Optimal Calibration for Multiple Testing against Local Inhomogeneity in Higher Dimension

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 nearest-neighbor statistics. It does not require any preliminary information about the multivariate densities such as compact support, strict positivity or smoothness and shape properties. The properly adjusted multiple testing procedure is shown to be sharp-optimal for typical arrangements of the observation values which appear with probability close to one. The proof relies on a new coupling Bernstein type exponential inequality, reflecting the non-subgaussian tail behavior of a 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 H\"older classes. The exact minimax risk asymptotics are obtained in terms of solutions of the optimal recovery

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 H\"{o}lder classes. Furthermore, its relative asymptotic efficiency with respect to an asymptotically minimax optimal test under sup-norm loss is close to 1 in case of homoscedastic Gaussian errors within a broad range of H\"{o}lder classes simultaneously.Comment: Published in at http://dx.doi.org/10.1214/009053607000000992 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Cognitive-Behavioral Treatment for Depression in Adolescents

The goal of this article is to briefly review and summarize the rationale and research support for cognitivebehavioral therapy (CBT) as a treatment for depressed adolescents. A primary focus of the paper is on our group CBT treatment for adolescent depression, entitled “The Adolescent Coping with Depression Course”. In addition, initial findings from a large, recently-completed study contrasting individual CBT with fluoxetine for depressed adolescents (Treatment of Adolescents with Depression Study) are presented. Although the research support for CBT as a treatment for depressed adolescents is generally encouraging, we need to better understand which depressed adolescents benefit from CBT, how and when to incorporate medication and family-based interventions into CBT treatment, how to treat depressed adolescents with comorbid psychiatric conditions, and how CBT interventions fare with non-European-American depressed adolescents