Nonparametric estimation of multidimensional density with the use of wavelet estimates of univariate projections

The paper explores the computerized tomography method of inverting the Radon transformation for obtaining statistical estimates of multidimensional probability densities. This method utilizes nonlinear wavelet estimators of univariate projections to construct the multidimensional density estimate. Nonlinear wavelet estimators possess the ability to adapt to the local properties of the estimated density function and, therefore, are less sensitive to the singular points than linear estimators. Another important practical feature of the considered method is its parallel structure, which allows a considerable speedup of constructing the estimates on the computers supporting parallel processing. It is also proved that under some regularity conditions, the uniform distance between the constructed estimate and the true multidimensional probability density converges to zero in the mean, and some estimates of the rate of this convergence are obtained.