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4 Statistical applications

On fractal distribution function estimation and applications

Theoretical background for affine IFS

Theorem 3 (Forte and Vrscay, 1995)

Theorem 4 (Iacus and La Torre, 2001).

Theorem 5 (Iacus and La Torre, 2001)

Theorem 8 (Forte and Vrscay, 1998)

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

4.1 Asymptotic results for the quantile-based IFS estimator

Theorem 12 (Gill and Levit, 1995)

4.2 Characteristic function and Fourier density estimation

It is rather natural to propose two estimators for a distribution function, the Markov operator *M ^{N}* with wavelets maps WL

for x = x_{i}, i = 1, . . . ,*N* + 1 and a linear interpolant between

and

Thus apparently, the worst thing one can do with the estimator *T _{F}* 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 *T _{P}* to

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 q_{i} the empirical

and

In both cases

is any member of

Stefano M. Iacus, Davide La Torre

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

Stefano M. Iacus, Davide La Torre