Supersmooth testing on the sphere over analytic classes

Abstract

We consider the nonparametric goodness-of-fit test of the uniform density on the sphere when we have observations whose density is the convolution of an error density and the true underlying density. We will deal specifically with the supersmooth error case which includes the Gaussian distribution. Similar to deconvolution density estimation, the smoother the error density the harder is the rate recovery of the test problem. When considering nonparametric alternatives expressed over analytic classes, we show that it is possible to obtain original separation rates much faster than any logarithmic power of the sample size according to the ratio of the regularity index of the analytic class and the smoothness degree of the error. Furthermore, we show that our fully data-driven statistical procedure attains these optimal rates.

Keywords:

• spherical deconvolution
• fully data-driven procedure
• minimax hypothesis testing
• nonparametric alternatives
• supersmooth error
• analytic classes
• rotational harmonics

AMS Subject Classification:

• Primary: 62G10
• Secondary: 62H11

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

Peter T. Kim. Supported in part by the NSERC (Canada), DG-2011-46204. Ja-Yong Koo. Supported in part by NRF (Korea), 2009-0075827 and 2013R1A1A2008619.