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 ψ of f. An advantage of their estimate fn is that an approximate standard error can be provided. A disadvantage is that fn is less accurate, on the average, than more usual kernel estimates. The reason is that fn is not sufficiently smooth. As improvement, a smoothed analogue fn(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 fn(m) has smaller L1-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 fn(m) provided look ‘surprisingly accurate’. The main advantage of fn(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 fn(m