On Locally Adaptive Density Estimation

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Abstract
Theoretical and practical aspects of the sample-point adaptive positive kernel density estimator are examined. A closed-form expression for the mean integrated squared error is obtained through the device of preprocessing the data by binning. With this expression, the exact behavior of the optimally adaptive smoothing parameter function is studied for the first time. The approach differs from most earlier techniques in that bias of the adaptive estimator remains O(h 2) and is not “improved” to the rate O(h 4). A practical algorithm is constructed using a modification of least squares cross-validation. Simulated and real examples are presented, including comparisons with a fixed bandwidth estimator and a fully automatic version of Abramson's adaptive estimator. The results are very promising.

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