities and do not need complex conjugation. Using Equation (18) in Lacour and Pham Ngoc (2014) which gives the expression of the variance of U K , we get…”

“…Following Fan (1991), Fan and Koo (2002), Healy et al (1998), Kim and Koo (2002), Lacour and Pham Ngoc (2014), we will appropriately define the smoothness of the error spectrally but with a modification due to the fact that the error W belongs to SO(3). This can be done by considering convolution as an operator (see Kim and Koo 2002).…”