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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

4 Statistical applications

It is rather natural to propose two estimators for a distribution function, the Markov operator MN with wavelets maps WL1 and the TN IFS. By Corollary 6 one can easily note that using the sample quantiles, it is not possible, in general, to achieve a precision ε = 1/N if the sample size n is less than N. But when n = N than, in the most defavorable case ε = 1/N, we just have the empirical distribution function for which we have the identity

for x = xi, i = 1, . . . ,N + 1 and a linear interpolant between

and

Thus apparently, the worst thing one can do with the estimator TF is to estimate the unknown distribution function with a linearized version of

The target of having ε = 1/100 means that at least 100 quantiles are needed and, non asymptotically, this is a to severe condition because, even having n = 100 observations, the empirical centiles are not good estimates of the true centiles. As we have seen in the previous section, Table 1, for having an error of order ε = 1/50 only 14 quantiles are needed around 1/3 of ε. So, as a rule of thumb we suggest to use a number of quantiles between n/2 and n/3. In oure monte carlo analysis we convain to use n/2. This strategy it is computationally heavy when n is large as the time to calculate the estimator increases too much, thus from a certain n it is better to use a fixed amount of quantiles. Our experience shows that N = 50 for large sample sizes is big enough, but for large sample sizes we suggest to use the empirical distribution function. Moreover, it has to be reminded that for N = 50 one can attend, in the worst case an error in sup-norm of 2%. Later on, we will give some theoretical results on the speed of convergence of TP to F with respect to N.

The two estimators are the fixed points of the following IFS:
a) The Markov-Wavelets IFS

where

and the

are solutions of the quadratic problem (Q) with vector of empirical moments ^g instead of g. The number of empirical moments (m = N + 1) used is linked to the number of wavelet maps

b) The quantile-based IFS

where

with qi the empirical

and

In both cases

is any member of

Stefano M. Iacus, Davide La Torre

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