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Filtered kernel density estimation

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Abstract This article describes a multiple‐bandwidth version of the kernel estimator for nonparametric probability density estimation, in which the bandwidths are chosen using a set of functions, called filter functions, which determine the support of the density appropriate to the different bandwidths. These filter functions are usually defined using a normal mixture fit to the data. Thus the estimator uses different bandwidths in different regions of the support of the distribution, as controlled by the filter functions. Copyright © 2009 John Wiley & Sons, Inc. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Density Estimation

A probability density consisting of a mixture of two normals. We illustrate four estimators computed on 200 random variates drawn from this model: the filtered kernel estimator, using a two component mixture of normals fit to the data to define the filtering functions: a standard kernel estimator; the kernel estimator with the bandwidth inflated or reduced in order to match the two components.

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Log‐normal example: 100 observations drawn from a log‐normal distribution. The figures are as in Figure 2.

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Log‐normal example: 100 observations drawn from a log‐normal distribution. The top figure shows the data compared to two mixture models. The first, in red, was selected purely by eye as compared to the histogram, intended to indicate the mode and tail of the distribution. The second, in blue, was fit using the EM algorithm with the first mixture as a starting point. The bottom figure shows the two filtered kernel estimators using the two mixture models compared to a standard kernel estimator.

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Statistical and Graphical Methods of Data Analysis > Density Estimation

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