Rate of Strong Consistency of Two Nonparametric Density Estimators

Abstract

Two nonparametric density estimators, based on Fourier series and the Fejer kernel, are presented. One of them $(\tilde{f}_N)$ is appropriate when the unknown density $f$ vanishes outside a known bounded interval; the other $(f_N^\sharp)$ is applicable without any assumptions about the support of $f$. The estimator $f_N^\sharp$ is of the type studied by Watson and Leadbetter (Sankhya A 26, 1964) and $\tilde{f}_N$ is almost of that type: both may be said to be of the "$\delta$-sequence type". If $f$ satisfies a certain Lipschitz condition at $x$ and the "number of harmonics" used in $\tilde{f}_N$ is asymptotically proportional to $N^\frac{1}{3}$, and $\rho_N/\log N \rightarrow \infty$, then $(N^{\frac{1}{3}}/\rho_N) \cdot |\tilde{f}_N(x) - f(x)| \rightarrow 0$ a.s.; a similar result holds for $f_N^\sharp$.