Data-Driven Smoothing Based on Convexity Properties

A theorem on uniform convergence for densities with a fixed number of inflection points is proved. As an application, a method of automatic choice of bandwidth parameters in density estimation is proposed. It is based on the idea of making equal the numbers of inflection points of the estimates and that of the target density. The resulting estimates are almost surely uniformly consistent. This method is closely connected with previous ideas of Silverman (1981, 1983) and provides, in many practical situations, a natural procedure for the bump-hunting problem. A Monte Carlo study is included.

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