Confidence sets for the optimal approximating model --- Bridging a gap between adaptive point estimation and confidence regions
- Rohde, Angelika
- Dümbgen, Lutz
2010 Mathematics Subject Classification
- 62G15 62G20
- Adaptivity, confidence sets, coupling, exponential inequality, model selection, multiscale inference, risk optimality
In the setting of high-dimensional linear models with Gaussian noise, we investigate the possibility of confidence statements connected to model selection. Although there exist numerous procedures for adaptive point estimation, the construction of adaptive confidence regions is severely limited (cf. Li, 1989). The present paper sheds new light on this gap. We develop exact and adaptive confidence sets for the best approximating model in terms of risk. Our construction is based on a multiscale procedure and a particular coupling argument. Utilizing exponential inequalities for noncentral $chi^2$--distributions, we show that the risk and quadratic loss of all models within our confidence region are uniformly bounded by the minimal risk times a factor close to one.