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For the entropy estimation (in section 5.3),
it is necessary to estimate and, consequently, to use
information from distances .
The likelihood function for the doubly censored
set of data is given by (cf. [Kendall and Stuart, 1979, eq. 32.37]):

(5.22) 
The likelihood equations are:

(5.23) 
and

(5.24) 
so that

(5.25) 
Note that depends on the reference distance.
Substitution of eq. (5.23) into (5.24)
and dividing all distances by yields:

(5.26) 
which is very similar to Ellner's result. The difference is that
the total number of distances () is fixed here instead of .
(That is why we had to use eq. (5.23) as well).
The asymptotic variance of maximum likelihood estimators
of a function of parameters
(here and ) is given by [Kendall and Stuart, 1979, eq. 17.87]:

(5.27) 
where is the information matrix

(5.28) 
Like eq. (5.9), eq. (5.27) takes account only of terms
in the variance [Kendall and Stuart, 1979, §17.24].
Solving eq. (5.27) for yields:

(5.29) 
The elements of the information matrix are obtained from
the second partial derivatives of the likelihood function:

(5.30) 

(5.31) 

(5.32) 
Thus

(5.33) 

(5.34) 

(5.35) 
since
,
and
.
Note that

(5.36) 
Now
If we substitute the estimated values and
we obtain:

(5.38) 
which is once again similar to Ellner's result.
Thus, asymptotically (because is fixed instead of ),
the ``doubly censored'' estimator of the dimension
is the same as Ellner's.
Next: The Neerven estimator
Up: The estimation of the
Previous: The ``doubly truncated'' case
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