Density estimation with non–parametric methods

One key issue in several astrophysical problems is the evaluation of the density probability function underlying an observational discrete data set. We here review two non-parametric density estimators which recently appeared in the astrophysical literature, namely the adaptive kernel density estimator and the Maximum Penalized Likelihood technique, and describe another method based on the wavelet transform. The efficiency of these estimators is tested by using extensive numerical simulations in the one-dimensional case. The results are in good agreement with theoretical functions and the three methods appear to yield consistent estimates. However, the Maximum Penalized Likelihood suffers from a lack of resolution and high computational cost due to its dependency on a minimization algorithm. The small differences between kernel and wavelet estimates are mainly explained by the ability of the wavelet method to take into account local gaps in the data distribution. This new approach is very promising, since smaller structures superimposed onto a larger one are detected only by this technique, especially when small samples are investigated. Thus, wavelet solutions appear to be better suited for subclustering studies. Nevertheless, kernel estimates seem more robust and are reliable solutions although some small-scale details can be missed. In order to check these estimators with respect to previous studies, two galaxy redshift samples, related to the galaxy cluster A3526 and to the Corona Borealis region, have been analyzed. In both these cases claims for bimodality are confirmed at a high confidence level.