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On fractal distribution function estimation and applications

Introduction

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

On fractal distribution function estimation and applications

Theorem 4 (Iacus and La Torre, 2001)

Under conditions i) to v):

1. Tp is an operator from

to itself.
2. Suppose that wi(x) = x, P i = p, and δi ≥ −p, then

3.


then Tp is a contraction on

with contractivity
constant c.

4. Let

such that TpF1 = F1 and Tp* F2 = F2. Then

where c is the contractivity constant of Tp. The theorem above assures the IFS nature of the operator Tp that can be denoted, as in the previous section, as a N-maps IFS(w, p) with obvious notation. The goal is again the solution of the inverse problem in terms of p. Consider the following convex set:

open full size image

then we have the following results:

... in Theorem 5

Stefano M. Iacus, Davide La Torre

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