NONPARAMETRIC DENSITY-ESTIMATION FOR CLASSES OF POSITIVE RANDOM-VARIABLES

In this study, a kernel-based density estimator for positive random va riables is proposed and analyzed. In particular, a nonparametric estim ator is developed which takes ad vantage of the fact that positive ran dom variables can be represented as the norms of random vectors. By ap propriately choosing the dimension of the assumed vector space, the es timator can be structured to exploit a priori knowledge about the dens ity to be estimated; The asymptotic properties (e.g., pointwise and L( 1)-consistency) of this density estimator are investigated and found t o be similar to the desirable features of the standard kernel estimato r. An upper bound on the expected value of ene L(1) error is also deri ved which provides insight into the behavior of the estimator. Upon us ing this upper bound, the optimal form for the estimator (i.e., the ke rnel function, the smoothing factor, etc.) is selected via a minimax s trategy. In addition, this upper bound is used to compare the asymptot ic performance of the proposed estimator to that of the standard kerne l estimator and to boundary-corrected kernel estimators. Numerical exa mples illustrate that the proposed scheme outperforms the standard and boundary-corrected estimators for a variety of density types.