Methods of density estimation on the Grassmann manifold

Abstract

The Grassmann manifold G k, m− k consists of k-dimensional hyperplanes in R m and is equivalent to the manifold P k, m− k of all m× m orthogonal projection matrices idempotent of rank k. This paper develops a method of semiparametric density estimation on the manifold P k, m− k , designed to nonparametrically correct a parametric model by its linear function, to obtain better density estimators than the ordinary kernel density estimator. We suggest two procedures to estimate the correction factors. Comparing with the ordinary kernel density estimator, for small smoothing parameter matrix and/or for large sample size n, the suggested semiparametric density estimator is seen to have approximately the same variance to the order of approximation used but a smaller bias. A one-to-one transformation of P k, m− k into R k(m−k) is of use in the asymptotic investigation. The general discussion is applied and examined for special kernel function, discrepancy measures for matrices on P k, m− k and starting parametric model.