Next: The ``doubly truncated'' case
Up: The estimation of the
Previous: The Takens estimator
Contents
Now suppose that the scaling behaviour is distorted in the smallest
distance region due to noise.
Ellner showed that ``unreliable''
distances can still be used for the
estimation of the dimension, by ``censoring on the left'' [Ellner, 1988].
It is called ``Type I censoring'' because it takes place
at a fixed point (here ) [Kendall and Stuart, 1979, §32.16].
The likelihood function is given by [Kendall and Stuart, 1979, cf. exer. 32.15]:

(5.15) 
where
all distances have been divided by .
The likelihood equation is:

(5.16) 
and the estimator of the dimension:

(5.17) 
with variance ( is fixed)

(5.18) 
Eqs (5.17) and (5.18)
are the same as Ellner derived
(see also [W.L. Deemer, Jr. and D.F. Votaw, Jr., 1955], [Kendall and Stuart, 1979, exer. 32.16]).
Now suppose that the scaling region extends down to zero, but that
we set anyway. In that case, the variance of the Ellner
estimator will be larger than the estimator of Takens; this is
caused by the loss of information about observations in .
Next: The ``doubly truncated'' case
Up: The estimation of the
Previous: The Takens estimator
Contents
webmaster@rullf2.xs4all.nl