2012/08/09 by Laëtitia Comminges, Comminges, Laëtitia, Arnak S. Dalalyan +2
Mathematics · #Advanced Statistical Methods and Models #FOS: Mathematics #Statistical Methods and Inference #Statistics Theory (math.ST) #math.ST #stat.TH
paper · pdf · doi:10.48550/arxiv.1208.1823
openalex publication_date 2012/08/09 · arxiv created 2013/01/04 · arxiv updated 2013/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the problem of testing a particular type of composite null hypothesis under a nonparametric multivariate regression model. For a given quadratic functional Q, the null hypothesis states that the regression function f satisfies the constraint Q[f]=0, while the alternative corresponds to the functions for which Q[f] is bounded away from zero. On the one hand, we provide minimax rates of testing and the exact separation constants, along with a sharp-optimal testing procedure, for diagonal and nonnegative quadratic functionals. We consider smoothness classes of ellipsoidal form and check that our conditions are fulfilled in the particular case of ellipsoids corresponding to anisotropic Sobolev classes. In this case, we present a closed form of the minimax rate and the separation constant. On the other hand, minimax rates for quadratic functionals which are neither positive nor negative makes appear two different regimes: "regular" and "irregular". In the "regular" case, the minimax rate is equal to n-1/4 while in the "irregular" case, the rate depends on the smoothness class and is slower than in the "regular" case. We apply this to the issue of testing the equality of norms of two functions observed in noisy environments.