Estimating a density by adapting an initial guess

摘要

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 ψ 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 ψ of f. Under certain conditions, the resulting density estimate f_n~((m)) has smaller L_1-error, on the average, than 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)).