Kernel density estimation based on the mean integrated squared error approach

Abstract

Kernel density estimation is one of the most useful nonparametric density estimation technique. This estimation technique is based on two parameters, the bandwidth and the kernel function. In order to obtain the "best" possible parameters, a minimization error criterion must be considered. So far, most of the efforts in literature are based on the Mean Integrated Squared Error (NIISE). vVhen using this error criterion results show that the kernel function choice is suboptimal when compared to the bandwidth choice. Due to this, research in this area focuses mostly on the choice of the bandwidth parameter. Results have shown that under mild conditions on the shape of the kernel function and the density function, the existence of the optimal bandwidth is guaranteed. Moreover, the sequence of optimal bandwidth will also satisfy special asymptotic results which turn out to be very useful in the estimation section of this method. In this dissertation we shall go through all these important results, with detailed explanation on kernel density estimation. Another important aspect of kernel density estimation is the choice between various bandwidth selectors. These selectors are data-driven techniques used to obtain an estimate of the optimal bandwidth. Every estimator comes with its own asymptotic theory explaining the relative rate of convergence, the latter being one of the techniques used to compare different bandwidth selectors together. From the practical side, the most effective way to compare bandwidth selectors is by means of simulation studies. In literature this technique is very popular because it shows how different bandwidth estimators behave especially when sample sizes are relatively small. In this project we shall study in detail, six of the most recommended bandwidth selectors. These will be compared by using their asymptotic result and also through a simulation study based on 15 benchmark normal mixture densities. At the end of this comparison we give our recommendations on which bandwidth selectors have the most desirable performance. The last part of this project will then consider a hypothesis testing method which checks the significance of modes in the density estimate. We shall use this technique to check the statistical significance of several density estimates which were made by Parman (2007) on OIB/MORB 4He/3He data set.