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Confidence sets for the optimal approximating model
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.
Adaptive goodness-of-fit tests based on signed ranks
Within the nonparametric regression model with unknown regression function $l$ and independent, symmetric errors, a new multiscale signed rank statistic is introduced and a conditional multiple test of the simple hypothesis $l = 0$ against a nonparametric alternative is proposed. This test is distribution-free and exact for finite samples even in the heteroscedastic case. It adapts in a certain sense to the unknown smoothness of the regression function under the alternative, and it is uniformly consistent against alternatives whose sup-norm tends to zero at the fastest possible rate. The test is shown to be asymptotically optimal in two senses: It is rate-optimal adaptive against Hölder classes. Furthermore, its relative asymptotic efficiency with respect to an asymptotically minimax optimal test under sup-norm loss is close to one in case of homoscedastic Gaussian errors within a broad range of Hölder classes simultaneously.
Sharp-optimal adjustment for multiple testing in the multivariate two-sample problem
Based on two independent samples $X_1, ...,X_m$ and $X_m+1, ...,X_n$ drawn from multivariate distributions with unknown Lebesgue densities p and q respectively, we propose an exact multiple test in order to identify simultaneously regions of significant deviations between p and q. The construction is built from randomized nearestneighbor statistics. It does not require any preliminary information about the multivariate densities such as compact support, strict positivity or smoothness and shape properties. The adjustment for multiple testing is sharp-optimal for typical arrangements of the observation values which appear with probability close to one, and it relies on a new coupling Bernstein type exponential inequality, reflecting the nonsubgaussian tail behavior of the combinatorial process. For power investigation of the proposed method a reparametrized minimax set-up is introduced, reducing the composite hypothesis ''$p = q$'' to a simple one with the multivariate mixed density $(m/n)p + (1 ? m/n)q$ as infinite dimensional nuisance parameter. Within this framework, the test is shown to be spatially and sharply asymptotically adaptive with respect to uniform loss on isotropic Hölder classes.