Abstract

this article. We introduce the classic nonparametric estimator, the histogram, and outline its theoretical properties as well as good practice. We demonstrate how to improve the histogram, leading to our discussion of popular kernel methods. We conclude with a bivariate example, a way of choosing smoothing parameters, and new directions that promise further improvements. Why choose nonparametric over parametric density estimation? Parametric density estimation requires both proper specification of the form of the underlying sampling density, f ` (x), and estimation of the parameter vector `. Parametric modeling entails two risks of bias: in estimation of ` and incorrect specification of f ` . Nonparametric density estimation provides a consistent algorithm for nearly any continuous density and avoids the specification step. Although the cumulative distribution and probability density functions carry the same information, densities are more easily interpreted than distributions, especially in more than one dimension, so our focus on the density is appropriate. Density estimation is broadly applicable for exploring data relationships, presenting data summaries, and constructing sophisticated nonparametric models of biostatistical data. Graphical representation of data is a powerful tool for summarization. Three simple exploratory graphical summaries are the box-and-whiskers plot (or boxplot), the stem-andleaf plot, and the histogram. Consider the cholesterol levels of 320 males with diagnosed coronary artery disease (Scott et al., 1978). Figure 1 displays a boxplot of these data. The data appear symmetric with a few outliers. The various percentiles displayed in the boxplot do not hint of any unusual feature such as we see in Figure 2 in the right histogram, which show...

Citations

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