Statistica Sinica 11(2001), 691-704

THRESHOLDING FOR WEIGHTED $\chi^2$

Iain Johnstone

Stanford University

Abstract: Given data from a spherical Gaussian distribution with unknown mean vector $\theta$, estimates of quadratic functionals are constructed by thresholding. Mean squared error bounds are derived via a comparison with those already available for a suitable noncentral $\chi^2$ variate. By way of illustration, the resulting inequalities are used to yield an optimal rate adaptivity result for estimation of integrated squared derivatives in the white noise model of nonparametric function estimation.

Key words and phrases: Adaptive estimation, integrated squared derivative, noncentral X², quadratic functional.