Density estimation on the spaces of symmetric and rectangular matrices

This paper develops the theory of density estimation on the space m of all m × m symmetric matrices and on the space Rm, p of all m × p rectangular matrices. We consider the two methods, kernel density estimation and density estimation by orthogonal series, and illustrate them with examples for the normal kernel density functions and for the Hermite orthonormal bases with symmetric and rectangular matrix arguments on m and Rm, p, respectively. The problem of estimating unknown joint density functions of multiple random matrices is also discussed.