Estimating mass and shape of domains in pet imaging

We find optimal rates of estimating the mass of a predefined domain in a PET image. We show that the optimal rate of convergence is (n/log n)^1/2. We introduce a family of estimators that depend on a smoothing kernel and a smoothing parameter. The asymptotic distribution of the estimator does not depend on the kernel or its bandwidth, as long as the latter converges to 0 at the right rate. It is efficient in a strong sense for 'nice' shapes. The convergence, however, is not uniform, even over simple family or regions. On the other hand, the mass of a region, defined by the image itself as a region of high concentration, can be estimated only at a slower rate of convergence.