Kernel Density Estimates in a Non-standard Situation

Kernel density estimate is an integral part of the statistical tool box. It has been widely studied and is very well understood in situations where the observations {xi}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{x_i\}$$\end{document} are i.i.d., or is a stationary process with some weak dependence. However, there are situations where these conditions do not hold. For instance, while the eigenvalue distribution of large-dimensional random matrices converges, the eigenvalues themselves are highly correlated for most common random matrix models. Suppose {Fn}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{F_n\}$$\end{document} is a sequence of empirical distribution functions (usually random) which converges weakly to a non-random distribution function F with density f in some probabilistic sense. We show that under mild conditions on the kernel K and the limit density f, the kernel density estimate f^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{f}$$\end{document} based on Fn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_n$$\end{document} converges to f in suitable probabilistic senses. This demonstrates the robustness of the kernel density estimate. We show how the rate of convergence of f^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{f}$$\end{document} to f can be linked to the rate of convergence of Fn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_n$$\end{document} and E(Fn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{E}}(F_n)$$\end{document} to F. Using the above general results, we establish the consistency of the kernel density estimates, including upper bounds on the rate of convergence, for two popular random matrix models. We also provide a few simulations to demonstrate these results and conclude with a few open questions.