Nonparametric estimation of distribution and density functions in presence of missing data: an IFS approach

Abstract

In this paper we consider a class of nonparametric estimators of a distribution function F, with compact support, based on the theory of IFSs. The estimator of F is tought as the fixed point of a contractive operator T defined in terms of a vector of parameters p and a family of affine maps W which can be both depend of the sample (X_1, X_2, ...., X_n). Given W, the problem consists in finding a vector p such that the fixed point of T is ``sufficiently near'' to F. It turns out that this is a quadratic constrained optimization problem that we propose to solve by penalization techniques. If F has a density f, we can also provide an estimator of f based on Fourier techniques. IFS estimators for F are asymptotically equivalent to the empirical distribution function (e.d.f.) estimator. We will study relative efficiency of the IFS estimators with respect to the e.d.f. for small samples via Monte Carlo approach. For well behaved distribution functions F and for a particular family of so-called wavelet maps the IFS estimators can be dramatically better than the e.d.f. (or the kernel estimator for density estimation) in presence of missing data, i.e. when it is only possibile to observe data on subsets of the whole support of F. This research has also produced a free package for the R statistical environment which is ready to be used in applications.