Problems in density estimation for independent and dependent data

Abstract

The aim of this thesis is to provide two extensions to the theory of nonparametric kernel density estimation that increase the scope of the technique. The basic ideas of kernel density estimation are not new, having been proposed by Rosenblatt [20] and extended by Parzen [17]. The objective is that for a given set of data, estimates of functions of the distribution of the data such as probability densities are derived without recourse to rigid parametric assumptions and allow the data themselves to be more expressive in the statistical outcome. Thus kernel estimation has captured the imagination of statisticians searching for more flexibility and eager to utilise the computing revolution. The abundance of data and computing power have revealed distributional shapes that are difficult to model by traditional parametric approaches and in this era, the computer intensive technique of kernel estimation can be performed routinely. Also we are aware that computing power can be harnessed to give improved statistical analyses. Thus a lot of modern statistical research involves kernel estimation from complex data sets and our research is concordant with that momentum. The thesis contains three chapters. In Chapter 1 we provide an introduction to kernel density estimation and we give an outline to our two research topics. Our first extension to the theory is given in Chapter 2 where we investigate density estimation from independent data, using high order kernel functions. These kernel functions are designed for bias reduction but they have the penalty of yielding negative density estimates where data are sparse. In common practice, the negative estimates would arise in the tails of the density and we provide four ways of correcting this negativity to give bona fide estimates of the probability density. Our theory shows that the effects of these corrections are asymptotically negligible and thus opens the way for the regular use of bias reducing, high order kernel functions. We also consider density estimation of continuous stationary stochastic processes and this is the content of Chapter 3. With this problem, the dependent nature of the data influences the accuracy of the kernel density estimator and we provide theory regarding the convergence of the kernel estimators of the density and its derivatives to the true functions. An important result from this study is that nonparametric density estimators from dependent processes can have the same rates of convergence as their parametric counterparts yet retain the flexibility of being independent of parametric assumptions. Our other results indicate that the convergence rate of the density estimator can be quite slow if there are large lag dependencies amongst the data and suggests that large samples would be required for reliable inference about such data. The flexibility of kernel density estimation for continuous and discrete data, independent and dependent observations, means that it is a useful statistical tool. The techniques given in this thesis are not restricted to the analysis of simple sets of data but may be employed in the construction of statistical models for complex data with a high degree of structure.