Nonparametric density estimation for classes of positive randomvariables

A kernel-based density estimator for positive random variables is proposed and analyzed. In particular, a nonparametric estimator is developed which takes advantage of the fact that positive random variables can be represented as the norms of random vectors. By appropriately choosing the dimension of the assumed vector space, the estimator can be structured to exploit a priori knowledge about the density to be estimated. The asymptotic properties (e.g., pointwise and L₁-consistency) of this density estimator are investigated and found to be similar to the desirable features of the standard kernel estimator. An upper bound on the expected value of the L₁ error is also derived which provides insight into the behavior of the estimator. Upon using this upper bound, the optimal form for the estimator (i.e., the kernel function, the smoothing factor, etc.) is selected via a minimax strategy. In addition, this upper bound is used to compare the asymptotic performance of the proposed estimator to that of the standard kernel estimator and to boundary-corrected kernel estimators. Numerical examples illustrate that the proposed scheme outperforms the standard and boundary-corrected estimators for a variety of density types