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

摘要

In this paper we consider a class of nonparametric estimators of adistribution function F, with compact support, based on the theory of IFSs. Theestimator of F is tought as the fixed point of a contractive operator T definedin terms of a vector of parameters p and a family of affine maps W which can beboth depend of the sample (X_1, X_2, ...., X_n). Given W, the problem consistsin 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 problemthat we propose to solve by penalization techniques. If F has a density f, wecan also provide an estimator of f based on Fourier techniques. IFS estimatorsfor F are asymptotically equivalent to the empirical distribution function(e.d.f.) estimator. We will study relative efficiency of the IFS estimatorswith respect to the e.d.f. for small samples via Monte Carlo approach. For well behaved distribution functions F and for a particular family ofso-called wavelet maps the IFS estimators can be dramatically better than thee.d.f. (or the kernel estimator for density estimation) in presence of missingdata, i.e. when it is only possibile to observe data on subsets of the wholesupport of F. This research has also produced a free package for the R statisticalenvironment which is ready to be used in applications.