Adaptive Kernel Density Estimators

This technique combines the data-adaptive philosophy (the philosophy of varying the bin width) with the kernel approach (Silverman 1986: 100-110). The intent, of course, is to gain the advantages of both. A necessary first step is to get some rough idea of the local density around each datum point. Almost any estimator works for such a pilot estimate; the standard kernel estimator with fixed bin width is a common choice. Next, assign a bin width to each datum point, tailoring that width to the local density as given by the pilot estimate. Then choose a kernel (e.g. Fig. 1b) and estimate the entire probability distribution in a way essentially like that described above for the kernel estimator. Authors have suggested ways to fine-tune one or more aspects of the procedure.

Figure 1: Basic concepts of kernel probability estimation. (a) Hypothetical probability density (the entire curve) for a group of measurements, showing bin width ε, data point x on which a bin is centered, and a neighboring point x_i. = (b) Popular kernels (assumed local probability distributions).