Deconvolution density estimation on the space of positive definite symmetric matrices

Motivated by applications in microwave engineering and diffusion tensor imaging, we study the problem of deconvolution density estimation on the space of positive definite symmetric matrices. We develop a nonparametric estimator for the density function of a random sample of positive definite matrices. Our estimator is based on the Helgason-Fourier transform and its inversion, the natural tools for analysis of compositions of random positive definite matrices. Under several smoothness conditions on the density of the intrinsic error in the random sample, we derive upper bounds on the rates of convergence of our nonparametric estimator to the true density.