On the Kernel Estimation of the Mode of the Density Function

Abstract

In this thesis, we study the kernel estimation of the probability density function and its mode. The problem of estimating the mode of a probability density function has been considered firstly by Parzen [17]. Parzen [17] has proposed a kernel estimator of the mode depends on a single bandwidth. Since the bandwidth plays an important role in the kernel estimator, the selection of a variable bandwidth that depends on the point where we estimate the probability density function will improve the estimation of the mode. In this thesis, the Parzen estimator has been improved by considering a kernel estimator with variable bandwidth for the mode of the density function. The asymptotic consistency and normality of the two mode kernel estimators has been shown under some conditions. The performance of the two estimators is tested via simulation studies and it is shown that the variable bandwidth estimator is more efficient than the Parzen’s estimator.