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On fractal distribution function estimation and applications

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  • Stefano Maria Iacus
  • Davide La Torre

Abstract

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.

Suggested Citation

  • Stefano Maria Iacus & Davide La Torre, 2002. "On fractal distribution function estimation and applications," Departmental Working Papers 2002-07, Department of Economics, Management and Quantitative Methods at Università degli Studi di Milano.
    • Handle: RePEc:mil:wpdepa:2002-07
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    1. Stefano Maria Iacus & Davide La Torre, 2002. "Approximating distribution functions by iterated function systems," Departmental Working Papers 2002-03, Department of Economics, Management and Quantitative Methods at Università degli Studi di Milano.
    2. Sam Efromovich, 2001. "Second Order Efficient Estimating a Smooth Distribution Function and its Applications," Methodology and Computing in Applied Probability, Springer, vol. 3(2), pages 179-198, June.
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