On fractal distribution function estimation and applications

In this paper we review some recent results concerning the approximations of distributionfunctions and measures on [0, 1] based on iterated function systems. The twodifferent approaches available in the literature are considered and their relations areinvestigated in the statistical perspective. In the second part of the paper we proposea new class of estimators for the distribution function and the related characteristicand density functions. Glivenko-Cantelli, LIL properties and local asymptotic minimaxeciency are established for some of the proposed estimators. Via Monte Carlo analysiswe show that, for small sample sizes, the proposed estimator can be as efficient oreven better than the empirical distribution function and the kernel density estimatorrespectively. This paper is to be considered as a first attempt in the construction ofnew class of estimators based on fractal objects. Pontential applications to survivalanalysis with random censoring are proposed at the end of the paper.