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From the previous subsections it follows that the
class boundaries and the number of classes can be chosen optimally only if
we know the dimension and
the number of distances within the scaling
region. However, for efficiency reasons, the computed distances should
be grouped at an earlier stage.
In the current implementation (computer program)
of the estimators (see section 5.7) the entire
possible distance range is divided in ``bins'' as follows:

(5.63) 
where is the right boundary of the first bin and
is the number of bins. The bin boundaries () do not satisfy
eq. (5.61), and has to be specified at
the beginning of the computer program, so
this procedure does not lead to optimal classes. It would be
possible however, to do the computations twice: first compute the
correlation integrals and identify the scaling region
and estimate the dimension and then choose the
bins according to eq. (5.61) and eq. (5.62).
For each of the different dimension estimators, i.e. linear
least squares and the
maximum likelihood estimators, the test can be applied. The
number of degrees of freedom of (see eq. 5.59)
depends on the included parts outside
the scaling region and the number of estimated parameters.
A summary is given in table 5.1.
Estimator 
Estimated parameters

Included regions 
Degr. of freedom 
linear least squares 
dimension, yintercept 
region (in ) 

Takens 
dimension 


Ellner 
dimension 
region 

double truncation 
dimension 


double censoring 
dimension and 
and
regions 

Table 5.1: Summary of test options.
Here is the number of bins in the scaling region.
For the Takens estimator,
the class boundaries can be chosen in an appropriate
way (see section 5.7.6)
in the range , even if we chose , to be able to
perform all tests on the same set of data simultaneously.
Next: Monte Carlo simulations
Up: Goodnessoffit tests
Previous: The number of classes
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