In this paper we review some recent results concerning the approximations of distribution functions and measures on [0, 1] based on iterated function systems.

The two different approaches available in the literature are considered and their relation are investigated in the statistical perspective. In the second part of the paper we propose a new class of estimators for the distribution function and the related characteristic and density functions. Glivenko-Cantelli, LIL properties and local asymptotic minimax efficiency are established for some of the proposed estimators. Via Monte Carlo analysis we show that, for small sample sizes, the proposed estimator can be as efficient or even better than the empirical distribution function and the kernel density estimator respectively. This paper is to be considered as a first attempt in the construction of new class of estimators based on fractal objects. Pontential applications to survival analysis with random censoring are proposed at the end of the paper.

Stefano M. Iacus Davide La Torre
Universit`a degli Studi di Milano
Dipartimento di Economia Politica e Aziendale
Via Conservatorio 7 - I-20122 Milano
March 8, 2002

Summary:

• 1 Introduction

• 2 Theoretical background for affine IFS

• 2.1 Minimization approach

• Theorem 1

• Theorem 2 (Collage theorem)

• Theorem 3 (Forte and Vrscay, 1995)

• 2.2 Direct approach

• Theorem 4 (Iacus and La Torre, 2001)

• Theorem 5 (Iacus and La Torre, 2001)

• Corollary 6

• Theorem 7.

• 2.3 The choice of affine maps

• 3 Fourier analysis results

• Theorem 8 (Forte and Vrscay, 1998)

• Theorem 9 (Collage Theorem for FT, (Forte and Vrscay, 1998))

• 4 Statistical applications

• 4.1 Asymptotic results for the quantile-based IFS estimator

• Theorem 10

• Theorem 11

• Theorem 12 (Gill and Levit, 1995)

• Theorem 13

• 4.2 Characteristic function and Fourier density estimation

• 5 Monte Carlo analysis

• 5.1 Applications to survival analysis

• Final remarks

• References