We construct honest confidence regions for a Hilbert space-valued parameter in various statistical models. The confidence sets can be centered at arbitrary adaptive estimators, and have diameter which adapts optimally to a given selection of models. The latter adaptation is necessarily limited in scope. We review the notion of adaptive confidence regions, and relate the optimal rates of the diameter of adaptive confidence regions to the minimax rates for testing and estimation. Applications include the finite normal mean model, the white noise model, density estimation and regression with random design.

: Quadratic regression functionals are important for the bandwidth selection of nonparametric regression techniques and for nonparametric tests. Based on local polynomial regression, we propose estimators for weighted integrals of squared derivatives of regression functions. The rates of convergence in mean square error are calculated with appropriate values of the smoothing parameter and the amount of smoothness assumed. The asymptotic distribution of the estimators is also considered with the Gaussian noise assumption. It is shown that when the estimators are pseudo-quadratic (linear components dominate quadratic components), asymptotic normality with the n \Gamma1 rate can be achieved. Key words and phrases. Asymptotic normality, estimation of quadratic functionals, local polynomial regression. Abbreviated title: Estimation of Quadratic Regression Functionals. AMS 1991 subject classification. Primary 62G07; Secondary 60F05. 1 Introduction Let (X 1 ; Y 1 ); : : : ; (X n ; Y n ) ...