Wavelet methods to estimate an integrated quadratic functional: Adaptivity and asymptotic law

Using wavelet thresholding methods, we give an adaptive estimator of [theta]=[integral operator]f2, where f is coming from the white noise model. We estimate the random bias term, which is the dominating term in the decomposition of the quadratic error, and state a central limit theorem. This estimator has two advantages: it is centered around [theta] and the rate does not require the knowledge on the regularity of f. Moreover, using our procedure, we provide explicit confidence intervals and critical regions for parametric tests on [theta].