Estimating a density by adapting an initial guess

De Bruin et al. (Comput. Statist. Data Anal. 30 (1999) 19) provide a unique method to estimate the probability density f from a sample, given an initial guess psi of f. An advantage of their estimate f(n) is that an approximate standard error can be provided. A disadvantage is that f(n) is less accurate, on the average, than more usual kernel estimates. The reason is that f(n) is not sufficiently smooth. As improvement, a smoothed analogue f(n)((m)) is considered. The smoothing parameter m (the degree of a polynomial approximation) depends on the supposed quality of the initial guess psi of f. Under certain conditions, the resulting density estimate f(n)((m)) has smaller L-1-error, on the average, than kernel estimates with bandwidths based on likelihood cross-validation. The theory requires that the initial guess is made up a priori. In practice, some data peeping may be necessary. The f(n)((m)) provided look 'surprisingly accurate'. The main advantage of f(n)((m)) over many other density estimators is its uniqueness (when the procedures developed in this article are followed), another one is that an estimate is provided for the standard error of f(n)((m)) (C) 2002 Elsevier Science B.V. All rights reserved.