Data-Based Optimal Smoothing of Orthogonal Series Density Estimates

Abstract

Let $f$ be a density possessing some smoothness properties and let $X_1,\cdots, X_n$ be independent observations from $f$. Some desirable properties of orthogonal series density estimates $f_{n,m,\lambda}$ of $f$ of the form $f_{n,m,\lambda}(t) = \sum^n_{u = 1} \frac{\hat{f}_u}{(1 + \lambdau^{2m})} \phi_u(t)$ where $\{\phi_u\}$ is an orthonormal sequence and $\hat{f}_u = (1/n)\sum^n_{j=1} \phi_u(X_j)$ is an estimate of $f_u = \int \phi_u(t)f(t) dt$, are discussed. The parameter $\lambda$ plays the role of a bandwidth or "smoothing" parameter and $m$ controls a "shape" factor. The major novel result of this note is a simple method for estimating $\lambda$ (and $m$) from the data in an objective manner, to minimize integrated mean square error. The results extend to multivariate estimates.