On the convergence of kernel estimators of probability density functions
The properties of the characteristic function of the fixed-bandwidth kernel estimator of a probability density function are used to derive estimates of the rate of almost sure convergence of such estimators in a family of Hilbert spaces. The convergence of these estimators in a reproducing-kernel Hilbert space is used to prove the uniform convergence of variable-bandwidth estimators. An algorithm employing the fast Fourier transform and heuristic estimates of the optimal bandwidth is presented, and numerical experiments using four density functions are described.
KeywordsHilbert Space Probability Density Function True Density Kernel Estimator Optimal Bandwidth
Unable to display preview. Download preview PDF.
- Scott, D. W. (1976). Nonparametric density estimation by optimization theoretic techniques, Doctoral dissertation at Rice University, Houston, Texas.Google Scholar
- Taylor, R. L. Complete convergence for weighted sums of arrays of random elements, preprint.Google Scholar
- Taylor, R. L. Convergence of weighted sums of arrays of random elements in typep spaces with application to density estimation, preprint.Google Scholar
- Woyczynski, W. A. On Marcinkiewicz-Zygmund laws of large numbers in Banach spaces and related rates of convergence, preprint.Google Scholar