ABSTRACT
The problem of providing an adequate non-parametric density for a data set is much more difficult than that of providing an adequate non-parametric regression function. The claims made for any particular method are correspondingly modest and this applies to the method described below. There are several reasons for this. Firstly, it is more of an ill-posed problem so to speak than non-parametric regression as it involves differentiation. Secondly, a density is a measure of closeness of points whereas non-parametric regression has an additive error. Thirdly, the additive error is often modelled by some parametric family which makes matters easier as the approximation region can be based on this. There is no equivalent for the density problem. Fourthly, although distribution free metrics such as the Kuiper metrics can be used to measure the closeness of the density as expressed by its distribution function to the empirical distribution function, the resulting bounds are typically too weak to pick up relevant features of the density. The situation is not satisfactory and although the taut string procedure to be described works well it is not entirely clear why this is so.
