Optimum Kernel Estimators of the Mode

Abstract

Let $X_1, \cdots, X_n$ be independent observations with common density $f$. A kernel estimate of the mode is any value of $t$ which maximizes the kernel estimate of the density $f_n$. Conditions are given restricting the density, the kernel, and the bandwidth under which this estimate of the mode has an asymptotic normal distribution. By imposing sufficient restrictions, the rate at which the mean squared error of the estimator converges to zero can be decreased from $n^{-\frac{4}{7}}$ to $n^{-1+\varepsilon}$ for any positive $\varepsilon$. Also, by bounding the support of the kernel it is shown that for any particular bandwidth sequence the asymptotic mean squared error is minimized by a certain truncated polynomial kernel.