Data-Based Optimal Smoothing of Orthogonal Series Density Estimates

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 $f_{n,m,\lambda}(t) = \sum^n_{\nu = 1} \frac{\hat{f}_\nu}{(1 + \lambda\nu^{2m})} \phi_\nu(t)$ where $\{\phi_\nu\}$ is an orthonormal sequence and $\hat{f}_\nu = (1/n)\sum^n_{j=1} \phi_\nu(X_j)$ is an estimate of $f_\nu = \int \phi_\nu(t)f(t) dt$, 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.