Kernel Smoothing in Matlab

The following sections are included:

• Preface

• Contents

The following sections are included:

• Kernels and their properties

• Use of MATLAB toolbox

• Complements

The following sections are included:

• Basic definition

• Statistical properties of the estimate

• Choosing the shape of the kernel

• Choosing the bandwidth

• Density derivative estimation

• Automatic procedure for simultaneous choice of the kernel, the bandwidth and the kernel order

• Boundary effects

• Simulations

• Application to real data

• Use of MATLAB toolbox

• Complements

The most commonly used nonparametric estimation of a distribution function F is an empirical distribution function F_n. But F_n is a step function even in case that F is continuous. Nadaraya (1964) proposed a smooth non-parametric alternative to F_n, namely, kernel distribution estimator. This estimator is obtained by integrating the well known Rosenblatt-Parzen kernel density estimate (2.2). More generally, having a nonparametric estimate of f one can obtain a nonparametric estimate of F by integrating …

Consider the following situation. Each of a set of objects is known to belong to one of two classes. An assignment procedure assigns each object to a class on the basis of information observed about this object. But such a procedure need not to be perfect and errors could be made and the object is assigned to an incorrect class. Thus it is necessary to evaluate the quality of performance of the procedure. There are many possible ways to measure the performance of the classification rules. It is often very helpful to have a way of displaying and summarizing performance over a wide range of conditions. This aim is fulfilled by ROC (Receiver Operating Characteristic) curve. It is a single curve summarizing the distribution functions of the scores of two classes…

In recent years considerable attention has been paid to methods for analyzing data on events observed over time and to the study of factors associated with occurrence rate for these events. The problem of analyzing such data sets arises in a number of applied fields, such as medicine, biology, public health, epidemiology, technical sciences etc. Such data are generically referred to as survival data. A special feature of survival data which renders standard methods inappropriate is they contain censored observations…

The aim of regression analysis is to produce a reasonable analysis of an unknown regression function m. By reducing the observational errors it allows the interpretation to concentrate on important details of the mean dependence of Y on X…

The present chapter is devoted to the extension of the univariate kernel density estimate to the multivariate setting. This extension is not without any problem. The most general smoothing parameterization of the kernel estimator in d-dimensions requires the specification of entries of a positive definite bandwidth matrix. The bandwidth matrix controls both the smoothness and the orientation of the multivariate smoothing. The multivariate kernel density estimator we are going to deal with is a direct extension of the univariate estimator (see, e.g., Wand and Jones (1995)). Papers Chacón et al. (2011); Duong et al. (2008) also investigated general kernel estimators of multivariate density derivative using general (or unconstrained) bandwidth matrix selectors. In this chapter we focus on estimates of both multivariate density and its gradient…

The following sections are included:

• Notation

• Bibliography

• Index